Lecture II Search Method for Lax Pairs of Nonlinear Partial Differential Equations

Transcription

1 Lecture II Search Method for Lax Pairs of Nonlinear Partial Differential Equations Usama Al Khawaja, Department of Physics, UAE University, 24 Jan First International Winter School on Quantum Gases Algiers, January 21-31, 2012

2 Outline I Introduction Overview Lax pair definition Lax pair example Lax pairs and exact solutions II Search method III Applications IV Integrability

3 Introduction Overview Main goal: Finding new exact solutions of nonlinear partial differential equations (PDEs) using the Darboux transformation method. This requires knowing the Lax pair associated with the PDE. Typically, Lax pairs are constructed either by trial and error or by setting a Lax pair and then finding out what PDE it corresponds to. Here, we present a systematic method to derive the Lax pair of a given PDE. The method indicates if the Lax pair does not exist. This is regarded as a new sense of integrability of the PDE. Main focus in this lecture will be on the class of one-dimensional nonlinear Schrödinger equations known as the Gross-Pitaevskii equation (GPE). This equation describes the dynamics of light pulses (optical solitons) in optical fibers and cold atomic gases (including matter-wave solitons).

4 Introduction Definition of Lax pair The nonlinear partial differential equation Consider a nonlinear partial differential equation of the form F [ ψ(x, t), p ψ(x, t) t p, q ψ(x, t) x q ], W (x, t), cc = 0, (1) where p and q are arbitrary positive integers and W (x, t) is an arbitrary function. Lax pair definition The Lax pair is a pair of matrices, U[ψ(x, t), p ψ(x, t)/ t p, q ψ(x, t)/ x q, W (x, t), cc], (2) and V[ψ(x, t), p ψ(x, t)/ t p, q ψ(x, t)/ x q, W (x, t), cc], (3) that defines a system of linear equations for an auxiliary field Ψ = ψ 1(x, t) ψ 2 (x, t) ϕ 1 (x, t) ϕ 2 (x, t) (4) as follows Ψ t = UΨ, Ψ x = VΨ, (5) The consistency condition Ψ xt = Ψ tx can be easily shown to be equivalent to U t V x + [U, V] = 0. (6)

5 Linking the Lax pair to the PDE U t V x + [U, V] = 0 F F 0 = 0, (7)

6 Introduction Lax pair example The PDE: i ψ(x, t) t + 2 ψ(x, t) x ψ(x, t) 2 ψ(x, t) = 0. (8) The Lax pair: V = U = c ψ ψ c 2 i c2 + i ψ 2 2 i cψ + i ψ x 2 i c ψ + i ψx 2 i c 2 i ψ 2, (9), (10) where c is an arbitrary constant. The link: = U t V x + [U, V] 0 i ψ t + ψ xx + 2 ψ 2 ψ i ψt + ψxx + 2 ψ 2 ψ 0 = 0, (11)

7 Introduction Lax pair and exact solutions The Darboux transformation method Generalized Lax pair Ψ x = U 0 Ψ + U 1 Ψ Λ, (12) Ψ t = V 0 Ψ + V 1 Ψ Λ + V 2 Ψ Λ 2, (13) Λ = λ 1 0, (14) 0 λ 2 where λ 1,2 are constants (spectral parameters). Consistency condition Ψ xt = Ψ tx leads to U 0t V 0x + [U 0, V 0 ] = 0, (15) U 1t V 1x + [U 0, V 1 ] + [U 1, V 0 ] = 0, (16) V 2x + [V 2, U 0 ] + [V 1, U 1 ] = 0, (17) [U 1, V 2 ] = 0, (18) Link with PDE U 0t V 0x + [U 0, V 0 ] = 0 F F 0 = 0. (19)

8 Introduction Lax pair and exact solutions The Darboux transformation method-cont. Darboux transformation 1. Start with a seed solution of PDE: ψ 0 (x, t) (can be the trivial solution). 2. Get the seed solution of the linear system: Ψ 0 (x, t). 3. Construct the Darboux transformation: where Ψ[1] is the transformed field and Ψ[1] = Ψ Λ σψ, (20) σ = Ψ 0 Λ Ψ 0 1. (21) Covariance Ψ[1] x = U 0 [1] Ψ[1] + U 1 [1] Ψ[1] Λ, (22) Ψ[1] t = V 0 [1] Ψ[1] + V 1 [1] Ψ[1] Λ + V 2 [1] Ψ[1] Λ 2. (23) U 0 [1] t V 0 [1] x + [U 0 [1], V 0 [1]] = 0 F = 0. F (24) 0 This requires U 0 [1] = σ U 0 σ 1 + σ x σ 1, (25) in addition to other relations for U 1, V 0, V 1, V 2. This means that Q[1] is a new solution of the same differential equation that Q 0 is a solution for,

9 Introduction Lax pair and exact solutions Example ψ(x, t) i t = 2 ψ(x, t) x λ ( λ γ(t) 2 γ(t) ) x 2 ψ(x, t) 2ae λγ(t) ψ(x, t) 2 ψ(x, t), (26) Consider the following linear system: Ψ x = ζj Ψ Λ + U Ψ, (27) Ψ t = iζ 2 J Ψ Λ Λ + ζ (iu + x γ(t) J) Ψ Λ + V Ψ, (28) where, Ψ(x, t) = ψ 1(x, t) ψ 2 (x, t) ϕ 1 (x, t) ϕ 2 (x, t), J = , Λ = λ λ 2, U = 0 aq(x, t) aq (x, t) 0, V = ia Q(x, t) 2 /2 aλx γq(x, t) + i aqx (x, t)/2 aλx γq (x, t) + i aq x(x, t)/2 ia Q(x, t) 2 /2,

10 Introduction Lax pair and exact solutions Example-cont. ζ(t) = exp (γ(t)), and λ 1 and λ 2 are arbitrary constants. For convenience, we presented the matrices in terms of the function Q(x, t) which is related to the wave function as follows Q(x, t) = ψ(x, t)e (γ(t)+i γ(t)x2 )/2. The linear system of 8 equations reduces to an equivalent system of 4 equations with nontrivial solutions by making the following substitutions: λ 1 = λ 2, ψ 2 = ϕ 1, and ϕ 2 = ψ1. Using the trivial solution, ψ 0(x, t) = 0, as a seed, the linear system will have the solution ψ 1 (x, t) = c 1 e 2iλ2 1 ϕ 2 (x, t) = c 2 e 2iλ2 1 where c 1 and c 2 are real arbitrary constants of integration. e 2γ(t) dt+e γ(t) λ 1 x, (29) e 2γ(t) dt e γ(t) λ 1 x, (30) σ = Ψ 0 Λ Ψ 0 1 U[1] = σ U σ 1 + σ x σ 1, ψ(x, t) = ψ 0 (x, t) + 2 a (λ 1 + λ 1)e i γ(t)x2 /4+γ(t)/2 ϕ 1 ψ 1 ϕ ψ 1 2.

11 Lax pair search method Expand the Lax pair in powers of ψ, p ψ/ t p, and q ψ/ x q, as follows N U = U 11i U 12i, (31) with and i=1 U 21i N V = V 11i i=1 V 21i U 22i V 12i V 22i, (32) U 11i = f 1i ψ i + ( p ψ/ t p ) i f 2i + ( q ψ/ x q ) i f 3i, (33) U 12i = f 4i ψ i + ( p ψ/ t p ) i f 5i + ( q ψ/ x q ) i f 6i, (34) U 21i = f 7i ψ i + ( p ψ/ t p ) i f 8i + ( q ψ/ x q ) i f 9i, (35) U 22i = f 10i ψ i + ( p ψ/ t p ) i f 11i + ( q ψ/ x q ) i f 12i. (36) V 11i = g 1i ψ i + ( p ψ/ t p ) i f 2i + ( q ψ/ x q ) i g 3i, (37) V 12i = g 4i ψ i + ( p ψ/ t p ) i f 5i + ( q ψ/ x q ) i g 6i, (38) V 21i = g 7i ψ i + ( p ψ/ t p ) i g 8i + ( q ψ/ x q ) i g 9i, (39) V 22i = g 10i ψ i + ( p ψ/ t p ) i g 11i + ( q ψ/ x q ) i g 12i. (40) where f ji, g ji, j = 1 12, i = 1, 2,..., are unknown functions of x and t. The upper limit of the summations, N, is to be set by inspecting the specific differential equation being solved.

12 Substitute these expansions in the consistency condition U t V x + [U, V] = 0 F F 0. (41) Equating the coefficients of terms of equal powers of ψ, p ψ/ t p, and q ψ/ x q, we obtain a finite number of linear differential equations for the unknown coefficients. Solving these equations determines the Lax pair. Example As an explicit example, we find the Lax pair of the GP equation ψ(x, t) i t + 2 ψ(x, t) x λ(x)2 ψ(x, t) + 2a(t) ψ(x, t) 2 ψ(x, t) = 0, (42) where a(t) = a 0 e γ(t)t and λ(x) and γ(t) are assumed to be independent general functions of x and t, respectively. The method is summarized as follows: 1. Expand U and V in powers of ψ: V = U = f 1 + f 2 ψ f 5 + f 6 ψ f 3 + f 4 ψ f 7 + f 8 ψ, (43) g 1 + g 2 ψ + g 3 ψ x + g 4 ψψ g 5 + g 6 ψ + g 7 ψ x + g 8 ψψ g 9 + g 10 ψ + g 11 ψx + g 12 ψψ g 13 + g 14 ψ + g 15 ψx + g 16 ψψ where f 1 8 (x, t) and g 1 16 (x, t) are unknown function coefficients. (44),

13 2. Require the consistency condition U t V x + [U, V ] = 0 to give rise to the GP equation (Eq.(42)). This results in 24 equations for the 24 coefficients: f 2 = f 3 = f 5 = f 8 = g 2 = g 3 = g 5 = g 8 = g 9 = g 12 = g 14 = g 15 = 0, f 4 = f 6 = a 0, g 7 = g 11 = a 0 i, g 4 = g 16 = a 0 i. Using these constant values, the equations for the rest of the coefficients simplify to g 10x + (f 7 f 1 )g 10 + a 0 (g 13 g 1 ) g 10 = g 6, (45) f 1t g 1x = 0, (46) f 7t g 13x = 0, (47) a 0 [ iλ 2 /4 (γ 2if 2 x + γ t t + 2f xx )/2 ] = 0, (48) g 10x (f 7 f 1 )g 10 a 0 (g 13 g 1 ) + a 0 [ iλ 2 /4 + (γ + 2if 2 x + γ t t + 2f xx )/2 ] = 0, (49) g 10 + i a 0 (f 1 f 7 ) + 2 a 0 f x = 0. (50) 3. Solve these equations to obtain the following Lax pair: V = U = f 1 a0 Q a 0 Q α 1 f 1 g 1 + ia 0 Q 2 g 10 Q + i a 0 Q x g 10 Q + i a 0 Q x α 1 g 1 ia 0 Q 2, (51), (52) where ψ(x, t) = e iλ(x) γ(t)t/2 Q(x, t). (53)

14 iη 2 f 1 (x, t) =, (54) 4λ 2 (α 1 1)η 1 g 1 (x, t) = i [ ] (c 2 2ζ 4 + c 2 3)η 4 2c 2 c 3 ζ 2 η 5 16(α 1 1)λ 2, (55) 2 η2 1 a0 η 6 g 10 (x, t) =, (56) 4λ 2 η 1 where η 1 = c 3 + c 2 ζ 2, η 2 = 4λ 2 f x η 1 + (λ 1 + 2λ 2 2x)η 3, η 3 = c 3 c 2 ζ 2, η 4 = λ 2 1 4λ 0 λ 2 2, η 5 = λ λ 2 2(4λ 0 + 8λ 1 x + 8λ 2 2x 2 ), η 6 = 4λ 2 f x η 1 + (λ 1 + 2λ 2 2x)η 2, and ζ = exp (λ 2 t). Calculating the consistency condition using this Lax pair, we obtain the Gross-Pitaevskii equation ψ(x, t) i t + 2 ψ(x, t) x (λ 0 + λ 1 x + λ 2 2x 2 )ψ(x, t) 2a 0 + c 2 e λ 2t + c 3 e λ 2t ψ(x, t) 2 ψ(x, t) = 0, (57) where c 2 and c 3 are arbitrary constants. Reference: Lax pairs of time-dependent GrossPitaevskii equation U. Al Khawaja, J. Phys. A: Math. Gen. 39 (2006)

15 Applications Linear Inhomogeneity Using the previously-described method, we found the Lax representation for the equation ψ(x, t) i t = ] [ 2 x 2 + x p2 ψ(x, t) 2 ψ(x, t), (58) namely Ψ x = UΨ + JΨΛ, (59) iψ t = VΨ + 2(ζJ + U)ΨΛ + 2JΨΛ 2, (60) where, Ψ(x, t) = ψ 1(x, t) ψ 2 (x, t) ϕ 1 (x, t) ϕ 2 (x, t), J = , (61) Λ = λ λ 2, U = ζ p q(x, t)/ 2 p r(x, t)/ 2 ζ, (62) V = (ζ 2 x/2)j + 2ζU J(U 2 U x ) (63) ζ(t) = it/2, and λ 1 and λ 2 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 ψ 0 (x, t) = A exp (iϕ 0 ) we obtain for the solution of

16 the repulsive interactions case (p = ±i) ψ(x, t) = e iϕ 0 [ A ± i 8λ 1r 2u+ r cosh θ 2iu + i sinh θ + ( u ) cos β + i( u + 2 ] 1) sin β ( u + 2 1) sinh θ + 2u + i sin β and for the attractive interactions case (p = 1), the solution is ψ(x, t) = e iϕ 0 [ A 8λ 1r 2u+ r cosh θ 2iu + i sinh θ + ( u ) cos β + i( u + 2 1) sin β ( u ) cosh θ + 2u + r cos β where ϕ 0 = t(p 2 A 2 (t 2 /3 + x)), θ = 2 [ r (t 2 + x) + 2( r λ 1i i λ 1r )t ] δ r, β = 2 [ i (t 2 + x) + 2( i λ 1i + r λ 1r )t ] + δ i, u ± = 8 p A/b ±, b ± = 4λ 1 ±, = 2λ 12 p 2 A 2, and δ is an arbitrary constant. (64) ] (65) Reference: Exact solitonic solutions of the Gross-Pitaevskii equation with a linear potential U. Al Khawaja, Phys. Rev. E 75, (2007).

17 Applications Periodic potential The following Gross-Pitaevskii equation f(x) 2 [ ie t x a 2ω 2 (a 2 cos (ωx) sin (ωx)) ke 3f(x)/2 Q(x, t) 2 ] Q(x, t) = 0, (66) where k = ±1 allows for attractive interatomic interactions (k = 1) and repulsive interatomic interactions (k = 1). Lax pair: U 0 = 3 4 a 2ω cos (ωx) i kq i 3 2 ke 4 f(x) Q 0 V 0 = i 2 kef(x)/2 Q 2 e f(x) q 1 e f(x)/2 q 2 i 2 kef(x)/2 Q 2, (67), (68) where q 1 = 1 8 a 2ω k cos(ωx)q kqx and q 2 = 1 4 a 2ω k cos(ωx)q + kq x. Reference: Integrability of a general GrossPitaevskii equation and exact solitonic solutions of a BoseEinstein condensate in a periodic potential U. Al Khawaja, Physics Letters A 373 (2009)

18 Applications Time-dependent potential and nonlinearity The GPE: ψ(x, t) i t Lax pair: = [ 2 x λ ( λ γ(t) 2 γ(t) ) ] x 2 2ae λγ(t) ψ(x, t) 2 ψ(x, t), (69) Ψ x = ζ(t)jψλ + UΨ, (70) where, Ψ t = 2iζ(t) 2 JΨΛ 2 + ζ(t)(2iu + λx γ(t)j)ψλ + VΨ, (71) J = 1 0, Λ = λ λ 2 0 aq(x, t) U =, aq (x, t) 0 V =, ia Q(x, t) 2 aλx γ(t)q(x, t) + i aqx (x, t) aλx γ(t)q (x, t) + i aq x(x, t) ia Q(x, t) 2 ζ(t) = exp (λγ(t)), and λ 1 and λ 2 are arbitrary constants. The function Q(x, t) is related to the wave function through Q(x, t) = ψ(x, t) ζ(t)e iλ γ(t)x2 /4., Reference: Soliton localization in BoseEinstein condensates with time-dependent harmonic potential and scattering length U. Al Khawaja, J. Phys. A: Math. Theor. 42 (2009) (19pp)

19 Applications The GPE: Cubic-Quintic nonlinearity ψ(x, t) i t = 2 x 2 ψ(x, t) + ψ(x, t) 2 ψ(x, t) + g 2 ψ(x, t) 4 ψ(x, t), (72) Lax pair: Lax pair DOES NOT EXIST for this PDE

20 Integrability To investigate the integrability of a very general GPE: ] 2 [i + f(x, t) t x 2 + g(x, t) Ψ(x, t) 2 + v(x, t) + i γ(x, t) Ψ(x, t) = 0, (73) Painlevé integrability conditions [1]: f(x, t) = f(t), g(x, t) = g(t), γ(x, t) = γ(t), (74) v(x, t) = v 0 (t) + v 1 (t) x + v 2 (t) x 2, (75) where v 0 (t) and v 1 (t) are arbitrary and v 2 (t) is given by 4f 3 g 2 v 2 + fg( fġ + f g) + g 2 ( f 2 f f) 2f 2 ġ 2 = 0. (76) [1] He et al., Phys. Rev. E 79, (2009). Similarity transformation method: The integrability condition (76) can be obtained by requiring the transformation ( Ψ(x, t) = exp ) β(x, t) + i θ(x, t) Q(X(x, t)), (77) where β(x, t), θ(x, t), and X(x, t) being real functions, to transform Eq. (73) into the following time-independent homogeneous equation ( ) p(x, t) ϵ Q XX (X) + δ Q(X) 2 Q(X) = 0, (78)

21 v(x, t) = ċ 5 + c 6 2 δ 2 fċ 2 2 4ϵ 2 g 2 ( c 6 δ (fg c 2 + ċ 2 g )) f 2fġ 2ϵfg 2 x + fg ( fġ + f g Lax pair integrability conditions: ) + g 2 ( f f f 2 ) + 2f 2 ġ 2 4f 3 g 2 x 2, (79) and f(x, t) = c 1(t) g(x, t) 2, (80) γ(x, t) = g t(x, t) g(x, t) 1 ċ 2 (t) 2 c 2 (t), (81) fg 3 (f t (g t 2gγ) f tt g) + f 2 t g 4 + 2f 3 g 3 (gv xx g x v x ) + f 2 g 2 ( g (4g t γ + g tt ) 2g 2 t 2g 2 ( γ t + 2γ 2)) ( ( + f 4 36gx 4 48gg xx gx g 2 g xxx g x + g 2 6gxx 2 gg (4))) = 0. (82) Special cases: Special case I: Constant and linear external potential With the choices: g(x, t) = 1 and c 1 (t) = c 2 (t) = 1, Eq. (73) takes the form ( ) Ψ xx + Ψ 2 Ψ + c 3 (t) + c 4 (t) x Ψ + i Ψ t = 0, (83) where c 3 (t) and c 4 (t) are arbitrary real functions. Special case II: Harmonic potential and gain/loss term

22 For g(x, t) = 1, c 1 (t) = 1, and c 2 (t) = e α t, where α is a real constant, Eq. (73) takes the form ( α Ψ xx + Ψ 2 2 Ψ + 4 x2 α ) 2 i + c 3(t) + c 4 (t) x Ψ + i Ψ t = 0. (84) Special case III: Harmonic potential and time-dependent nonlinearity For g(x, t) = e α t, c 1 (t) = c 2 (t) = e 2α t, Eq. (73) takes the form ( ) α Ψ xx + e α t Ψ 2 2 Ψ + 4 x2 + c 3 (t) + c 4 (t) x Ψ + i Ψ t = 0. (85) Special case IV: x n -dependent coefficients For g(x, t) = x n, where n is an integer, and c 1 (t) = c 2 (t) = 1, Eq. (73) takes the form + 2 [i + x 2n t x 2 + xn Ψ 2 ( 1 4 n(n + 2)x 2(n+1) + c )] 3(t) n + 1 xn+1 + c 4 (t) x Ψ = 0. (86) This is our first counter example to the conclusion of the Painlevé test; an integrable NLS equation with space-dependent coefficients.

23 Special case V: Time-dependent coefficients For the special case of g(x, t) = g(t), γ(x, t) = 0, the above Painlevé and similarity transformation conditions are retrieved. Reference:A comparative analysis of Painlevé, Lax pair, and similarity transformation methods in obtaining the integrability conditions of nonlinear Schrödinger equations U. Al Khawaja, JOURNAL OF MATHEMATICAL PHYSICS 51, (2010).

24 Conclusions The Lax pair of nonlinear partial differential equations can be generated (if it exists) using the systematic method described in this lecture. Lax pairs can be used to derive new solutions to partial differential equations through the Darboux transformation method. The existence of Lax pair can be used to judge on the integrability of differential equations. An equation can be integrable in one sense and nonintegrable in another.