· Web viewOrdinary differential equation (ODE) of the n-th order is an equation of the form y n t =f t,y t , y ' t , y '' t ,…, y n-1 t (*) Its solution is n-time differentiable

„ZPR PWr – Zintegrowany Program Rozwoju Politechniki Wrocławskiej”

Differential equations (for Big Data Analytics)

Andrzej Janutka

UNITS 1 Cauchy problems part I (first-order ODEs and Sturm-Liouville problem)...p 22 Cauchy problems part II (systems of first-order-linear ODEs)............................p 73 Lyapunov stability of autonomous systems of first-order ODEs.........................p 94 Second-order ODEs reducible to first-order ODEs; basics of Hamilton

formalism...........................................................................................................................................p 125 Nonlinear oscillations – Duffing oscillator......................................................................p 176 Lotka-Volterra competive models (and their extensions)....................................p 227 Models of macro-economic growth.....................................................................................p 268 Cauchy problems for second-order linear PDEs..........................................................p 299 Nonlinear PDEs of soliton solutions (example: nonlinear Schrodinger

equation in 1+1D)..........................................................................................................................p 3610 Ginzburg-Landau equation and topological solitons (domain walls)…......…p 40

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„ZPR PWr – Zintegrowany Program Rozwoju Politechniki Wrocławskiej”UNIT 1

Cauchy problems part I (first-order ODEs) and Sturm-Liouville problem

Cauchy problem

Ordinary differential equation (ODE) of the n-th order is an equation of the form

y ( n) ( t )= f (t , y (t ) , y ' ( t ) , y ' ' ( t ) ,…, y (n−1) (t ) ) (*)

Its solution is n-time differentiable function y(t), of the Cn class on a given area of the variable. The equation (*), together with the initial conditionsy (t 0 )= y0 , y

' ( t0 )= y1, y' ' (t 0 )= y2, ..., y

( n−1 ) (t 0 )= yn−1

constitute the Cauchy problem, the solution of whom is y(t).

First-order ODEs

The Cauchy problem for the first-order ODE is formulated withy‘(t)=f(t,y(t)) ˄˄ y (t ¿¿0)= y0 ¿Another form of the first-order ODE can be obtained via writing

f ( t , y (t ) )=−h (t , y (t ) )g ( t , y (t ) )

,

thus, h (t , y (t ) )d t+g ( t , y ( t ) )d y=0.

For several types of ODEs, there are establiched integrating schemes

(i) Separable ODEs

Let h(t,y)=h(t) and g(t,y)=g(y), thus, h (t )d t+g ( y )d y=0.Theorem: provided g, h are continous, and g(y0)≠0, there exists an area (a,b) R: for ϵt0 (a,b), the Cauchy (initial) problem ϵy‘(t)=-h(t)/g(y(t)) y(t˄ 0)=y0

has a unique solution y: (a,b)→R given with G(y)=H(t)-H(t0)+G(y0), where G,H are antiderivatives (indefinite integrals) of g, and h, respectively.

Among separable ODEs, especial case of h=1, g(y)=-1/f(y) is called an autonomous ODEy‘(t)=f(y(t)).

(ii) Exact ODEs

Let f(t,y(t))=-h(t,y(t))/g(t,y(t)), (thus, h(t,y(t))dt+g(t,y(t))dy=0),

and ∂g∂t

=∂h∂ y .

Theorem: provided g(y0)≠0, there exists an area (a,b) R: for tϵ 0 (a,b), the Cauchy ϵ(initial) problem y‘(t)=- h(t,y(t))/g(t,y(t)) ∂g/∂t=∂h/∂y y(t˄ ˄ 0)=y0

2

„ZPR PWr – Zintegrowany Program Rozwoju Politechniki Wrocławskiej”has a unique solution given by the equipotential lineV(y,t)=C, where h(t,y(t))=∂V/∂t, g(t,y(t))=∂V/∂y.

Proof: dVdt

=∂V∂ t

+ ∂V∂ y

dydt

=h (t , y ( t ) )+g (t , y ( t ) ) y ' ( t )=0 provided V(t,y(t))=const.

Note: some non-exact ODEs can be tranformed into the form of the exact ODE via multiplying the equation by an integrating factor.

Example: f(t)=t is an integrating factor of the equation (3 ty+ y2 )+( t2+ty ) dydt

=0, thus,

(3 t 2 y+ty2)+ (t3+t 2 y ) dydt

=0 is an exact ODE.

(iii) Homogeneous ODEs

Let f(t,y(t))=f(y(t)/t).Theorem: let f: (a,b)→R is continuos, and y0/ t 0 ϵ(a ,b). Then, the Cauchy problem for the homogenuos ODE y‘(t)=f(y(t)/t) y(t˄ 0)=y0

has a unique solution. It is obtainable via a transform u(t):=y(t)/t into a separable ODE u‘(t)=[f(u(t))-u(t)]/t.

(iv) Linear ODEs

Let f(t,y(t))=-g(t)·y(t)+h(t).Theorem: let f: (a,b)→ R is continuos, and t 0∈ (a ,b ) .Then, the Cauchy problem for the linear ODEy‘(t)+g(t)·y(t)=h(t) y(t˄ 0)=y0

has a unique solution. It is obtainable with: a method of integrating factor (a), a method of undetermined coefficients (b), method of variation of constant (c).

(a) The method of integrating factor.Multiply the linear ODE by the exponential of G(t), where G(t) is an indefinite integral to g(t); eG(t)[y‘(t)+g(t)·y(t)]=eG(t)h(t). (*) Hence d/dt[eG(t)y(t)]=eG(t)h(t) and y(t)=e-G(t) eʃ G(t)h(t)dt

Note: equation (*) is an exact ODE, thus, eG(t) is an integrating factor of the linear ODE. The potential for (*) is of the form V ( t , y )=−eG (t) y+∫eG (t ) h ( t )d t .

(b) The method of undetermined coefficients.Solve the homogeneous linear equation yh‘(t)+g(t)·yh(t)=0.Its solution of the formyh(t)=Ce-G(t)

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„ZPR PWr – Zintegrowany Program Rozwoju Politechniki Wrocławskiej”contribues to the solution of the nonhomogeneous linear equation y(t)=yh(t)+yp(t),where yp(t) is any particular solution (yp(t)=e-G(t) eʃ G(t)h(t)dt), allowing for the initial condition to be satisfied.

(c) The method of variation of constantSolve the homogeneous linear equation yh‘(t)+g(t)·yh(t)=0.Transform its solution of the formyh(t)=Ce-G(t)

via changing the constant C into a fnction C(t), thus, obtaining the secondary differential equationC‘(t)e-G(t)=h(t),thus, C(t)= eʃ G(t)h(t)dt=Y(t)+C1.

(v) Bernoulli ODEs

Let f(t,y(t))=-g(t)·y(t)+h(t)·yp(t), where p<0 or p>1. Dividing the Bernoulli equationy‘(t)+g(t)·y(t)=h(t)·yp(t)by yp(t), one transforms it into the linear ODE u' (t)1−p

+g (t ) ·u (t )=h (t ) , where u(t):=y1-p(t). Hence the Bernoulli ODE is uniquely soluble.

(vi) Ricatti ODEs

Let f(t,y(t))=-g(t)·y(t)-h(t)·y2(t)+k(t).Via transforming y(t)=x(t)+w(t), where w(t) is any special solution to the Ricatti equation, one obtains the secondary equation of the Bernoulli formx‘(t)+[g(t)+2h(t)w(t)]·x(t)=-h(t)·x2(t). Finding w(t) requires divining.

Linear ODEs of any order

– Method of Laplace transforms

Define, for any f:[0,∞)→R and s R, the (Laplace) transform ϵ F ( s)=L[ f ]≡∫0

∞

f ( t ) e−st d t.

Useful formula:L [ tn eαt ] (s )= n !(s−α )n+1 , in particular; L [eαt ] (s )= 1

s−α .

Theorem 1: provided f‘ is continuous on [0,∞) and there exist constants M>0, R, αϵt0⩾0 such that |f|, |f‘|⩽Me-α for t [ϵ t0,∞), for s ( ,∞), the Laplace transforms ϵ α F(s)=ℒ[f], G(s)=ℒ[f‘] are established and G(s)=sF(s)-f(0).

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„ZPR PWr – Zintegrowany Program Rozwoju Politechniki Wrocławskiej”Theorem 2: provided f‘,f‘‘ are continuous on [0,∞) and there exist constants M>0, R, αϵt0⩾0 such that |f|, |f‘|, |f‘‘|⩽Me-α for t [ϵ t0,∞), for s ( ,∞), the Laplace transforms ϵ αF(s)=ℒ[f], G(s)=ℒ[f‘‘] are established and G(s)=s2F(s)-sf(0)-f‘(0)....Theorem n: provided f‘,f‘‘, ..., f(n) are continuous on [0,∞) and there exist constants M>0,

R, αϵ t0⩾0 such that |f|, |f‘|, ...,|f(n)|⩽Me-α for t [ϵ t0,∞), for s ( ,∞), the Laplace transforms ϵ αF(s)=ℒ[f], G(s)=ℒ[f(n)] are established and G(s)=snF(s)-sn-1f(0)-sn-2f‘(0)-...-sf(n-2)(0)-f(n-1)(0).

Theorem: assume functions f,g are continuous on [0,∞) and there exist the Laplace transforms F(s)=ℒ[f], G(s)=ℒ[g] for s ( ,∞). If ϵ α F(s)=G(s), then f(t)=g(t) for t [0,∞).ϵ

Example: solve the initial problem y‘‘(t)+2y‘(t)+y(t)=et+e-t y(0)=1 y‘(0)=0˄ ˄ . . ℒ[y‘‘+2y‘+y](s)=s2F(s)-sy(0)-y‘(0)+2sF(s)-2y(0)+F(s) ˄ℒ[et+e-t]=1/(s-1)+1/(s+1) . (s2+2s+1)F(s)-(s+2)= 1/(s-1)+1/(s+1) . F(s)=(s+2)/(s+1)2+1/(s-1)(s+1)2+1/(s+1)3

=1/(s+1)+1/(s+1)2+1/(s+1)[1/(s-1)- 1/(s+1)]/2+1/(s+1)3

=1/(s+1)+1/(s+1)2/2+[1/(s-1)-1/(s+1)]/2+1/(s+1)3

=1/(s+1)/2+1/(s-1)/2+1/(s+1)2+1/(s+1)3

=ℒ[e-t/2+et/2+te-t+t2e-t/2] Solution: y(t)= et/2+e-t/2+te-t+t2e-t/2

- Method of Green functions

Define the Green function G(x,s) of a linear operator L=L(x) via the condition LG(x,s)= (x-s)δ , where the RHS function is the Dirac delta, and consider linear ODE of the formLy(x)=f(x).Since the properties of the Dirac delta, one writes

∫−∞

∞

LG ( x , s ) f ( s)d s= f (x),

thus, ∫−∞

∞

LG ( x , s ) f ( s)d s=L(∫−∞

∞

G ( x , s ) f (s )d s)=Ly ( x ) .

The (so called fundamental) solution to ODE can be written as

y (x )=∫−∞

∞

G ( x , s ) f (s )d s .

The Green functions are useful to boundary-value problemsLy(x)=f(x) Dy=0.˄

Example: a Sturm-Liouville boundary-value problem. Assume f(x) to be continuous on [0,l], and

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L= ddx

p ( x ) ddx

+q(x ) ˄ Dy=(α1 y ' (0 )+β1 y (0)α2 y

' (l )+β2 y (l) )Theorem: there is a unique solution to the Sturm-Liouville problem

y ( x )=∫0

l

G (x , s ) f (s )d s ,

and the Green function G(x,s) satisfies the following:- G(x,s) is continuous with respect to x and s,- LG(x,s)=0 for x≠s- DG(x,s)=0 for s≠0- G‘(s+,s)-G‘(s-,s)=1/p(s)- G(x,s)=G(s,x)

Let us specify a physical problem. Let p(x)=0, q(x)=k2=const, y(0)=y( /2k)=0, thus, we are πsearching for the profile of a standing linear wave in 1D, while disturbed with any function f(x). For x≠s, one finds G‘‘(x,s)+k2G(x,s)=0, [in general, G‘‘(x,s)+k2G(x,s)= (x-s)],δthus, for x<s, G(x,s)=A1(s)sin(kx)+B1(s)cos(kx) G(0,s)=B˄ 1(s)=0 for x>s, G(x,s)=A2(s)sin(kx)+B2(s)cos(kx) G( /2k,s)=A˄ π 2(s)=0

The continuity conditions G(s-,s)=G(s+,s), G‘(s+,s)-G‘(s-,s)=1/p(s) imply A1(s)sin(ks)=B2(s)cos(ks),-A1(s)k cos(ks)-B2(s)k sin(ks)=1,Hence,

A1 (s )=−cos ( ks)k

,B2 (s )=−sin (ks)k

,

and the Green function is determined.

Notes:the existance of the solutions to the Cauchy problem for the first-order ODEy‘(t)=f(t,y(t)) ˄ y (t ¿¿0)= y0¿is guaranteed under the conditions of Peano Theorem:Let the function f(t,y): Rm+1→Rm be continuos in Q={(t,y):tϵ[t0,t0+a],|y-y0|⩽b} and sup(t,y) Qϵ |f(t,y)|=M. Then, the Cauchy problem for the first-order ODE has a solutin in [t0,t0+α], where α=min(a,b/M).

The uniqueness of the solution is related to Picard-Lindelof Theorem: Assume the Lipschitz condition |f(t,y01)-f(t,y02)|⩽L|y01-y02| is satisfied for a constant L [0,∞) for any pairs (ϵ t,y01), (t,y02) belonging to Q={(t,y):|t-t0|⩽a,|y-y0|⩽b}$, and f(t,y0): Rm+1→Rm is continuous in Q and sup(t,y)ϵQ|f(t,y)|=M. Then, the Cauchy problem for the first-order ODE has a unique solution in |t-t0| , ⩽αwhere <min(a,b/M,1/L).α

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UNIT 2

Cauchy problems part II (systems of first-order-linear ODEs)

Consider a system of first-order ODEs

{y1' ( t )= f 1(t , y1 ( t ) , y2 (t ) ,…, yn ( t ))y2' ( t )=f 2 (t , y1 ( t ) , y2 (t ) ,…, yn ( t ) )

⋮yn' ( t )= f n(t , y1 ( t ) , y2 (t ) ,…, y n ( t ))

and the initial conditions

{ y1 (t 0 )= y01y2 (t 0 )= y02

⋮yn ( t0 )= y0n

The initial problem can be concisely written withy‘(t)=f(t,y(t)) ˄ y(t0)=y0

We focus ourselves on the case of the systems of linear ODEsf(t,y(t))=A(t)·y(t)+h(t),where A(t) denotes n x n matrix.

Systems of homogeneous linear ODEs

Let h(t)=0.Theorem: provided A(t) is continuous (each of the components are continuous functions) for t I ϵ ⊆ R and t0 I,ϵ the set of solutions to any system of homogeneous linear ODEs is n-dimensional linear space. In other words, any solution is a linear combination of n solutions which constitute a basis.

Any basis {y1(t),y2(t),...,yn(t)} of the solution space is called a fundamental system of solutions while the matrix Y(t)=[ypq(t)] is called a fundamental matrix and it satisfies the equation Y‘(t)=A(t)·Y(t).

Theorem: let A(t) is continuous for t I R.The solutions ϵ ⊆ y1(t), y2(t), ..., yn(t) constitute a fundamental system if and only if the Wronski determinant

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„ZPR PWr – Zintegrowany Program Rozwoju Politechniki Wrocławskiej”W(y1(t), y2(t), ..., yn(t))=det(Y(t))≠0for t I.ϵ

Systems of homogeneous linear ODEs with constant coefficients

Consider a system of homogeneous linear ODEs with constant coefficients A(t)=A=const.

(i) Method of matrix exponential

Let us define e A≡∑i=0

∞ Ai

i ! , where A0=1. The sum is convergent since A is a limited

operator.Lema: provided AB=BA, e A+B=eA ∙ eB.Lema: (e t A )'=Ae t A

Theorem: e t A is a fundamental matrix to y‘(t)=A·y(t). Comment: the columns of etA are the basic solutions to the homogeneous linear ODE.

(ii) Euler method (for non-degenerate eigenvalues of A)

Theorem: let all the eigenvalues ( : R λ λ λϵ ⌄ Im >0) of λ A are non-degenerate and v ϵ Cn/{0} denotes the eigenvector relevant to the eigenvalue . Then, each of n columns λof the fundamental matrix is related to a single eigenvalue and it is of the form e tλ v or Re(e tλ v) or Im(e tλ v). Comment: notice that we consider y(t) Rϵ n only.

Systems of non-homogeneous linear ODEs

Theorem: provided A(t), h(t) are continuous (each of the components are continuous functions) for t I R and ϵ ⊆ t0 I, there is a unique solution to any system of linear ODEs ϵwith constant coefficients. The general solution is obtinable with the method of variation of constant.

Method of variation of constantsLet Y(t) be a fundamental matrix of the of the associated homogeneous equation, thus, a solution to Y‘(t)=A(t)·Y(t). We look for the solutions to the non-homogeneous equation in the form y(t)=Y(t)·C(t),where C(t)=[C1(t),C2(t),...,Cn(t)]. Substituting this ansatz to the primary non-homogeneous equation, one arrives at the primary equationY(t)·C‘(t)=h(t).

Systems of non-homogeneous linear ODEs with constant coefficients

Consider a system of non-homogeneous linear ODEs with constant coefficients A(t)=A=const and h(t)=h=const≠0. Then, C(t)= eʃ -tA·hdt, Thus

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„ZPR PWr – Zintegrowany Program Rozwoju Politechniki Wrocławskiej”y(t)=etA· eʃ -tA·hdt

UNIT 3

Lyapunov stability of autonomous systems of first-order ODEs

Consider the Cauchy problem for an autonomous system of ODEsy‘(t)=f(y(t)) ˄ y(t0)=y0.Consider ys Rϵ n such that f(ys)=0. Then ys=y0 and we call the solution y(t)=y0=const a stationary (or equilibrium) solution, while ys is called the equilibrium point.

Assume ys is the equilibrium point of an autonomous system of ODEs and there exists >0 such that the initial problemδ

y‘(t)=f(y(t)) ˄ y(t0)=y1

has a unique solution for any y1: ||y1-ys||< δ and t [ϵ t0,∞).Definition: we call the equilibrium point ys stable (Lyapunov stable) if, for any >0, εthere exists such δ1>0 that, for any initial value(s) y1: ||y1-ys||<δ1, the solution satisfies ||y(t)-ys||< ε in the area t [ϵ t0,∞).Definition: we call the equilibrium point ys asymptotically stable if it is stable and, there exists such δ2>0 that, for any initial value(s) y1: ||y1-ys||<δ2, the solution satisfies limt→∞ y(t)→ys.Definition: we call the equilibrium point ys exponentially stable if it is asymptotically stable and, there exist such >0, >0, α β δ3>0 that, for any initial value(s) y1: ||y1-ys||<δ3, the solution satisfies ||y(t)-ys||< ||α y1-ys||e- tβ in the area t [ϵ t0,∞).

Comment: the Lyapunov stability means that solutions which are a litle-bit shifted from the equilibrium point at the evolution beginning do not diverge out of the vicinity of this point. Evolving asymptotically-stable solutions, tend to the equilibrium point in addition, while the exponentially-stable solutions tend to equilibrium with an increaed rate.

Let all the first-order partial differentials of the components of f(y) [of the functions fi(y1,y2,...,yn)] are continuous in an area X⊆Rn and let ys X is the equilibrium point of the ϵautonomous system of ODEs. The Jacobi matrix of f at ys is denoted with J(ys): Jij(ys)≡[∂fi/∂yj]|y=ys.Theorem: if the real part of each eigenvalue of J(ys) is negative, the equilibrium point ys is asymptotically stable.If the real part of at least one eigenvalue of J(ys) is positive, the equilibrium point ys is unstable.

9


Note that, for the homogeneous system of linear ODEs with constant coefficients, the Jacobi matrix of f is just A matrix.

Definition: Lyapunov (candidate) function V(y): Rn →R is any function that satisfies the following

1) V(y)=0 ⇔ y=ys

2) V(y)>0 ⇔ y≠ys

3)dVd t

=∑i=1

n ∂V∂ y i

f i ( y )=∇V ∙ f ( y )≤0 for every y≠ys

Theorem: assume f to be of C1 class in X⊆Rn and ys X. If the Lyapunov function of ϵ f exists, the equilibrium point ys is Lyapunov stable. Additionally, if

dVd t

=∇V ∙ f ( y )<0 for every y≠ys,

the equilibrium point ys is asymptotically stable.

Consider the case of the Lyapunov-stable equilibria which are not asymptotically stable; the time-differential of the Lyapunov function satisfies

dVdt

=∇V ∙ f ( y )=0 for every y≠ys.

Definition: the Lyapunov function of the above property is called the first intergral of the (system of) ODE(s).

Therefore, the existence of first integrals is

Example: finding the first integral of the autonomous systems in R2

{d y1d t

=f 1( y1, y2)

d y2dt

=f 2( y1, y2) .

Dividing the equations, one obtains d y2d y1

=f 2( y1, y2)f 1( y1 , y2)

,

thus, f 1 ( y1 , y2 )d y2−f 2 ( y1 , y2 )d y1=0 .Its general solution can be written in the formV(y1,y2)=const,while the exact ODE itself in the form∂V∂ y1

f 1 ( y1 , y2 )+ ∂V∂ y2

f 2 ( y1 , y2 )=0.

Hence, V(y1,y2) satisfies the definition of the first integral of the system of autonomous ODEs.

Note: different systems of ODEs can relate to similar first integrals. For example consider the pair

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{ y1= y1y2= y2

and { y1= y1(1− y2)y2= y2(1− y2)

Note: there exists the definition of the Lyapunov function V(t,y) for non-autonomous system of QDEs and the definition of the first integrals of non-autonomous system of ODEs. One of the first integrals is a Lyapunov function that is constant on any integral

curve, thus, dVd t

=∂V∂ t

+∇V ∙ f (t , y )=0.

Theorem: let y0 is a non-singular point of an autonomous system of ODEs, which means f(y0)≠0. If f is of the C1 class in the vicinity X⊂Rm of y0, there exist m-1 independent first integrals V1, V2,...,Vm-1 of the system, of the C1 class, in the vicinity of y0. The criterium of the independence: rank of the Jacobi matrix [J(y)]ij=∂Vi/∂yj] is m-1.

Theorem: exponential stability theorem. A vector ys is the exponentially-stable equilibrium of an autonomous system if and only if there exists >0 and a Lyapunov εfunction V(y) which satisfies

(i) α 1‖y− ys‖2≤V ( y )≤α 2‖y− ys‖

2,

(ii) V ¿ y=f ( y)≤−α3‖y− ys‖2,

(iii) ‖∇V‖≤α 4‖y− y s‖,where α1,α2,α3,α4>0, for ‖y− ys‖<ε.

Example: the damped linear oscillator m y+2β y+ky=0 .Obviously, ys≡ ( ys ,m ys )=0 is the exponentially stable equilibrium point of the relevant

system of Hamilton equations. It is instructive to check that energy V ( y , y )=12m y2+1

2k y2

can play a role of the Lyapunov function. Its time derivative V=m y y+ky y=−2 β y2is negative definite, however, it tends to zero for t→∞. Thus, it points out the stability of the equilibrium point while not the asymptotic stability.

Let us consider another Lyapunov function V ( y , y )=12m y2+1

2k y2+εmy y which is

positive definite provided is a sufficiently-small positive number. Simultaneously,ε V=−(2β−εm ) y2−εk y2−ε 2βy y is negative definite, hence, y=0 is the asymptotically-stable equilibrium. Morreover, one can check applying the exponential-stability theorem, that ys=0 is the exponentially-stable equilibrium.

Lyapunov functions for systems of the linear first-order homogeneous ODEs with constant coefficients in 2D

Let {x '=ax+byy '=cx+dy

For ad−cb>0 and a ,d<0 and b , c ≠0, the expression x2+k y2 is a Lyapunov function in a neighbourhood of an equilibrium if k=b /c provided b and c are of the same sign, whereas, k=−b /c provided b and c are of opposite signs.

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For ad−cb<0 and a+d<0, the expression x2+Bxy+C y2 is a Lyapunov function in a

neighbourhood of an equilibrium if B=d−ac and C= (a+d )2−2bc

2c2 provided c ≠0, or

B=0 and C> b2

4ad if c=0.

The isolated (singular) equilibria can be: saddles, knots, focal points or centers.For saddles, the eigenvalues of the characteristic marix at the equilibrium are real and of different signes,For knots, the eigenvalues are of the same sign; positive for unstabble knots andnegative for stable knots.For focal points, the eigenvalues are complex, whereas the real parts of them are negative for stable focal pints while positive for unstable focal points,For centers, the eigenvalues are purely imaginary,

UNIT 4

Second-order ODEs reducible to first-order ODEs; basics of Hamilton formalism

Variational calculus and Lagrange formalism

The simplest class of variational problems relates to looking for the trajectories x(t) Cϵ 1[a,b] which minimize the objective functional (action)

L [x ]=∫a

b

L(x , x , t)d t

while satisfying the Dirichlet boundary conditions x(a)=xa ˄ x(b)=xb, („a problem with invariable ends of the trajectory“).

The minimization condition is analogous to the condition of function minimiumf ( x+d x )−f ( x )≡d f =0→L [ x+δ x ]−L [ x ]≡δL=0 Here, δ x ( t )=ϵ v ( t )˄v (a )=v (b )=0, and

δL=∫a

b d Ldϵ |

ϵ=0ϵd t=∑

i=1

n

∫a

b

( ∂ L∂ x iv i+

∂ L∂ x i

v i)d t ϵIntegrating the second term by parts, one arrives at

δL=∑i=1

n

¿¿

Since vi are free, the intergrands have to be equal to zero, which leads to the system of n Lagrange equations (ODEs of the second order)

∂L∂x i

− dd t

∂ L∂ x i

=0.

12

„ZPR PWr – Zintegrowany Program Rozwoju Politechniki Wrocławskiej”Example: the Newton’s equation for a conservative-force problem m r=−∇U is

equivalent to the system of the Lagrange equations of the Lagrangian L=m2r ∙ r−U (r)

Exampe: Fermat Principle. The time that a point of the front of an electromagnetic wave needs to cover a path between the positions A and B is given by

T= 1c∫A

B

nds=1c∫t A

tB cvdsdt

dt=Sc , where n denotes the refractive index. Parametrizing the

wavefront motion with a Cartesian coordinate z, one writes

S=∫z A

z B

n(x ( z ) , y ( z ) , z)√ x2 (z )+ y2(z )+1dz , where x≡dx /dz , y ≡dy /dz, and formulates

(following Fermat) the principle of minimum length of the pathδS=0 while noticing the form of the optical Lagrangian L(x , y , x , y , z )=n (x , y , z )√ x2+ y2+1 .

Hamilton formalism

The purpose is the reduction of the n second-order Lagrange equations into the system of 2n first-order ODEs (Hamilton equations). Despite the number of ODEs increases twice with that, the advantage of reducing the order of the equations cannot be stressed enough.

Definition: the generalized momenta pi(t)≡∂ L∂ xi

, for i=1,2,...,n.

Definition: the Hamilton function (Hamiltonian) H (x , p , t)≡∑i=1

n

x i pi−L(x , p , t).

The variational principle L [x+δ x , p+δ p ]−L [ x , p ]≡δL=0 , (here, x and p are indepednent sets of variables, δ x (t )=ϵ v (t ), δ p (t )=ϵ u (t ) ˄ v (a )=v (b )=0) leads to

δL=∫a

b d Ldϵ |

ϵ=0ϵd t=∑

i=1

n

∫a

b

( pi δ x i (x , p ,t )+ x iδ pi−∂H∂x i

δ x i−∂ H∂ p i

δ p i)d t=∑i=1

n [∫ab

(− p iδ x i+ x i δ p i−∂H∂ xi

δ xi−∂ H∂ pi

δ pi)d t+p i δ xi|ab]=∑

i=1

n

∫a

b [(− pi−∂ H∂x i ) v i+( x i−

∂H∂ pi )ui]d t ϵ

Since vi, ui are free, one arrives at the system of 2n Hamilton equations

{− pi−∂ H∂ x i

=0

x i−∂ H∂ p i

=0

i=1,2,…n

Dissipative systems

Defining the variational functional derivative δδ x i

≡ ∂∂ xi

− dd t

∂∂ x i

, one writes the

Lagrange equatons with δ Lδ x i

=0.

13

„ZPR PWr – Zintegrowany Program Rozwoju Politechniki Wrocławskiej”Rayleigh dissipation function R ( x , x ) is positive definite and relates to the dissipaton

functional R [ x ]=∫a

b

Rd t . Completing the minimum-action principle by a principle of

maximum dissipation δL−δR=0

leads to the completed Lagrange equations δ Lδ x i

− δ Rδ xi

=0 . The relevant Hamilton

equations take the form

{p i=−∂ H∂ x i

+ δ R

δ ∂ H∂ pi

x i=∂ H∂ pi

i=1,2,…,n

Phase portraits

Definition: phase space X of the autonomous system is an open subset in R2n such that all curves of the solutions to the initial problem y(t,y0), (e.g. [x(t,x0,p0),p(t,x0,p0)] of Hamilton equations), are confined to X.

Definition: flow is a continous one-parameter family of transformations Tt : X→X with the property Tm+n(y)=Tn(Tm(y)) for m (-∞,∞). If it is for ϵ m [0,∞) only, we callϵthe family of the interactions a semi-flow.

Definition: given y X, ϵ the set {Tn(y)|n Zϵ } is called orbit or trajectory of y. If one restrics to n>0, the set of transformation is called a semi-orbit.

Orbit is a set of solutions to the autonomous system which can be obtained from each other via the operations of the flow.

Theorem: Each point of the phase space belongs to one and only one orbit.

Hence, the orbit is identical with an area of the phase space.

Example: consider the system (in Cartesian and polar coordinates)

¿ → {r=r (1−r2)φ=1

The solution is

r (t )=r0 e

t

√r 02e2 t−r02+1

, φ (t )=t+φ0,

thus, r (φ )= peφ

√ p2 e2φ−p2+1.

The orbits are: (i) the point r=0, (ii) the circle r=1, (iii) the ring 0<r<1, (iv) the ring r>1.

14

„ZPR PWr – Zintegrowany Program Rozwoju Politechniki Wrocławskiej”Definition: the point pϵX of the property f(p)=0 is called the critical (or stationary) point of the flow.

Thus, the critical point is an equilibrium point of the system of ODEs.

In the above example, the point r=0 is a critical point. The circle r=1 is a closed (periodic) orbit. The closed orbit is just a periodic solution to the system of ODEs.

Theorem (Poincare-Bendixson): in R2, if the orbit contains a critical point, it is the critical point or a closed (periodic) orbit.

Definition: homoclinic orbit is the orbit of solutions „starting and ending“ at a single critical point lim ¿t→∞ y ( t )=lim ¿t→−∞ y ( t )=p ¿¿

Example: the solution to { x1=x2x2=x1+x1

2

is given with the Lyapunov function via V (x1 , x2 )=x22

2−

x12

2−x13

3=C . The critical points are

p1=(0,0), p2=(-1,0). Linearizing ODEs in the vicinity of p1, x1≅ x2 , x2≅ x1, one finds the p1 to be a saddle point of V(x1,x2), whereas, the linearization in the vicinity of p2, x1≅ x2 , x2≅−x1 shows p2 to be a local maximum of V(x1,x2), (see the contour plot of V and the phase portrait below). The loop starting and ending at p1 is a homoclinic orbit.

Definition: heteroclinic orbit is the orbit of solutions starting at one critical point and ending at another one lim ¿t→−∞ y ( t )=p1 , lim ¿ t→∞ y (t )=p2¿¿

ODEs with parameter; bifurcations

Consider an autonomous system with a parameter μy '=f ( y ( t ) , μ)

Any qualitative change of the phase portrait of the system with changing μ is called bifurcation, and the value of the parameter =μ μ0 is called bifurcation point.

Example: let

¿ → {r=r (μ−r 2)φ=1

,

15

„ZPR PWr – Zintegrowany Program Rozwoju Politechniki Wrocławskiej”in Cartesian and polar coordinates , respectively. For <0, the solution isμ

r ( t )=√−μr0e

μt

√−r02 e2μt+r0

2−μ,φ ( t )= t+φ0,

and r=0 is a stable equilibrium point. For =0, μ r (t )=¿, thus, r=0 is an (asymptotically) stable equilibrium. For >0, there are two equilibria; stable (limit cycle) μ r=√μ, and unstable point r=0. The stability of r=√μ follows from r>0 for r<√μ while r<0 for r>√μ. Hence, (Hoopf) bifurcation point is =0, and the bifurcation diagram is given below. In μgeneral, the Hoopf bifurcation is related to the appearance of a stable limit cycle when going from a stable singular point.

Example: consider scalar (logistic) ODE of the population growth with a parameter (harvesting rate) b≥0; y=ry (1− y /N )−b= f ( y ,b). For b=0, it has two equilibria: unstable y=0˄f ' (0,0 )=r>0 ,∧stable y=N ˄f ' (N ,0 )=−r<0, then,

y (t )=N y0

y0+(N− y0 )e−rt . For b>0, unstable and stable equilibria are y= Nr−√Nr (Nr−4b )2 r

and y=√Nr (Nr−4b )+Nr2 r

, respectively, and there is a saddle-node bifurcation in the

diagram below

16


UNIT 5

Nonlinear oscillations – Duffing oscillator

The Duffing equation (of a nonlinear damped-driven oscillator) x+δ x−βx+α x3=γ cos (ωt )

(of α>0¿describes e.g. the oscillations of the cantilever magnetometer (figure below), where, the driving is provided by a sinusoidal-vibration generator.

We reduce the second order ODE to the pair of first-order ODEs and write the equivalent autonomous system

{ x=vv=βx−αx3−δv+γ cos (φ)

φ=ω

17

„ZPR PWr – Zintegrowany Program Rozwoju Politechniki Wrocławskiej”In the case of δ , γ ,ω=0, we find its Lyapunov function (the energy)

H ( x , v )= v2

2−β x2

2+α x4

4For <0 (weak magnets), there is one stable equilibrium β (x,v)=0, whereas, for >0, thereβ are an unstable equlibrium point (x,v)=0 and two stable ones (x , v¿=(±√ β /α ,0) . The point β=0 is a pitchfork bifuration shown in the diagram below

The phase portraits (below) are plotted for the case δ,γ,ω=0, α=1. The left portrait relates to β<0, the right to β>0. When δ≠0, the closed (periodic) orbits turn into spirals winding the stable equilibria.

Transition to chaos In the case γ,ω≠0, the character of the driven oscillations depends on the initial position, amplitude and frequency of the driving field. In plots below: the solutions of the Duffing equation with the parameters α=1, β=10, δ=2, ω=5 and the initial conditions x (0 )=1, x (0 )=5 (a)-(b), and x (0 )=3 (c)-(d). The initial condition for y(0) favor the oscillations around the unstable equilibrium y=0 (a),(b) or around stable equilibria x=±√10 (c),(d). Deterministic chaos appears via increasing the driving amplitude

=2 (a),(c) → =10 (b),(d), which enables the dynamical variable to jump between γ γdifferent orbits of the basic autonomous system.

18


Another approach to chaos follows from the analysis of the critical points of the extended system of variables (x,v,φ). The extended system is autonomous, thus, its orbits are periodic according to Poincare-Bendixson theorem. This requires the frequencies of the periodic motion to be multiple of the driving frequency, and the claim of relapsing to the backward states enables finding the equilibria of x(t, )γ with the conditions βx−αx3+γ cos (φ0+n )=0, n=1,2 ,… . Below we plot the critical lines for n=1,2,...,90 and =1, α β=10, φ0=0.001 ,showing chaos to result from widespreading stable and unstable equilibria.

19


For β<0, iregular oscillations are seen at the beggining stage of the evolution only (below: the evolution with =1, α β=-10, δ=2, γ=10, ω=5, x(0)=3, x(0)=1) . All the equilibria are stable (below: the plot of 90 critical lines for n=1,2,...90, β=-10, φ0=0.001)

Nonlinear resonance – method of averaging

Consider the case of weak nonlinearity; |β /α|≫1. Via the van der Pol transform

u¿ x cos (φ )− vωsin (φ ) , w=−x sin (φ )− v

ωcos (φ), and/or the inverse transform

x=ucos (φ )−w sin (φ ) , v=−ω [u sin (φ )+w cos (φ ) ] ,which relate to a rotation of the plane (x,v) relative to the plane (u,w), one arrives at the system of first-order non-autonomous ODEs

¿

20


We average the RHSs of the above system over single period of time with

⟨ A ⟩= 1T ∫

0

T

A ( t )d t , T ≡2π /ω, obtaining

{ u= 12ω {( β+ω2 )w−3 α

4(u2+w2 )w−δωu },

w= 12ω {− (β+ω2 )w+ 3α

4(u2+w2 )u−δωu−γ },

which can be written with polar coordinates r=√u2+w2, Φ=arctan (wu

)

{ r= 12ω [−ωδr−γ sin (Φ)] ,

r Φ= 12ω [−(β+ω2) r+ 3α

4r3−γ cos (Φ)] .

The equilibrium values of (r, )Φ , for / =0.2, / =2.5, / =0.0005 (blue) or 0.05 (rose) δ β γ β α βor 0.15 (beige) are plotted below with dependence on the driving frequency; the

contour plots of δ 2ω2r 2+[−(β+ω2 ) r+ 3α4

r3]2

=γ2 and

(β+ω2 )ω2 γδsin (Φ )−3 α

4 [ γδ sin (Φ )]3

=γ ω3cos (Φ). The dashed lines represent unstable

(saddle) equilibria.

With the plots above, one determines the conditions of the resonance. With the inverse van der Pol transform, one finds the interpretation of the solutions via x (t )=r (t ) cos [ωt+Φ (t )], thus, r(t) is the amplitude of the driven oscillations of the primary coordinate while (t)Φ is their „initial“ phase.

21


UNIT 6

22


Lotka-Volterra models (and their extensions)

The most general form of the Lotka-Volterra system is y i(t)= y i (t ) f i( y (t )).

A prototype model of the population dynamics (logistic equation)

Building the simplest model of the growth of a population, one starts from the linear ODE x=γx, where γ denotes the rate of the exponential growth. The rate is dependent of a „birth coefficient“ (the number of borns in a time unit) and on the „decease coefficient“ (the number of deceases in a time unit), =b-dγ . Thus γ can be positive or negative.Considering the effect of the population size on the decease number, we modify the decease coefficient with d=ax. The resulting ODE (of the Bernoulli type) x=x (b−ax )

is called the logistic equation, the solution of whom x (t )= b/a

1+( ba x0

+1)e−bt

is plotted for ax0/b=0.01

Including the effect of harvesting leads to an additional modification of the model

x=x (b−ax )+h xD+x . In the large population limit, x=x (b−ax )+h is called the constant-

rate harvesting model, while in the low-population case, x=x (b−ax )+hx /D is called the proportional-harvesting model. The effect of predation is beeing included with another

term x=x (b−ax )−p x2

D2+x2.

„Rabbit-sheep“ model of competition

The model of the population dynamics of two competing species takes the form of a „vector logistic equation“

{x1=x1 (b1−a11 x1−a12 x2 )x2=x2 (b2−a21 x1−a22 x2 )

where the birth coefficients satisfy b1>0, b2>0.

Predator-prey model

23

„ZPR PWr – Zintegrowany Program Rozwoju Politechniki Wrocławskiej”The population dynamics of the predator and prey species is obtained from the above by changing positive rates into negative ones for the predator (the birth and decease

coefficients exchange their roles) {x1=x1(b1−a11 x1−a12 x2D+x1 )

x2=x2 (−d2+a21 x1−a22 x2 )Consider a simplification of the predation effect and neglect the second-order effects of the decease

{ x1=x1 (b1−a12 x2 )x2=x2 (−d2+a21 x1)

Numerical solution to the simplified version is periodic for b1=1.5, d2=3, a12=1, a21=1, x1(0)=1, x2(0)=0.5, (plotted below), showing cycles of the population increase for the predator and prey to overlap.

Below: celebrated data of hare and lunx pelt purchase from hunters by Hudson’s Bay Company of Canada. Mutually-shifted oscillations of the two species populations are clearly seen indicating the success of the predator-prey model.

Example: stability analysis of the predator-prey model.

24

„ZPR PWr – Zintegrowany Program Rozwoju Politechniki Wrocławskiej”There are two critical points (x1s,x2s)=(0,0) (the saddle-point equilibrium of the separatrices x1=0 and x2=0) or (x1s,x2s)=(d2/a21,b1/a12). Unfortunately, the point (d2/a21,b1/a12) is a center of the linearized system (a stable while not asymptotically stable equilibrium in the center a closed periodic orbit), hence, the linearization does not provide the information on the character of the critical point of the nonlinear system. Instead, one

finds a first integral via comparing the equations {a21 x1+a12 x2=b1a21 x1−d2a12 x2d2x1

x1+b1x2

x2=b1a21 x1−d2a21 x2,

thus obtaining

∫(a21 x1+a12 x2−d2x1x1

−b1x2x2 )d t=a21 x1+a12 x2−d2 ln x1−b1 ln x2=C . The function

g (x1 , x2 )=−x1d2 x2

b1 e−a21x1e−a12 x2=−C is the first integral. It takes minimum at (d2/a21,b1/a12) which appears to be a center of the nonlinear system.

Note: the predator-prey model is not strucurally stable, which means that a small perturbation of the ODE(s) leads to qualitative change of the phase portrait. For instance,

consider the modified system{x1=x1 (b1−a12 x2 )−ε x12

x2=x2 (−d2+a21 x1 ).

Linearizing the predator-prey system around (d2/a21,b1/a12) with {x1=−d2a12a21 (x2− b1

a12 )x2=

a12 b1a12 (x1− d2

a21),

one finds the solution in the form of the ellyptic trajectory

{ x1 ( t )=d2a21

K cos (√b1d2t+φ )+d2a21

x2 (t )=√b1d2a12

K sin (√b1d2 t+φ )+b1a12

,

Where K ,φ are determined by the initial conditions.

Approximate periodic solutions for the case of small population fluctuations

Assume ∆ x2/ x2≪1, where ∆ x2=max [ x2 ( t ) ]−min [ x2 (t )] and x2 denotes the average of x2 over time. Although the method is applicable to any two-component Lotka-Volterra equation, we consider the case of the predator-prey model. Differentiating the dynamical equation of x1 over time and substituting x2with x2, one arrives at x1=[b12+a12 (d2−2b1 )x2+a122 x2

2 ] x1−a12a21 x2 x12≡β x1+γ x1

2. In order to find the general periodic solution to the last equation expressed with Jacobi elliptic functions, we substitute x1 with C+ y2, thus, obtaining2 ( y y+ y2)=βC+γ C2+ (β+2 γC ) y2+γ y4.Comparing this equation to those which are satisfied by one of the Jacobi elliptic functions sn(t,k) or cn(t,k) or dn(t,k), lets say z=cn(t,k)

25

„ZPR PWr – Zintegrowany Program Rozwoju Politechniki Wrocławskiej”z=−(1−2k2 ) z−2 k2 z3, ( z )2=(1−z2 ) (1−k2+k2 z2) ,one finds y (t )=A cn (νt , k ), and βC+γ C2=2 A2 ν2 (1−k2 ), β+2γC=−4 ν2 (1−2k 2), γ / A2=−6ν2 k2. For given β ( x2 ) , γ (x2), the above equations, together with y(0)=y0, constitute the system for the constants A, C, k, ν.

For determining x2 of the predator-prey model, it can be proven that

x1,2≡1T ∫

0

T

x1,2 (t ) d t reads x1=d2a21

, x2=b1a12

, where T denotes the period of any oscillatory

solution.

May extension of the predator-prey model

Inclusion of the environment capacity (the maximum of the prey population m1 and the maximum of the predator population m2xx1; both are governed by the food accessibility) and of the effect of „predation saturation“ (the predators do not kill more preys than they are able to eat, while the predator population is relatively small) lead to a May model

{x1=b1 x1(1− x1m1 )−a12

x1 x21+a12 x1

x2=b2 x2(1− x2m2 x x1 )

Due to te assumption of small predator population, unlike in the usual predator-prey model, the birth coefficient is not determined by the population of preys. There are three critical points of the model is (0,0), (m1,0), and (x1*,x1*/m2x), where x1* is

the root of the equation a12m2 x x1¿=

−b1m1

( x1¿−m1 )(a12 x1¿+1 ) that satisfies x1*>0.

Analyzing the Jacobi matrix, one finds (0,0), (m1,0) to be saddle points, while (x1*,x1*/m2x) is stable provided Tr J(x1*,x1*/m2x)<0. In the opposite case, (x1*,x1*/m2x) is unstable, while Tr J(x1*,x1*/m2x)=0 relates to the Hopf bifurcation whose limit cycle appears to be asymptotically stable (thus, it is an atractor).

Model of epidemy

Let x1 denotes the population of unifected persons, x2 denotes the population of infected ones, a denotes the coefficient of the infection rate, and d is the rate of decease. The

differential model takes the form { x1=−a x1 x2x2=a x1x2−d x2

Kinetics of chemical reactions (chemical oscillations governed by Lotka-Volterra model)

The predator-prey model describes quantitatively one of the simplest autocatalytic reactions

26

„ZPR PWr – Zintegrowany Program Rozwoju Politechniki Wrocławskiej”A+X→2 XX+Y→2YY→E

with the rate equationsd [X ]dt

=kx [A ] [ X ]−k y [ X ] [Y ]

d [Y ]dt

=k y [ X ] [Y ]−kd [Y ]

The autocatalysis means one of the reaction products to be a catalyst of the same or coupled reaction.

Consider another (the most simple) autocatalytic reaction A+B⇆2B, that is described quantitatively with the rate equations ¿. The system is not of the Lotka-Volterra type.

UNIT 7

Models of macro-economic growth

Basic (Solow) model

The pace of the capital K(t) accumulation depends on the production Y(t) and the consumption C(t) according to the equationK=Y−C−δK ,where the (exponential) depreciation of the capital has been included with the rate , δ(in literature, the capital accumulation is also called the investment I (t)≡K).At the macro-scale, a significant fraction 1-s of the production is consumed (s is called a saving rate) and, limiting the model to a closed economy, one writes the constraint C (t)= (1−s )Y (t).The production is a composite function of labour impact L(t) (that is proportional to the population size), technological impact A(t) and capital; Y ( t )=Y (L ( t ) , A (t ) , K (t)). Growth of both impacts is governed by the rate equations A=gA , L=nL.Additionally, we assume (i) the production function to be homogeneous (of degree 1, scalable) relative to the capital and labour with 1L ( t )

Y (L (t ) , A (t ) , K (t))=Y (1 , A (t ) ,K ( t)/L(t)),

and (ii) Y is a concave function of K (the production growth becomes slower and slower with increasing the capital).

First, let us neglect the effect of the technological impact; Y (L (t ) , A (t ) ,K (t ) )=Y (L(t) ,K ( t )). For this case, denoting the capital per capita by k (t)≡K (t) /L(t ) and differentiating its logarithm over time, thus, obtaining kk= K

K− LL= sY

K−δ−n= sY

Lk−δ−n , one arrives at the fnal rate equation

k=sY (1 , k )−(δ+n ) k.

27


It has a single eqilibrium point k ¿=sY (1 , k¿)

δ+n whose stability is confirmed via

linearizing the rate equation around k*

k=[s Y ' (1 , k¿)−δ−n ] (k−k¿ ). The stability condition is sY ' (1 , k¿ )−δ−n<0, thus,

Y ' (1 , k¿ )<Y (1, k ¿)k¿ . The later inequality is satisfied according to our pevious assumption

(ii); Y“<0, thus, 0=Y ' (1,0 )<Y (1 , k¿)+Y ' (1, k ¿)(0−k¿). Hence, we have arrived at a simple model of the economic growth. Unfortunately, the existance of the stable equilibrium is not in agreement with data on economies. Therefore, the applicability of the model is limited to a certain time scale.

A production function in another (Douglas-Cobb) form Y ( t )=A ( t ) Kα ( t ) L1−α( t); 0<α<1,

satisfies (i) and (ii) since 1L∂Y∂K

= 1LαA Kα−1L1−α= ∂

∂ KA( KL )

α

= ∂∂K

Y (1 , A , KL

) and

∂2Y∂K 2=α (α−1 ) A K α−2L1−α<0. The rate eqation for the capital per capita is the same as in

the case of neglecting the technological impact A(t). The rate equation for the

production reads YY

= AA

+α KK

+(1−α ) LL=g+α sY−δK

K+(1−α )n.

With an additional assumption of the capital stock to be growing at a constant rate (the case of a steady-state growth), it has to satisfy Y/K=const (since K=sY−δK ).

Thus, YY =g+α Y

Y + (1−α )n and YY =

g1−α +n.

Therefore, the growth of the production per capita y (t )≡Y (t)/L(t) relates to the rate of

the technological impact via yy= g1−α .

Note: Lagrange formalism for systems with constraints. Method of Lagrange

multipliers. We look for the minimum of the action functional L [x ]=∫a

b

L(x , x , t)d t

with an additional condition (a constraint) f ( x , x , t )=0. Defining a secondary action

functional ~L [ x ]=∫a

b~L(x , x , t)d t ≡∫

a

b

[~L ( x , x ,t )−λ (t) f ( x , x , t ) ]d t , where is called the λ

Lagrange multiplier, one solves the Lagrange equations, finding a parametric solution x (t , λ (t)). Substituting x (t , λ (t)) into the constraint equation, one determines the

Lagrange multiplier. In the special case of the static constraint f ( x , t )=0, the Lagrange multiplier is just a constant.

(Ramsey) model with delayed consumption

Denote the consumption per capita by c ( t )≡C (t)/L(t ). Its evolution is governed by the requirement of maximizing the utility functional (a measure of welfare)

28


U [c (t ) ]=∫t

∞

u (c (t ' ))exp (−¿θ(t '−t ))d t ' ¿

that allows for inclusion of some delay of the consumption in time, u(c) is called the utility function and it measures a degree of temporal satisfaction of the consumers. Here, ϴ denotes the rate of predicted (by consumers) consumption increase.The rate equation of the capital per capita k= y (1 , A , k )−c ( t )−(n+δ )k−g (*) is a dynamical constraint for maximizing the utility functional. Applying the method of Lagrange multipliers, via modifying the utility into

~U [c ( t ) , k (t ) ]=U [c (t ) ]−∫t

∞

λ [ k− y (1 , A (t' ) , k (t' ) )+c ( t' )+(n+δ ) k (t ')+g ]exp(−θ (t '−t ))d t ' ,

one arrives at the Lagrange equations u' (c )−λ=0, λ−θλ+( y ' (1 , A , k )−n−δ ) λ=0 .Differentiating the first one over time and combining it with the second equation, we obtain u' ' (c ) c−θu' (c )+ ( y ' (1 , A , k )−n−δ )u' (c )=0,

thus, −u' ' (c ) cu ' (c )

cc≡ 1σcc=−θ+ y ' (1, A , k )−n−δ. (**)

Treating the last equation as a rate equation for a consumption per capita, one closed the Ramsey model in two equations (*) and (**). The eqilibrium of the Ramsey model (c¿ , k¿ ) is determined with y (1 , A ,k¿ )=c¿+ (n+δ ) k¿+g, y ' (1. A ,k¿ )=n+δ+θ.It appears to be a saddle point, which can be checked out via linearizing the Ramsey system around (c¿ , k¿)

{c=c¿σ ( c¿ ) y ' ' (1, A , k¿) (k−k¿ )k=−(c−c¿ )+θ (k−k¿)

The determinant of the linearized system c¿σ ( c¿ ) y ' ' (1 , A , k¿ ) is negative since y ' ' (1 , A ,k )<0. The eigenvalues and eigenvectors read

λ1(2)=12

(θ±√θ2−4c¿σ (c¿ ) y ' '(1 , A , k¿)) , (thus, λ1>0 , λ2<0), and

[1, λ1 (2 )c¿σ (c¿) y ' ' (1 , A , k¿) ], (which indicate the directions of separatrices).

In conclusion, the equilibrium of the Ramsey model is unstable. The system of linearized equations shows tendency of the capital to diverge via reducing the consumption to zero or the opposite; divergence of the consumption via reducing the capital to zero. The parameter g (the rate of the technological-impact growth which can be considered as a rate of the expenditure of the government money) is a crucial factor to maintain the

system on the separatix [c , λ1c¿σ (c¿) y ' ' (1 , A , k¿)

c ].

29


UNIT 8

Cauchy problems for second-order linear PDEs

Cauchy problem

A partial differential equation (PDE) of the second order is the equation of the form

F (x1 ,…, xn, u ,∂ u∂ x1

,…, ∂u∂ xn

, ∂2u∂x1

2 ,…, ∂2u

∂ xn2 ,

∂2u∂ x1∂ x2

,…, ∂2u∂ xn−1∂ xn )=0

the solution of whom u(x1 ,…,xn) is of the C2 class on a given area V⊂En. Together with

the (boundary) conditions ∀M∈∂V u (M )=φ (M )˄∀M∈∂V ( ∂u∂n )M=ψ (M ), the second-

order PDE constitute the Cauchy (initial) problem, the solution of whom is u(x1,...xn).

Method of characteristics (applicable to linear first- and second-order PDEs in 2D or 1+1D)

The method is devoted to finding general solutions to PDEs of the first and second

order. Consider PDE of the form A ∂2u∂x2

+2B ∂2u∂ x ∂ y

+C ∂2u∂ y2

+a ∂u∂ x

+b ∂u∂ y

+cu+d=0, (*)

30

„ZPR PWr – Zintegrowany Program Rozwoju Politechniki Wrocławskiej”where A,B,...,d are differentiable functions of x and y variables. One calls characteristics of the above equation the integral lines of the ODE of the formAdy2−2 Bdxdy+Cdx2=0, thus, the solutions to the equations dydx

=B−√B2−ACA

, dydx

=B+√B2−ACA

for A≠0, or dxdy

=B−√B2−ACC

, dxdy

=B+√B2−ACC

for C≠0. These are given by the comparison of the frst integrals f(x,y), g(x,y) of the pair of ODEs to constants. For B2−AC>0 (PDE of the hyperbolic type), f(x,y) and g(x,y) take real values. For B2−AC<0 (PDE of the elliptic type), f(x,y) and g(x,y) are complex functions and f ( x , y )=g(x , y). For B2−AC=0 (PDE of the parabolic type), there is only one first integral f(x,y) to the pair, while g(x,y) will be considered as an arbitrary function. Transforming the variables into

[ξ ( x , y ) , η ( x , y )]={ [ f ( x , y ) , g ( x , y )]B2−AC>0[ℜ f ( x , y ) ,ℑ f (x , y )]B2−AC<0

[ f (x , y ) , g ( x , y ) ]˅ [ g ( x , y ) , f ( x , y ) ]B2−AC=0,

one arrives at PDE (*) with exchanged variables x→ξ , y→η and coefficients

A→A0=A( ∂ξ∂ x )2

+2B ∂ξ∂x

∂ξ∂ y

+C( ∂ξ∂ y )2

, B→B0=A ∂ξ∂ x

∂η∂ x

+B( ∂ξ∂ x ∂η∂ y

+ ∂η∂ x

∂ξ∂ y )+C ∂ξ

∂ y∂η∂ y ,

D→D0=A ( ∂η∂ x )2

+2B ∂η∂ x

∂η∂ y

+C ( ∂η∂ y )2

, a→a0=A ∂2ξ∂ x2

+2B ∂2 ξ∂ x∂ y

+C ∂2ξ∂ y2

+a ∂ξ∂x

+b ∂ξ∂ y ,

b→b0=A ∂2η∂ x2

+2B ∂2η∂ x∂ y

+C ∂2η∂ y2

+a ∂η∂ x

+b ∂η∂ y , c→c0=c, d→d0=d .

Then, A0=B0=0 for B2−AC>0; A0=C0=0 forB2−AC<0; B0=C 0=0 for B2−AC=0.

Example: wave equation in 1+1D∂2u∂x2

− 1c2

∂2u∂ t2

=0

Here B=0 and B2−AC= 1

c2>0 ,thus , thewaveequation is of the hyperbolic type. The equation

of characteristics ( dtdx )2

= 1c2

leads to the pair dxdt

=c , dxdt

=−c. The characteristics

x−ct=C1 , x+ct=C2 correspond to the first integrals ξ ( x , t )=x−ct , η ( x , y )=x+ct . The coefficients of novel PDE take the values ofA0=C0=0 ,B0=2 , a0=b0=c0=d0=0, thus, the canonical form of the wave equation is ∂2u∂ξ∂η

=0. Integrating the later equation, once over ; ξ ∂u∂η

=C1 (η ) , and once over ;η

∂u∂ξ

=C2(ξ), one finds u (ξ ,η )=C1 (ξ )+C2 (η ). Hence,

u ( x , t )=C1 ( x−ct )+C2(x+ct) is a (general) solution to the wave equation.

Example physical realization. Consider the system Maxwell equations in the absence of electrical charges and currents (vacuum): (i) the Gauss law for the electric field ¿E=0, (ii)

the Gauss law for the magnetic field ¿B=0, (iii) Faraday’s law rot E=−∂B∂ t , (iv)

31


generalized Ampere’s law rot B= 1c2

∂E∂t . Performing the operation of „rot“ on the

equations (iii) and (iv), and utilizing the identity rot rot=−∆+grad÷¿ , together with (i) and (ii), one arrives at the secondary equations of the electromagnetic field

∆ E− 1c2

∂ E∂ t

=0˄∆ B− 1c2

∂B∂ t

=0. For a wave propagating along x direction (e.g. in a

linear waveguide) the linearly (circularly) polarized field E(x,t)=lEu(x,t) c˄ B(x,t)=lBu(x,t) ˄ lE⟘lB⟘OX of uϵR (Ey(x,t)+iEz(x,t)=u(x,t) B˄ y(x,t)+iBz(x,t)=iu(x,t)/c E˄ x=Bx=0 of u C)ϵsatisfies the secondary (wave) equation as well as the pirmary (Maxwell) equations provided the envelope function u(x,t) satifies the wave equation in 1+1D.

Example Cauchy problem for the wave equation; a plane wave from a periodic source. Let the point source of the wave be x=x0, and, u (x0 ,t )=u0sin (ωt+φ). Consider the propagation in the upper half line x>x0 under the condition of inflow at the boundary

( ∂u∂ x )x= x0

=−1c

du (x0 , t )dt

, thus, { C1 (x0−ct )+C2 (x0+ct )=u0 sin (ωt+φ)

C1' (x0−ct )+C2

' (x0+ct )=−u0ω

c cos (ωt+φ). Integrating the

second equation of the system over t, one obtains C1 (x0−ct )−C2 (x0+ct )=u0sin (ωt+φ). The system is soluable provided C2 (η )=0 and the

Cauchy problem solution is u ( x , t )=u0 sin [ω(t− x−x0c )+φ ].

Example: diffusion equation in 1+1D−∂u∂ t

+D ∂2u∂ x2

=0

The above PDE is of the parabolic type. The equation of characteristics is dt2=0, thus, t=const. The above diffusion equation is in its own canonical form.

Example of physical realization. Fick’s laws.

First Fick’s law: the density of the (mass, heat, and so on) diffusion current j is proportional to the gradient of the density (concentration) ϱ: j=−Dgrad ρ , where D is called the coefficient of diffusion. The density (ϱ r,t) can denote the mass density, temperature, density of the electric charge, and so on. In the first Fick’s law, the gradient of density plays a similar role to the electric field in the Ohm’s law.

Second Fick’s law: the density (concentration) of diffusing value (mass, heat, and so on)

satisfies the diffusion equation −∂ ρ∂t

+D ∆ ρ=0. The second law follows from the equation

of continuity (of the flow) ∂ ρ∂t

+¿ j=0, upon substitution of the expression

j=−Dgrad ρ (the first Fick’s law).

Example: Laplace equation in 2D∂2u∂x2

+ ∂2u∂ y2

=0

32

„ZPR PWr – Zintegrowany Program Rozwoju Politechniki Wrocławskiej”The Laplace equation is of the elliptic type. The equation of characteristics dx2+dy2=0 has the soltution x=C1, y=C2, thus, the Laplace equation is in its canonical form.

Method of variable separation

Consider PDE in (n+1)D of the form ∑i=1

n

M iu=Mu, where

M i≡1

ρi(x i) [ ∂∂ xi

ai ( xi )∂∂x i

+bi(x i)], M≡A ∂2

∂t 2+B ∂

∂t , ρi , bi∈C ,ai∈C1.

For simplicity, we consider problems in 2+1D only, while the method is applicable to any D. One looks for the solutions in the form

u ( x , y , t )= ∑m,n=1

∞

AmnXm ( x )Y n ( y )Tmn (t ),

where, X m,Y n are solutions to the eigenequations M x Xm−λm Xm=0, M yY n−λnY n=0, thus, T mn has to satisfy MT mn− λm λnTmn=0. Here, one utilizes the theorem: eigenfunctions belonging to different eigenvalues are

linearly independent (ortogonal vectors in L2, where ⟨ X i , X j ⟩≡∫a

b

ρ ( x )X i ( x ) X j ( x )dx=δij,

for instance with ρ ( x )=1).

One looks for functions that satisfy the boundary conditions of the first, second or third kindX m (a )=0 or X m

' (a )=0 or X m' (a )+α X m (a )=0, respectively, as well as X m (b )=0 or

X m' (b )=0 or X m

' (b )+ β Xm (b )=0.Consider the Hilbert space H0 (a subspace of L2 distinguished by boundary constraints),H 0= {f : f ∈ L2˄α1 f

' (a )+β1 f (a )=α 2 f' (b )+β2 f (b )=0} ,

where α 1β1≠0 and α 2β2≠0, and let the eigenfunctions Xm(x) which satisfy the boundary conditions are ortogonal to each other. Then, any function of H0 can be

expanded with f ( x )=∑k=1

∞

Ak X k (x ),

where Ak=1

‖Xk‖2∫

a

b

ρ1 ( x ) f ( x ) X k ( x ) dx , ‖X k‖2=∫

a

b

ρ1(x )|X k(x )|2dx .


+D ∂2u∂ x2

=0

One performs the expansion u ( x , t )=u0+∑n=1

∞

Xn (x )T n(t) and includes it into PDE. The

diffusion equation is satisfied if

33


X n' 'T n=D XnT n

' , thus Xn

' '

Xn=D

Tn'

Tn. Since x, and t are independent variables, the later requires

X n' '

Xn=−N n

2

Tn'

Tn=−1

DNn2, thus,

X n ( x )=Ancos (N n x )+Bn sin (Nn x)

T n ( t )=Cn exp (−Nn

2 tD

)

The boundary condions lead to the equationsα 1Nn [−Ansin (N na )+Bncos (Nna)]+β1 [ Ancos (Nna)+Bnsin (Nna)]=0

α 2Nn ¿for Nn ,Bn. The coefficients An ∙Cn are to be found with an initial condition u ( x ,0 )=g(x ).

Example of physical realization. We consider Mawell equations in the presence of a stationary current (and zero charge density)

¿E=¿H=0 , rot H=σ E ,rot E=−μ ∂ H∂t

, j=σ E

Performing the operation of „rot“ on the Maxwell equations, and utilizing the identity rot rot=−∆+grad÷¿ leads to the diffusion equation

∆ H=σμ ∂H∂ t

Example of Cauchy problem for the diffusion equation in 1+1D,

∂2H z

∂x2=σμ

∂ H z

∂ t.

We determine the distrbution of the magnetic field and the current density in an infinite metalic plate of a thickness 2a, x∈ (−a ,a ) along normal to the plate, assuming that the magnetic filed is constant and directed along z-axis at the surfaces of the plate. Thus, H z (−a ,t )=H z (a , t )=H 0>0. Let H z ( x ,0 )=0 for x∈ (−a ,a ) . (*)There is no reason for changing the initial directions of the fields in time, thus, H ( x , t )=[0,0 , H z ( x ,t ) ]. In the consequence of Maxwell equations E ( x ,t )=[0 , E y ( x ,t ) ,0 ] , j ( x ,t )= [0 , j y ( x , t ) ,0 ]. Applying the method of variable separation, one looks for the Hz in the form

H z ( x , t )=H0+∑k=0

∞

Xk (x )T k (t ) whose single-variable functions satisfy

X k' '+N k

2 X k=0 ,T k' + 1

σμN k2T k=0. General solutions to the later equations are

X k ( x )=A kcos (N k x )+Bk sin (Nk x ), T k (t )=C k exp(−N k

σμt).

The first of the conditions (*) have to be satisfied by each of the products X k (x)Tk (t ),

which leads to Bk=0 and, via cos (N k a )=0, to N k=π2a

(2k+1). Exchanging the products

AkCkinto Ak and applying the second of the conitions (*), one writes

34


∑k=0

∞

Ak cos[ (2k+1 ) πx2a ]=−H 0. Multiplying the latest equation by cos [ (2l+1 ) πx

2a ] and

utilizing ∫0

a

cos [ (2k+1 ) πx2a ]cos [(2 j+1 ) πx

2a ]d x∝ δkj , we obtain the expansion coefficients

Ak=−H 0∫

0

a

cos [ (2 k+1 ) πx2a ]d x

∫0

a

cos2[ (2k+1 ) πx2a ]d x

=(−1 )k 4H 0

(2k+1 )π.

Finally,

H z ( x , t )=H0+4H 0

π ∑k=0

∞ (−1 )k

2 k+1cos [ (2k+1 ) πx

2a ]exp[−(2 k+1 )2 π 2

4a2tσμ ],

and, via the Ampere’s law ( j=rot H),

j y ( x ,t )=−∂ H z

∂ x=2H 0

a ∑k=0

∞

(−1 )kcos [ (2k+1 ) πx2a ]exp[−(2k+1 )2 π 2

4 a2tσμ ].

Method of potentials

Knowing a specific solution to linear PDE relevant to the space except a singular point v(r,t), one can write any solution in the form u (r , t )=∫ ρ(ξ) v (r−ξ , t)dξ , where ρ(ξ) denotes a space distribution of a charge. The functio u(r,t) is called a potential while the integral kernel v(r,t) is a single-point potential up to a point charge.

Example: Laplace equation in 3D or 2D ∂2u∂x2

+ ∂2u∂ y2

+ ∂2u∂z2

=0∧∂2u∂x2

+ ∂2u∂ y2

=0

In 3D case, the single-point potential is v (r )=v (r )=C1+C2

r, where r ≡|r|, while the

condition (of integrability) limr→0

v (r )=¿0 ¿ leads to v (r )=1r . In 2D case, the single-point

potential is v (r )=v (r )=ln 1r .The potential u (r , t )=∫ ρ(ξ) v (r−ξ , t)dξ , where ξ∈V , is a

solution to the Laplace equation for r∈Ω provided Ω∩V=∅ .

Example: Poisson equation in 3D or 2D ∂2u∂x2

+ ∂2u∂ y2

+ ∂2u∂z2

=−4 πρ ( x , y , z )∨ ∂2u∂ x2

+ ∂2u∂ y2

=−2 πσ (x , y )

For r∈Ω, the potential of the Poisson in 3D (2D) equation is u (r )=∫ ρ' (ξ ') v(r−ξ ')dξ '+∫ ρ (ξ ) v (r−ξ )d ξ ,

(u (r )=∫ σ ' (ξ ')v (r−ξ ')d ξ'+∫σ (ξ ) v (r−ξ ) d ξ), where ξ '∈V , and Ω∩V=∅ , whileξ∈Ω.

Example: wave equation in 3+1D ∂2u∂x2

+ ∂2u∂ y2

+ ∂2u∂z2

− 1c2

∂2u∂ t2

=0

The single-point (retarded) potentials (basic solutions) to the wave equation in 3+1D are

35

„ZPR PWr – Zintegrowany Program Rozwoju Politechniki Wrocławskiej”v1 (r , t )=Φ1 ¿¿ and v2 (r , t )=Φ2 ¿¿.

Example: diffusion equation in 3+1D or 1+1D∂2u∂x2

+ ∂2u∂ y2

+ ∂2u∂ z2

=D ∂u∂ t

∨ ∂2u∂ x2

=D ∂u∂ t

The single-point (retarded) potential (a basic soltion) to the diffusion equation in 3+1D (1+1D) is

v (r , t )= 1( t−t0 )3/2

exp[ −r2

4D (t−t 0 ) ], (v ( x ,t )= 1√ t−t0

exp[ −x2

4D (t−t 0 ) ]), where t>t0.

Method of Green functions

Consider PDE of the form Lu (x , y , z ,t )=f (x , y , z ,t ) and define its Green function withLG ( x , y , z , t ; x ' , y ' , z ' , t ' )=−δ (x−x ')δ ( y− y ')δ (z−z ')δ (t−t ' ), thus,

∫−∞

∞

∫LG (r ,t ; r ' ,t ' ) f (r ' , t ' )d r ' dt '=L(∫−∞

∞

∫G (r , t ;r ' , t ' ) f (r ' ,t ' )dr ' dt ' )=Lu(r , t),

the fundamental solution to PDE takes the form

u (r , t )=∫−∞

∞

∫G (r , t ; r ' , t ' ) f (r ' , t ' )d r ' dt ' .

The Green function is a sum of single-point potential of the homogeneous equation and

some function g (x , y , z , t ; x ' , y ' , z ' ,t ' ), and satisfies lim(r ,t )→P

G (r , t ; r ' ,t ' )=0 for (r , t ) , (r ' ,t ' )∈Ω

and P∈∂Ω as well as G (r , t ; r ' ,t ' )=G(r ' , t ' ; r ,t ).

The Green functions can also be used in order to include initial and boundary conditions when solving the homogeneous PDE.


+D ∂2u∂ x2

=0

The Green function is of the form G (x , t ; x ' ,t ' )= 1√ t−t '

exp[ −(x−x' )2

4D (t−t ' ) ]+g (x , t ; x ' ,t '),

where x , x '∈(b , c) and t>t‘. The function g(x,t;x‘,t‘) satisfies the diffusion equation and the relationships

limx'→b

g (x , t ; x ' , t' )=¿ 1√t−t '

exp [ −(x−b)2

4D (t−t ' ) ]¿, limx'→c

g (x ,t ; x' , t' )=¿ 1√ t−t '

exp[ −(c−x)2

4D (t−t ' ) ]¿.

The Green function satisfieslim

x'→b (x '→c)G (x ,t ; x ' ,t ' )=0, and G (x , t ; x ' ,t ' )=G( x' , t ; x , t ') for t>t‘.

Consider the initial and boundary conditions

36


{u ( x ,0 )=h(x)u (b , t )=f 1(t )u (c ,t )=f 2(t )

Since G(x,t;x‘,t‘) satisfies the primary PDE, the function

u ( x , t )= 12√D √π∫b

c

h ( x' )G(x ,t ;x ' ,0)d x'+ 12√D√π∫0

t

[f 1 (t ' ) ∂G∂ x' |x'=b

−f 2(t') ∂G∂ x '|x '=c]d t '

is the specific solution to the diffusion equation.

If b=∞ and c=-∞, the Green function for t>t‘ is G (x , t ; x ' ,t ' )= 1√ t−t '

exp[ −(x−x' )2

4D (t−t ' ) ].Then, for u ( x , t )=h ( x ), the specific solution reads

u ( x , t )= 12√D √π ∫

−∞

∞

h (x ' )G(x , t ; x ' ,0)d x '.

An example nonlinear PDEs: diffusive Lotka-Volterra models

A general form of the (two-component) diffusive Lotka-Volterra system of PDEs (in 1+1D) is

{∂u∂ t =D1∂2u∂ x2

+u (b1−a11u−a12 v )

∂v∂ t =D2

∂2 v∂ x2

+v (b2−a21u−a22v )

With positive coefficients, it models the competition in the presence of the diffusion and it is known as a competition-diffusion model. Let us rescale the coefficients, thus, simlying the the system into

{ ∂u∂ t

=∂2u∂ x2

+u (1−u−cv )

∂v∂ t =d ∂2 v

∂ x2+v (a−bu−v )

The asymptotics of solutions depends on the relationships between the coefficients in the following way

(i) if a<min(b,1/c), then limt→∞

[u ( x , t ) , v ( x , t ) ]=[1,0],

(ii) if b<a<1/c, then limt→∞

[u ( x , t ) , v ( x , t ) ]=[1−ac1−bc

, a−b1−bc ],

(iii) if 1/c<a<b, then [1,0] and [0,a] are locally-stable rest points,

(iv) if a>max(b,1/c), then limt→∞

[u ( x , t ) , v ( x , t ) ]=[0 , a ].We say that the regime (iii) corresponds to the bistability of the system.

Note: the Euler-Lotka PDEs are useful in describing many chemical „reaction-diffusion“ processes. Extended models including polynomial nonlineartities describe a Grey-Scott

37

„ZPR PWr – Zintegrowany Program Rozwoju Politechniki Wrocławskiej”model of cubic autocatalytic reaction and a Belousov-Zhabotinskii reaction (whitin Noyes-Filed model). Traveling-wave solutions

Consider specific solutions of the traveling-wave type [u(x,t),v(x,t)]=[U(z),V(z)], where z=x- tη and η denotes the velocity of the propagation. The primary PDEs are thus transformed into secondary ODEs of the form

{−η dUdz

=d2Ud z2

+U (1−U−cV )

−η dVdz =d d

2Vd z2

+V (a−bU−V )

In the bistability regime of the coefficients (iii), the following boundary conditions are

satisfied limz→−∞

[U ( z ) ,V ( z ) ]= [0 , a ] , limz→∞

[U ( z ) ,V ( z ) ]=[1,0]. Finding exact analytical

solutions is possible with additional relationships between the coefficients.

For instance, with d=13c , b=2+

5 a3 -ac, η=

−2+ac√2ac , one finds

U ( z )=12 [1+ tanh(√2ac4 z )]

V ( z )=a4 [1−tanh(√2ac4 z )]

Standing-wave solutions

In the regime (iv), we look for the standing-wave solutions (of η=0), thus,

[U,V]=[U(x),V(x)]) which satisfy the boundary conditions limx→∓∞

[U ( x ),V ( x ) ]=[0 , a ],(pulse solutions).

For instance, with d=a−b

−1+ac , one finds

U ( x )= 36e√ρ x

(1+6e√ρ x )2

V ( x )=a [1− 36 e√ ρ x

(1+6 e√ ρ x )2 ]where ρ≡−1+ac .

There are solutions which do not belong to the classes (i)-(iv).

For instance, with d=13c , a=50−435bc+261b

2 c2

3α, α≡−200+165bc−9b2 c2, one finds a

periodic solution of the standing-wave type

U ( x )=25(−5+3bc)α

+ 100 k √ β√8 α2

sn( √5 β√α

,k )V ( x )=

−3 (25−30bc+3b2 c2 )cα

−156k √2 β

√α2sn (√5 β√α

, k )+10 k 2βcαsn(√5β√α

,k )2

38


where β≡(5−3bc)2

1+k2.

UNIT 9

Nonlinear PDEs of soliton solutions (example: nonlinear Schrodinger equation in 1+1D)

We focus on a single (Hirota) method of finding the soliton solutions to nonlinear Schrodinger equation (NLS)

i ∂u∂t

+ β ∂2u∂ x2

+δ|u|2u=0

that is the simplest and useful to many other nonlinear PDEs that posses soliton solutions. The Hirota method (also known as a direct method) is based on a representation of the solution with a pair of functions which are sums of exponents, the exponentials of whom are linear in x an t varialbles (for some problems other functions than the exponents are appilcable).

39

„ZPR PWr – Zintegrowany Program Rozwoju Politechniki Wrocławskiej”For NLS, N-soliton solution is obtainable via a transform

u ( x , t )=G ( x , t )F ( x ,t )

, (*)

which leads to secondary (bilinear) PDEs

{ [ i( ∂∂t − ∂∂ t ' )+β ( ∂

∂ x− ∂∂x ' )

2] F ( x , t )G(x ' , t ')|t '=t , x'=x=0

β ( ∂∂x

− ∂∂x ' )

2

F (x , t ) F¿(x ' , t ')|t '=t ,x '=x−δG ( x , t )G¿ ( x , t )=0

Therefore, the Hirota method is called a bilinearization method as well.

First, we consider the case of βδ>0 (which corresponds to describing „self-focusing“ solitons, called as bright solitons as well). The solutions to the bilinear equations G(x,t) and F(x,t) can written using an ansatz (Hirota expansion). It reads,for N=1,

F ( x ,t )=1+a11eγ 1 (x , t )+γ 1

¿(x ,t )

G ( x , t )=eγ 1(x ,t )

And, for N=2,F ( x ,t )=1+a11e

γ 1 (x , t )+γ 1¿ (x , t )+a22e

γ 2 (x , t )+γ 2¿ (x , t )+a12e

γ 1( x ,t ) +γ 2¿ ( x ,t )+a21 e

γ 2( x ,t ) +γ1¿ ( x ,t )

+a1122 eγ1 ( x ,t )+ γ1

¿ ( x ,t )+ γ2 ( x ,t )+ γ2¿ ( x ,t )

G ( x , t )=eγ 1(x ,t )+eγ 2(x ,t )+a122 eγ1 ( x ,t )+ γ2 ( x ,t )+ γ2

¿ (x ,t )+a112eγ 1 (x , t )+γ 2 (x , t )+γ 1

¿(x ,t )

Here,

γi (x , t )≡ pi x+ i pi2t−γi

0, a i j≡δ2 β

1( pi+ p j

¿)2=a j i

¿ , and the coefficients of the higher-

order terms can be expressed with a ij≡2βδ ( p i−p j )

2=ai j¿

in the following way

a ijk=aij ai k a j k , a1122=a12a11a22a12a21a12 .

The coefficients pi are arbitrary complex numbers independent of parameters of the primary PDE. Thus, they are constants of motion, which reveals an important property of the nonlinear PDEs relevant to soliton solutions; there exists infinite number of the first integrals to those equations. Also, the presence of many first integrals is related to the asymptotics of the soliton solutions. For instance, consider the limits of the two-soliton solution for ℜk i ∙ℑ ki<0

limt→−∞

u ( x ,t )= eγ 1(x , t)

1+a11 eγ1 ( x ,t )+ γ1

¿ (x, t )+eγ 2(x ,t )

1+a22 eγ2 ( x ,t )+γ 2

¿( x, t )

¿a11

−1 /2 eℑ γ1 (x ,t )

sech (ℜ γ1 ( x , t )+log a112 )

+a22

−1 /2 eℑ γ2 (x ,t )

sech (ℜ γ2 (x , t )+log a222 )

¿Hence, the only consequence of the soliton collision is a „jumpwise“ shifting the phases and centers of the pulses. However, the pulse widths, velocities, rotation frequencies, and amplitudes are conserved during the collision since they are governed by values of the constants pi.

40

„ZPR PWr – Zintegrowany Program Rozwoju Politechniki Wrocławskiej”Let us serch for pulse solutions to the NLS of βδ<0 (which corresponds to describing „self-defocusing“ solitons, called as dark solitons as well). Equivalently, we write the self-defocusing NLS in the form

i ∂u∂t

−α ∂2u∂ x2

+δ|u|2u=0

where αδ>0. Via transform (*), one decouples it into the bilinear system

{ [i( ∂∂ t

− ∂∂ t ' )−α ( ∂

∂ x− ∂∂ x ' )

2

−λ] F ( x ,t )G(x ' , t ')|t '=t , x '= x=0

[α ( ∂∂x

− ∂∂ x' )

2

+λ ]F (x , t )F ¿(x ' , t ')|t '=t ,x '=x+δG ( x , t )G¿ ( x , t )=0

where is a constant to be determined. With the ansatzλF ( x , t )=1+ χ eθ1( x ,t )

G ( x , t )=c ( x , t ) (1+ χ Z1eθ1( x, t))

one obtains a single-pulse solution (a dark-soliton), while two-soliton solution is obtained with

F ( x , t )=1+eθ1 ( x ,t )+eθ2(x , t)+b12eθ1 (x ,t ) +θ2(x , t)

G ( x , t )=c (1+Z1 eθ1(x ,t )+Z2eθ2(x , t)+b12Z1Z2 e

θ1 ( x ,t )+θ2 (x ,t ))

Here c ( x , t )=Ce i [lx−(λ−l2 )t ] , λ=−δα

CC¿, Z j=

−p j+i( 2δα CC¿−p j2)1/2

p j+i(2δα C C¿−p j2)1/2

θ j ( x , t )=p j x−[( 2δα CC¿− p j2)1/2

−2 l ]p j t+θ j0,

b12=( p1−p2)

2+[( 2δα C C¿−p12)1 /2

−( 2δα C C¿−p22)1/2]

2

( p1+ p2 )2+[( 2δα CC ¿−p12)1 /2

−( 2δα CC ¿−p22)1 /2]

2 .

The single-pulse (dark-soliton) solution can be rewritten in the form

u ( x , t )=c ( x , t )1+ χ Z1e

θ1 ( x ,t )

1+ χ eθ1 ( x, t ) =Ce i [lx−( λ−l2) t ] {(1+Z1 )−(1−Z1 ) tanh [ θ1 ( x , t )2 ]}

Usually the square of the field module represents some intensity (e.g. the power denstity of the electromagnetic radiation) that is calculated in order to better understand physical meaning of the dark-soliton-like object

|u( x , t)|2=2|C|2 {2−(1−ℜZ1) sech2[ θ1 ( x ,t )

2 ]}Thus, the dark soliton describes a pulse-like whole in a uniformly-excited medium, in contrast to the bright soliton which relates to the excitation pulse of the intensity |u( x , t)|2∝ sech2 [γ 1 ( x ,t ) ].

41


UNIT 10

Ginzburg-Landau equation and simplest topological solitons (domain walls)

Genesis and domain-wall solutions

Consider the simplest nonlinear equation having a pitchfork bifurcation (with respect to a parameter a, at a=0) m=am (t )−bm3(t). (*) Let m(t)∈R be an order parameter of a complex system (e.g. a component of magnetization

42

„ZPR PWr – Zintegrowany Program Rozwoju Politechniki Wrocławskiej”vector), the bifurcation point is accompanied by a phase transition since the stationary solutions to (*) are m=0 for a>0 (a disordered, paramagnetic phase), and m=±√a/b for a<0 (an ordered, ferromagnetic phase).The Hamiltonian of (*), (which contains the potential energy only) takes the form (up to a multiplying constant)

Φ (m ,T )=V (−a (T )m2+ b(T )2

m4)+Φ0(T ),

where V denotes the volume of the system. It can be thought of as a thermodynamic potential (free energy) whose minima determine the preferable state of the system. Thus, its coefficients are dependent on the thermodynamic parameters, for instance, on the temperature T, with a∝T C−T , b=const . Here, TC denotes the critical temperature (of the phase-transition point). The time derivative in (*) is of the dissipative origin, since the Hamiltonian is independent of any variable canonically conjugate to m.

One includes local phase inhomogeneity via adding a gradient term to the density of the free energy (in 1+1D)

Φ (m , ∂m∂x

,T )=∫(c (T )( ∂m∂x )2

−a(T )m2+b (T )2

m4)dV +Φ0(T ),

in a way that the Hamiltonian density had a global minimum. Therefore, (*) is transformed into PDE (a Ginzburg-Landau equation)

∂m∂ t

=c ∂2m∂ x2

+am (x , t )−bm3(x , t)

whose differential part is similar to the diffusion equation.

Solving the stationary GL equation for a<0, one finds

m (x )=±√ ab tanh [√ a2c (x−x0 )]

which describes a domain wall (DW) of the width √2c /a , centered at x0 . The sign plus (minus) corresponds to a conserved (topological) charge q=1 (q=-1), where q≡√b/4 a [m (∞ )−m (−∞) ]. That object is called tail-to-tail DW (head-to-head DW) due to the relative orientation of the magnetization arrows of two neighboring domains. Because of the topological charge accompanied to the DW, it is a kind of topological soliton.

In the case of a complex order parameter ψ (x ,t )∈C, GL equation takes the form∂ψ∂ t

=c ∂2ψ∂ x2

+a1ψ (x , t )+a2ψ¿ ( x ,t )−b|ψ|2(x , t)ψ (x , t )

(e.g. relevant to describing a superconductor in the critical regime, close to the superconductor-metal phase transition). Is stationary solutions represent an Ising DW

ψ ( x )=±√ a1+a2btanh [√ a1+a2

2c (x−x0) ] and a Bloch DW

ψ ( x )=±√ a1+a2btanh [√ 2a2c (x−x0 )]±(∓)i 2√ a1−3a2b

sech[√ 2a2c (x−x0 )]. 43

„ZPR PWr – Zintegrowany Program Rozwoju Politechniki Wrocławskiej”The later is characterized by a chirality (a clockwise or anticlockwise rotationof the 2D vector [ℜψ,ℑψ ] inside DW) besides the charge q.The bifurcation (phase transition) point relates to a1+a2=0, whereas, the GL potential

Φ (ψ , ∂ψ∂ x

,T )=∫(c (T ) ∂ψ∂x

∂ψ¿

∂x−a1 (T )ψψ¿+

a2(T )2

(ψ2+ψ¿2 )+ b(T )2

(ψψ¿ )2)dV +Φ0(T )

favors Bloch DWs for a1−3a2>0, while Ising DWs for a1−3a2<0. The point a1=3a2 corresponds to the so-called Bloch-Ising transition (within the ordered phase).

Exact stationary states of two DWs are found for the case of the coexistence of one Ising DW and one Bloch DW. States of two Ising DWs or two Bloch DWs are more important because of the Bloch-Ising transition (the instability of one of the DW kinds), while they are treated with approximate (perturbative) methods only.

Field-driven DW motion

Add a constant term to GL equation ∂m∂ t

=c ∂2m∂ x2

+am (x , t )−bm3 ( x ,t )+H (**)

The last term on the RHS represents an external (magnetic) field and its form follows from adding the density of the Zeeman energy −m ( x , t )H to the GL potential. Using a stationary solutions we write an ansatz

m (x , t )=±√ ab tanh [√ a2c (x−x0( t))],

thus,

∂m∂ t

=− x0∂m∂ x .

While we do not manage to find strict solution with the above ansatz, for x in the vicinity of x0(t), utilizing the relationship tanh (x)≈ x, we establish the DW velocity

x0=∓√2bca

H .

Note that DWs of GL equation are not topological solitons in a strict sense. There are exact two-DW solutions to GL equation which describe pairs of one Bloch DW and one Neel DW, however, in the stationary state only. Proper topological solitons (stable against the collisions) are predicted with a Landau-Lifshitz equation of the magnetization-vector evolution.

Propagating phase fronts

For H=0, one finds another non-stationary localized solution to (**), an exact one

m (x , t )=±√ ab eμ( x, t)

1+eμ (x ,t ) ,

where μ ( x , t )≡±√ a2c ( x−x0 )+3a2 t .

It represents a phase front which propagates into an unstable disordered phase of zero “magnetization”. Ways of creating areas of disordered phase can be different, while it

44

„ZPR PWr – Zintegrowany Program Rozwoju Politechniki Wrocławskiej”has been suggested that the Bloch to Ising transition in a system of interacting DWs can lead to the appearance of a local disorder.

45

Documents

· Web viewOrdinary differential equation (ODE) of the n-th order is an equation of the form y n t =f t,y t , y ' t , y '' t ,…, y n-1 t (*) Its solution is n-time differentiable