Darboux Transformation, Lax Pairs,

Exact Solutions of Nonlinear Schrodinger

Equations,

and

Soliton Molecules

Usama Al Khawaja,

Physics Department,

United Arab Emirates University.

23 Jan. 2012

First International Winter School on Quantum Gases

Algiers, January 21-31, 2012

General Outline

Lecture I: Darboux Transformation (DT) and Lax Pair

• Linear and Nonlinear Partial Differential Equations

Lecture II: Lax Pair Search Method and Integrability

Lecture III: Single soliton solutions

• Vibrating harmonic trap

• Optical Lattice

• Rogue waves

Lecture IV: Two solitons solutions (Soliton Molecules)

• Instructive Form of the Exact solution

• Formation, Stability, and Dynamics of Soliton Molecules

• Methods of Stabilizing Soliton Molecules

Lecture I

Darboux Transformation (DT)

and Lax Pair

Outline

I Darboux Transformation (DT)

II DT for Linear Ordinary Differential Equations

• Schrodinger Equation with a Harmonic Potential

III DT for Nonlinear Partial Differential Equations

• The Korteweg-de-Vries (KdV) Equation

• The Zakharov-Shabat Method

• The Nonlinear Schrodinger (NS) Equation

IV Search for the Lax Pairs

V Exact Solitonic Solutions of the Gross-Pitaevskii Equation

Darboux Transformation

Consider a Sturm-Liouville equation

−Ψxx(x) + u(x)Ψ = λΨ(x) , (1)

with known fixed eigensolution Ψ1 and eigenvalue λ1.

The Darboux transformation on Ψ is defined as:

Ψ[1] =

(d

dx− Ψ1x

Ψ1

)Ψ . (2)

Theorem: The function Ψ[1] satisfies

−Ψ[1]xx + u[1]Ψ[1] = λΨ[1] , (3)

with

u[1] = u− 2

(Ψ1x

Ψ1

)x

. (4)

Eq.(1) is said to be covariant under the DT.

DT for Linear Ordinary Differential Equations

• Schrodinger Equation With a Harmonic Potential

Consider the equation

−Ψxx + x2Ψ = λΨ , (5)

with seed solution Ψ1 = exp (x2/2) and λ1 = −1.

Applying DT on Ψ and u = x2, we get

Ψ[1] =

(d

dx− x

)Ψ, u[1] = x2 − 2, (6)

which satisfy the equation

−Ψ[1]xx + x2Ψ[1] = (λ+ 2)Ψ . (7)

For example: Ψ = exp (−x2/2) with λ = 1, gives

Ψ[1] = −2x exp (−x2/2). (8)

with eigenvalue λ+ 2 = 3.

Applying the DT on Ψ[1], we get

Ψ[2] = (4x2 − 2) exp (−x2/2). (9)

with eigenvalue λ+ 2 = 5.

Applying the DT on Ψ[1] n-times, we get

Ψ[n] = −Hn exp (−x2/2). (10)

with eigenvalue λ+ 2 = 2n+ 1, where

Hn = (−1)n exp (x2/2) (d/dx− x)nexp (−x2/2) (11)

is the Hermite polynomial of order n.

DT for Nonlinear Partial Differential Equations

• The Korteweg-de-Vries (KdV) Equation

The KdV equation reads

ut = 6uux − uxxx . (12)

Lax introduced the pair of operators (denoted afterwards as the Lax

pair)

L = −∂2x + u, A = −4∂3x + 6u∂x + 3ux , (13)

such that the KdV equation can be written as

∂tL = [L,A] . (14)

The last equation can be considered as the consistency condition of the

following linear system for an auxiliary field Ψ

LΨ = λΨ

Ψt = AΨ, (15)

where the consistency condition is

(∂tL− [L,A])Ψ = 0 , (16)

is equivalent to the KdV equation.

The linear system (15) is covariant under the DT. Thus

L[1]Ψ[1] = λΨ[1]

Ψ[1]t = A[1]Ψ[1], (17)

and the consistency condition becomes

(∂tL[1]− [L[1], A[1]])Ψ[1] = 0 , (18)

which is equivalent to

u[1]t = 6u[1]u[1]x − u[1]xxx . (19)

This means that u[1] is a new exact solution of the KdV equation

which can be obtained from the known exact solution u via the DT.

DT for Nonlinear Partial Differential Equations

• The Zakharov-Shabat Method

Consider the linear system

Ψt = IΨΛ+ UΨ, Ψx = JΨΛ+ UΨ, (20)

where Λ, I, and J are constant matrices and [I, J ] = 0. The

consistency condition is

Ut − Vx = [I, J ], [I, U ] = [J, V ]. (21)

This linear system is found to be covariant under the following version

of the DT

Ψ[1] = ΨΛ− σΨ, σ = Ψ1Λ1Ψ−11 . (22)

Applying this DT on the linear system system, we get

U [1] = U + [J,Ψ1Λ1Ψ−11 ]. (23)

DT for Nonlinear Partial Differential Equations

• The Nonlinear Schrodinger (NS) Equation

Consider the linear system

Ψx = JΨΛ+ UΨ,

Ψt = 2JΨΛ2 + 2UΨΛ+ (JU2 − JUx)Ψ, (24)

where

U =

0 iq(x, t)

ir(x, t) 0

J =

i 0

0 −i

Λ =

λ1 0

0 Λ2

Ψ =

ψ1 ψ2

ϕ1 ϕ2

. (25)

The consistency condition results in the equation

iqt = qxx + 2q2r, irt = −rxx − 2r2q . (26)

Under the condition q = r∗, the last equation becomes

irt + rxx + 2|r|2r , (27)

which is a nonlinear Schrodinger equation known also as the

Gross-Pitaevskii equation (GP).

Applying the DT on the above linear system, we get

U [1] = U + [J,Ψ1ΛΨ−11 ] , (28)

which leads to

r[1] = r−2(λ21−λ11)ϕ1ϕ2(ψ1ϕ2−ϕ1ψ2)−1, Λ1 = diag[λ11, λ21] . (29)

Search for the Lax Pairs

• Liang et al. [Phys. Rev. Lett. 95, 050402 (2005).] found the Lax pair

for the following GP equation

i∂ψ(x, t)

∂t+∂2ψ(x, t)

∂x2+

1

4λ2x2ψ(x, t)+2a0e

λt|ψ(x, t)|2ψ(x, t) = 0. (30)

• We devised a method of searching for the Lax pair corresponding to a

general Schrodinger equation [U. Al Khawaja, J. Phys. A: Math. Gen. 39

(2006) 9679-9691.]

i∂ψ(x, t)

∂t+ K1(x, t)

∂2ψ(x, t)

∂x2+K2(x, t)ψ(x, t)

+ K3(x, t)|ψ(x, t)|2ψ(x, t) = 0, (31)

where K1(x, t), K2(x, t), and K3(x, t) are, in principle, arbitrary

functions corresponding to effective mass, external potential including

loss or gain, and nonlinearity, respectively.

• It turned out that Lax pairs exist only if certain relations between

the coefficients K1,K2,K3 are satisfied [V.N. Serkin et al.,IEEE

8,No.3(2002).].

• As an explicit example, we found the Lax pair of the GP equation

i∂ψ(x, t)

∂t+∂2ψ(x, t)

∂x2+

1

4λ(x)2ψ(x, t) + 2a(t)|ψ(x, t)|2ψ(x, t) = 0, (32)

where a(t) = a0eγ(t)t and λ(x) and γ(t) are assumed to be independent

general functions of x and t, respectively. For the special case of

λ(x) = λx and γ(t) = λ, the expulsive potential case, Eq. (30), is

retrieved.

• The method is summarized as follows:

1. We write ψ in the form

ψ(x, t) = e−if(x)−γ(t)t/2Q(x, t). (33)

2. We expand U and V in powers of Q:

U =

f1 + f2Q f3 + f4Q

f5 + f6Q∗ f7 + f8Q

∗

, (34)

V =

g1 + g2Q+ g3Qx + g4QQ∗ g5 + g6Q+ g7Qx + g8QQ

∗

g9 + g10Q∗ + g11Q

∗x + g12QQ

∗ g13 + g14Q∗ + g15Q

∗x + g16QQ

∗

,

(35)

where f1−8(x, t) and g1−16(x, t) are unknown function coefficients.

3. We require the consistency condition Ut − Vx + [U, V ] = 0 to give

rise to the GP equation (Eq.(32)). This results in 24 equations for the

24 coefficients:

f2 = f3 = f5 = f8 = g2 = g3 = g5 = g8 = g9 = g12 = g14 = g15 = 0,

f4 = −f6 =√a0, g7 = g11 =

√a0 i, g4 = −g16 = a0 i. Using these

constant values, the equations for the rest of the coefficients simplify to

g10 = −g6, (36)

f1t − g1x = 0, (37)

f7t − g13x = 0, (38)

g10x + (f7 − f1)g10 +√a0 (g13 − g1)

−√a0

[−iλ2/4− (γ − 2if2x + γtt+ 2fxx)/2

]= 0, (39)

g10x − (f7 − f1)g10 −√a0 (g13 − g1)

+√a0

[−iλ2/4 + (γ + 2if2x + γtt+ 2fxx)/2

]= 0, (40)

g10 + i√a0(f1 − f7) + 2

√a0 fx = 0. (41)

4. We solve these equations to obtain the following Lax pair:

U =

f1√a0Q

−√a0Q

∗ α1f1

, (42)

V =

g1 + ia0|Q|2 −g10Q+ i√a0Qx

g10Q∗ + i

√a0Q

∗x α1g1 − ia0|Q|2

, (43)

where

f1(x, t) =iη2

4λ2(α1 − 1)η1, (44)

g1(x, t) =i[(c22ζ

4 + c23)η4 − 2c2c3ζ2η5

]16(α1 − 1)λ22η

21

, (45)

g10(x, t) = −√a0η6

4λ2η1, (46)

where η1 = c3 + c2ζ2 , η2 = −4λ2fxη1 + (λ1 + 2λ22x)η3, η3 = c3 − c2ζ

2 ,

η4 = λ21 − 4λ0λ22, η5 = λ21 + λ22(4λ0 + 8λ1x+ 8λ22x

2),

η6 = 4λ2fxη1 + (λ1 + 2λ22x)η2, and ζ = exp (λ2t).

Calculating the consistency condition using this Lax pair, we obtain

the Gross-Pitaevskii equation

i∂ψ(x, t)

∂t+

∂2ψ(x, t)

∂x2+

1

4(λ0 + λ1x+ λ22x

2)ψ(x, t)

+2a0

c2eλ2t + c3e−λ2t|ψ(x, t)|2ψ(x, t) = 0, (47)

where λ0,1,2, c2 and c3 are arbitrary coefficients.

5. For the special case c2 = 1 and c3 = 0, the above Lax pair and GP

equation reduce to those of Liang et al..

6. A seed solution for the last GP equation can be derived for the

general case:

ψ(x, t) = A√sech [λ2(2c3 + t)]

× exp{c4 + i

[c1(t) + c2(t)x+ λ2 tanh [λ2(2c3 + t)]x2/4

]},

(48)

where c1(t) and c2(t) are given by

c1(t) = c6 +1

16λ32

{λ2(c7 − t)(λ21 − 4λ0λ

22)

+ 8c5λ1λ2(sech η1 − sech η2)

+ (λ21 − 16c25λ22)(tanh η1 − tanh η2)

+ 2A2e2c4g0

∫ t

c7

dt′γ(t′)sech [λ2(2c3 + t′)]

}, (49)

c2(t) = c5 sech η1 +λ14λ2

tanh η1, (50)

where η1 = λ2(2c3 + t), η2 = λ2(2c3 + c7), and c3−7 are constants of

integration.

Exact Solitonic Solutions of the Gross-Pitaevskii

Equation

• Linear Inhomogeneity

Using the previously-described method, we found the Lax

representation for the equation

i∂ψ(x, t)

∂t=

[− ∂2

∂x2+ x− p2|ψ(x, t)|2

]ψ(x, t), (51)

namely

Ψx = JΨΛ+UΨ, (52)

iΨt = WΨ+ 2(ζJ+U)ΨΛ+ 2JΨΛ2, (53)

where,

Ψ(x, t) =

ψ1(x, t) ψ2(x, t)

ϕ1(x, t) ϕ2(x, t)

, J =

1 0

0 −1

, (54)

Λ =

λ1 0

0 λ2

, U =

ζ p q(x, t)/√2

−p r(x, t)/√2 −ζ

, (55)

W = (ζ2 − x/2)J+ 2ζU− J(U2 −Ux) (56)

ζ(t) = it/2, and λ1 and λ2 are arbitrary constants and

q(x, t) = r∗(x, t) = ψ(x, t), and p = 1 for attractive interactions and

p = ±i for repulsive interactions.

Applying the DT, using the seed ψ0(x, t) = A exp (iϕ0) we obtain for

the solution of the repulsive interactions case (p = ±i)

ψ(x, t) = eiϕ0 [ A± i√8λ1r

× 2u+r cosh θ − 2iu+i sinh θ + (|u+|2 + 1) cosβ + i(|u+|2 − 1) sinβ

(|u+|2 − 1) sinh θ + 2u+i sinβ

](57)

and for the attractive interactions case (p = 1), the solution is

ψ(x, t) = eiϕ0 [ A−√8λ1r

× 2u+r cosh θ − 2iu+i sinh θ + (|u+|2 + 1) cosβ + i(|u+|2 − 1) sinβ

(|u+|2 + 1) cosh θ + 2u+r cosβ

](58)

where ϕ0 = t(p2A2 − (t2/3 + x)),

θ =√2[∆r(t

2 + x) + 2(∆rλ1i −∆iλ1r)t]− δr,

β = −√2[∆i(t

2 + x) + 2(∆iλ1i +∆rλ1r)t]+ δi, u

± =√8 pA/b±,

b± = 4λ∗1 ±∆, ∆ =√2λ∗1

2 − p2A2, and δ is an arbitrary constant.

Exact Solitonic Solutions of the Gross-Pitaevskii

Equation

• Linear Inhomogeneity: Properties of the Solution

Depending on the arbitrary constants, we have three types of solutions:

i) oscillatory solutions.

ii) nonoscillatory solutions, i.e., single-soliton solution.

iii) oscillatory with a localized envelope.

-7.5 -5 -2.5 0 2.5 5 7.5 10

x�∆

12345678

Ρ�Ρ0

HeL

-7.5 -5 -2.5 0 2.5 5 7.5 10

0.51

1.52

2.53

3.54

Hf L

-7.5 -5 -2.5 0 2.5 5 7.5 10

1

1.2

1.4

1.6

1.8

2HcL

-7.5 -5 -2.5 0 2.5 5 7.5 10

5

10

15

20

25

30HdL

-7.5 -5 -2.5 0 2.5 5 7.5 10

1

2

3

4

5HaL

-7.5 -5 -2.5 0 2.5 5 7.5 10

2

4

6

8

10HbL

Figure 1: Density ρ(x) = |ψ(x)|2 at time t = 0 for the case of attractive

interactions. The arbitrary constants chosen to generate these plots are: δr =

δi = 0 and A = 1 for all plots. In (a) λ1i = 0, λ1r = 0.29, in (b): λ1i = 0,

λ1r = −0.6, in (c): λ1i = 0, λ1r = 0.8, in (d): λ1i = 0, λ1r = −1.5, in (e):

λ1i = 2, λ1r = −0.6, and in (f): λ1i = 2, λ1r = 0.29.

Exact Solitonic Solutions of the Gross-Pitaevskii

Equation

• Linear Inhomogeneity: Properties of the Solution

Time evolution of solitons:

i) for single solitons: x = −t2 − 2(λ1i − λ1r∆i/∆r)t+ δr/√2

ii) for multiple solitons: x = −t2 − 2(λ1i + λ1r∆r/∆i)t+ δi/√2.

→ Parabolic with trajectory acceleration of -1.

In real units, this acceleration equals −F/m, where F is the force

constant of the linear potential and m is the mass of an atom.

-5

0

5

10

x�∆

0

2

4

t�Τ

0

1

2

3

Ρ �Ρ0

-5

0

5

10

x�∆

0

1

2

3

-7.5

-5

-2.5

0

2.5

x�∆

0

1

2

t�Τ

0

10

20

Ρ �Ρ0

-7.5

-5

-2.5

0

2.5

x�∆

Figure 2: Surface plots of the density ρ(x, t) = |ψ(x, t)|2 versus x and t for

the attractive interactions. The upper plot corresponds to Fig. 1(d) while the

lower figure corresponds to Fig 1(a).

Exact Solitonic Solutions of the Gross-Pitaevskii

Equation

• Linear Inhomogeneity: Properties of the Solution

The peak of the soliton is maximum at times:

t = nπ∆r/√8|∆|2λ1r, where n is an integer.

0 2 4 6 8 10

t�Τ

12.5

15

17.5

20

22.5

25

27.5

Ρ�Ρ0

HbL

0 2 4 6 8 10

2

4

6

8

10

HaL

Figure 3: The soliton density along its trajectory for attractive interactions.

The upper figure corresponds to Fig. 1(c) and the lower figure corresponds to

Fig. 1(d).

Exact Solitonic Solutions of the Gross-Pitaevskii

Equation

• Linear Inhomogeneity: Properties of the Solution

Phase of Solitons:

Phase difference equals π and is a constant of time.

-40 -20 0 20 40

x�∆

-2

0

2

4

6

8

10

Ρ�Ρ0

HbL

-40 -20 0 20 40

-2

0

2

4

6

8

10HaL

Figure 4: The density (solid curve) and phase (light curve) of a solitonic

solution for attractive interactions with δr = δi = 0, A = 1, λ1i = 0.03, and

λ1r = 0.6. The upper plot is for t = 0 and the lower plot is for t = 0.85τ.

Search is still ongoing for more exact

solutions...