Nonlinear transport phenomena: models, method of solving and unusual features (3)

Nonlinear transport phenomena:models, method of solving and unusual

features: Lecture 3

Vsevolod Vladimirov

AGH University of Science and technology, Faculty of AppliedMathematics

Krakow, August 6, 2010

KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 1 / 34

Traveling wave solutions supported by GBE

We are interested in the following generalization of the BE,called convection-diffusion - reaction equation:

τ ut t + ut + uux = ν [un ux]x + f(u) (1)

Our aim is to study the wave patterns, i.e. physicallymeaningful traveling wave (TW) solutions, having the form

u(t, x) = U(ξ), ξ = x− V t.

We are specially interest in the existence ofI solitons,I compactons,I some other solutions related to them

u(t, x) = U(ξ), ξ = x− V t.

Solitons and compactons are solitary waves, moving withconstant velocity V without change of their shape. The maindifference between them is seen on the graphs shown below:

Figure: Graph of the KdV soliton U(ξ) = ACosh2[B ξ]

Figure: Graph of the Rosenau-Hyman compacton

U(ξ) =

{ACos2[B ξ], if |ξ| < 2π,

0 otherwise

So how can we distinguish the solitary wave solutions andcompactons within the set of TW solutions?

To answer this question, we restore to the geometricinterpretation of these solutions.

So how can we distinguish the solitary wave solutions andcompactons within the set of TW solutions?

To answer this question, we restore to the geometricinterpretation of these solutions.

Soliton on the phase plane of factorized system

Both solitons and compactons are the TW solutions, having theform

u(t, x) = U(ξ), with ξ = x− V t.

We give the geometric interpretation of the soliton byconsidering the Korteveg-de Vries (KdV) equation

ut + uux + uxxx = 0

Inserting the TW ansatz into the KdV equation, we get:

[U(ξ) +

U2(ξ)

2− V U(ξ)

This equation is equivalent to the following dynamical system:

ut + uux + uxxx = 0

[U(ξ) +

U2(ξ)

2− V U(ξ)

ut + uux + uxxx = 0

[U(ξ) +

U2(ξ)

2− V U(ξ)

ut + uux + uxxx = 0

[U(ξ) +

U2(ξ)

2− V U(ξ)

{U(ξ) = −W (ξ),

W (ξ) = U2(ξ)2 − V U(ξ)

Lemma 1.The system (2) is a Hamiltonian system, with

H(U,W ) =W 2

6− V

2= Ekin(W ) + Epot(U). (3)

Lemma 2.Hamiltonian function H(U,W ) remains constantalong a trajectory of the dynamical system (2).Theorem 1. 1

I All stationary points of the system (2) are placed at theextremal points of the function Epot(U) = U3

6 − V U2

2 ;I local maximum of Epot(U) correspond to a saddle point,

while the local minimum corresponds to a center;I all possible coordinates of the phase trajectory corresponding

to H(U, W ) = C satisfy the inequalities C − Epot(U) ≥ 0.1Andronov, Vitt, Khaikin, Theory of oscillations, Nauka, Moscow, 1966

{U(ξ) = −W (ξ),

W (ξ) = U2(ξ)2 − V U(ξ)

H(U,W ) =W 2

6− V

2= Ekin(W ) + Epot(U). (3)

6 − V U2

{U(ξ) = −W (ξ),

W (ξ) = U2(ξ)2 − V U(ξ)

H(U,W ) =W 2

6− V

2= Ekin(W ) + Epot(U). (3)

6 − V U2

{U(ξ) = −W (ξ),

W (ξ) = U2(ξ)2 − V U(ξ)

H(U,W ) =W 2

6− V

2= Ekin(W ) + Epot(U). (3)

6 − V U2

{U(ξ) = −W (ξ),

W (ξ) = U2(ξ)2 − V U(ξ)

H(U,W ) =W 2

6− V

2= Ekin(W ) + Epot(U). (3)

6 − V U2

{U(ξ) = −W (ξ),

W (ξ) = U2(ξ)2 − V U(ξ)

H(U,W ) =W 2

6− V

2= Ekin(W ) + Epot(U). (3)

6 − V U2

{U(ξ) = −W (ξ),

W (ξ) = U2(ξ)2 − V U(ξ)

H(U,W ) =W 2

6− V

2= Ekin(W ) + Epot(U). (3)

6 − V U2

Graph of the function Epot(U) = U3

6 − V U2

Phase portrait of the system (2)

6 − V U2

Statement 1. The only trajectory that could correspond to thesoliton solution is the closed loop bi-asymptotic to the saddlepoint.

Statement 2. The solitary wave solution U(ξ) is nonzero forany ξ ∈ R.Proof. The linearized system U = −W, W = −V U isequivalent to the single equation U = V U . Solution to thisequation, that can approach zero, must have the formU(ξ) = C e±V ξ. It is obvious, that the function U(ξ) is nonzerofor any nonzero ξ, and can attain zero when ξ → ±∞.

Compactons on the phase plane of factorized system

Let us discuss the compacton TW solutions, basing onRosenau-Hyman equation

ut +(u2

)x x x

= 0. (4)

Inserting the TW ansatz u(t, x) = U(ξ), ξ = x− V t into (4),we get, after some manipulation, the dynamical system{

d Ud T = −2 U2W,d Wd T = U

[−V U + U2 + 2W 2

] , (5)

where dd T = 2U2 d

d ξ .

4U4 − V

3U3 + U2W 2.

ut +(u2

)x x x

= 0. (4)

[−V U + U2 + 2W 2

] , (5)

where dd T = 2U2 d

d ξ .

4U4 − V

3U3 + U2W 2.

ut +(u2

)x x x

= 0. (4)

[−V U + U2 + 2W 2

] , (5)

where dd T = 2U2 d

d ξ .

4U4 − V

3U3 + U2W 2.

Graph of the function Epot(U)

Statement 3. The solution of to (5) corresponding to thehomoclinic loop has a compact support.Sketch of the proof. It can be shown, that the saddleseparatrices enter (leave) the origin forming right angle with thehorizontal axis.Therefore, in proximity of the originW ≈ −C1 U

α with 0 < α < 1. From this we get the asymptoticequation −W = d U

d ξ = C1 Uα. It is evident, thus, that the

asymptotic solution takes the formU(ξ) ≈ C2 (ξ − ξ0)

γ , 1 < γ = 11−α . So the function U(ξ)

approaches zero as ξ → ξ0 ± 0.So we deal in fact with the glued (generalized) solution,and its nonzero part corresponds to the homoclinicloop.KPI, 2010, L3 Nonlinear transport phenomena: TW solutions to GBE 12 / 34

Very important conclusion

The localized wave patterns, such as solitary waves andcompactons are represented in the phase plane of thefactorized system by the HOMOCLINIC LOOP, i.e. thephase trajectory bi-asymptotic to a saddle point.

Factorization of the modeling system

Let’s return to the generalized C-R-D equation:

αutt + ut + uux − κ (un ux)x = f(u). (6)

Inserting the TW ansatz u(t, x) = U(ξ) ≡ U (x− V t) into thissystem,we get:

∆(U)U = ∆(U)W, (7)∆(U)W = −

[f(U) + κnUn−1W 2 + (V − U)W

where ∆(U) = κUn − αV 2.

∆(U)U = ∆(U)W, (7)∆(U)W = −

[f(U) + κnUn−1W 2 + (V − U)W

∆(U)U = ∆(U)W, (7)∆(U)W = −

[f(U) + κnUn−1W 2 + (V − U)W

∆(U)U = ∆(U)W, (7)∆(U)W = −

[f(U) + κnUn−1W 2 + (V − U)W

∆(U)U = ∆(U)W, (7)∆(U)W = −

[f(U) + κnUn−1W 2 + (V − U)W

The time-delayed C −R−D equation is of dissipativetype. Therefore:

I the factorized system is not Hamiltonian;

I the homoclinic trajectory will appear at the specificvalues of the parameters!!!

Our further strategy:

I we choose specific function f(u) such that thefactorized system will have at least two stationarypoints (U0, 0) and (U1, 0) with 0 ≤ U0 < U1;

I we state the condition for which the stationary point(U1, 0) becomes a center;

I we state, the conditions which guarantee that the otherpoint (U0, 0) simultaneously is a saddle;

I we analyze the appearance of the limit cycle with zeroradius at the point (U1, 0);

I we check numerically the possibility of homoclinicbifurcation appearance.

Further assumptions: We assume that

f(U) = (U − U0)m (U − U1) ψ(U), U1 > U0 ≥ 0,

where ψ(U)|<U0, U1> 6= 0.

Under these assumption our system has two stationary points(U0, 0) and (U1, 0) lying on the horizontal axis of the phasespace (U, W ), and no any other stationary point inside thesegment < U0, U1 >.

Creation of a stable limit cycle

Theorem 1.

1. If the following conditions hold

∆(U1) · ψ(U1) ≡(κUn

1 − αV 2)ψ(U1) > 0, (8)

and|∆(U1)| ϕ(U1) + κnUn−1

1 |ϕ(U1)| > 0, (9)

whereϕ(U) = (U − U0)

m ψ(U), ∆(U) = κUn − αV 2then invicinity of the stationary point (U1, 0) a stable limitcycle with zero radius is created, when the wave packvelocity V approaches the bifurcation value Vcr1 = U1.

2. Under these conditions, the other stationary point(U0, 0) is a topological saddle, or, at least, contains asaddle sector in the half-plane U > U0.

Theorem 1.

∆(U1) · ψ(U1) ≡(κUn

1 − αV 2)ψ(U1) > 0, (8)

1 |ϕ(U1)| > 0, (9)

Theorem 1.

∆(U1) · ψ(U1) ≡(κUn

1 − αV 2)ψ(U1) > 0, (8)

1 |ϕ(U1)| > 0, (9)

A peculiarity of our factorized system

∆(U)U = ∆(U)W, (10)∆(U)W = −

[f(U) + κnUn−1W 2 + (V − U)W

where f(U) = (U − U1) (U − U0)m ψ(U), is the presence of the

line of singularities ∆(U) = κUn − αV 2 = 0, moving alongthe phase plane, as the bifurcation parameter V ischanged.

This creates an extra mechanism of the limit cycle destruction,different from the homoclinic bifurcation.

But it is just the singular line, which makes possible thepresence of such TW solutions, as compactons, shock frontsand the peakons.

A necessary condition of their appearance reads as follows:the singular line must contain the topological saddle at themoment of the homoclinic bifurcation!

∆(U)U = ∆(U)W, (10)∆(U)W = −

[f(U) + κnUn−1W 2 + (V − U)W

∆(U)U = ∆(U)W, (10)∆(U)W = −

[f(U) + κnUn−1W 2 + (V − U)W

∆(U)U = ∆(U)W, (10)∆(U)W = −

[f(U) + κnUn−1W 2 + (V − U)W

To what kind of localized solution corresponds thehomoclinic loop?

Asymptotic study of the dynamical system, corresponding tosource equation

αutt + ut + uux − κ (un ux)x = (u− U0)m (u− U1) ψ(u) (11)

enable to state that:I homoclinic loop corresponds to the compactly-supported

TW if 0 < m < 1, and n ∈ N, and U∗ = U0, whereU∗ ⇔ ∆(U) = 0 ;

I homoclinic loop corresponds to the soliton-like TW ifm = 1, n ∈ N and U∗ < U0;

I homoclinic loop corresponds to a semi-compacton (orshock-like TW solution) if m ≥ 1, n ∈ N, and U∗ = U0.

Figure: Vicinity of the origin for various combinations of theparameters m, n

Numerical investigation of factorized system

I Numerical simulations of the system (10) were carriedout with κ = 1, U1 = 3, U0 = 1. The remainingparameters varied from one case to another.

I We discuss the results concerning the details of thephase portraits in terms of the reference frame(X, W ) = (U − U0, W ).

I The localized wave patterns are presented in ”physical”coordinates (ξ, U), where ξ = x− V t is the TWcoordinate

Figure: Homoclinic solution of the system (10) withϕ(U) = (U − U0)

1/2 (m = 1/2)(left) and the corresponding compactlysupported TW solution to Eq. (11) (right), obtained for n = 1,α = 0.12, Vcr2

∼= 2.68687 and U∗ − U0 = −0.133684

1/2 (m = 1/2) (left) and the corresponding TWsolution to Eq. (11) (right), obtained for n = 1, α = 0.13827,Vcr2

∼= 2.68892 and U∗ − U0 = 0.99973

1 (m = 1) (left), the corresponding tandem ofwell-localized soliton-like solutions to Eq. (11) (center), and thesoliton-like solution (right), obtained for n = 1, α = 0.06,Vcr2

∼= 2.65795 and U∗ − U0 = −0.576119

Figure: Homoclinic solution of the system (10) with ϕ(U) = (U −U0)1

(m = 1) (left), the corresponding tandem od solitary wave solutions toEq. (11) (center) and a single solitary wave solution (right), obtainedfor n = 1, α = 0.142, Vcr2

∼= 2.65489 and U∗ − U0 = 0.000878617

Figure: Tandems od shock-like solutions to Eq. (11), corresponding toϕ(U) = (U − U0)

1 (m = 1), U∗ ≈ U0, n = 2 (left), n = 3 (center), andn = 4 (right)

Figure: Shock-like solutions to Eq. (11), corresponding toϕ(U) = (U − U0)

3 (m = 3), U∗ ≈ U0, n = 1 (left), n = 3 (center), andn = 4 (right)

Figure: Shock-like solutions to Eq. (11), corresponding toϕ(U) = (U − U0)

m, U∗ ≈ U0, n = 4, m = 1 (left), m = 2 (center), andm = 3 (right)

Figure: Periodic solution of the system (10) with ϕ(U) = −(U − U0)1

(m = 1) (left) and the corresponding tandem of generalized peak-likesolutions to Eq. (11) (right), obtained for n = 1, α = 0.552,Vcr2

∼= 3.00593 and U∗ − U1 = 1.98765

Figure: Homoclinic solution of the system (10) withϕ(U) = −(U − U0)

1/2 (m = 1/2) (left) and the corresponding tandemof generalized peak-like solutions to Eq. (11) (right), obtained forn = 1, α = 0.562, Vcr2

∼= 3.14497 and U∗ − U1 = 2.55863

Summary

1. The time-delayed C −R−D system possesses a largevariety of the localized wave patterns.

2. The type of the pattern is strongly depend on thevalues of the parameters

3. An open question is the analytical description of thelocalized wave patterns within the given model.

4. Another open question is the stability and theattractive features of the localized wave patterns.

Summary

THANKS FOR YOUR ATTENTION

Delayed C-R-D equation can be formally obtained if weconsider the integro-differential equation

−∞exp[− t− t′

α] [κ (un ux)x − uux + f(u)] (t′, x)

instead ofut = κ (un ux)x − uux + f(u).

Integrating the integro-differential equation w.r.t. temporalvariable, w obtain the target equation

αut t + ut = κ (un ux)x − uux + f(u).

−∞exp[− t− t′

α] [κ (un ux)x − uux + f(u)] (t′, x)

−∞exp[− t− t′

α] [κ (un ux)x − uux + f(u)] (t′, x)

−∞exp[− t− t′

α] [κ (un ux)x − uux + f(u)] (t′, x)

Nonlinear transport phenomena: models, method of solving and unusual features (3)

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