New Exact Solutions to NLS Equation and Coupled NLS Equations

Transcription

1 Commun. Theor. Phys. (Beijing, China 4 (2004 pp c International Academic Publishers Vol. 4, No. 2, February 5, 2004 New Exact Solutions to NLS Euation Coupled NLS Euations FU Zun-Tao, LIU Shi-Da, LIU Shi-Kuo School of Physics, Peking University, Beijing 0087, China (Received May 6, 2003 Abstract A transformation is introduced on the basis of the projective Riccati euations, it is applied as an intermediate in expansion method to solve nonlinear Schrödinger (NLS euation coupled NLS euations. Many kinds of envelope travelling wave solutions including envelope solitary wave solution are obtained, in which some are found for the first time. PACS numbers: Ge Key words: projective Riccati euations, NLS euation, envelope travelling wave solution, envelope solitary wave solution Introduction A number of problems are described in terms of suitable nonlinear models, such as nonlinear Schrödinger euations (NLS in plasma physics, KdV euation in shallow water model, 2 so on, in branches of physics, mathematics, other interdisciplinary sciences. It is an interesting topic to seek exact solutions to these nonlinear models. Here we will consider two nonlinear models, i.e. NLS euation coupled NLS euations. The NLS euation reads iu t + αu xx + u 2 u = 0, ( the coupled NLS euations read iu t + αu xx + ( u v 2 u = 0, iv t + αv xx + ( v u 2 v = 0. (2 In Ref. 3, on the base of Lamé euation Lamé functions, we obtain the perturbed solutions to the NLS euation coupled NLS euations. In this paper, we will reconsider these two nonlinear evolution systems. A transformation is obtained from the well-known projective Riccati euations, 4 6 then this transformation is taken as an intermediate to solve these two systems, where many kinds of envelope travelling wave solutions including envelope solitary wave solutions are derived, among which some are found for the first time. 2 Exact Solutions to NLS Euation In order to solve E. (, we take the following transformation u = φ(ξ e i(kx ωt, ξ = x c g t, (3 where k is the wave number, ω is the angular freuency, c g is the group velocity, φ(ξ is a real function. Substituting E. (3 into E. ( results in the following ordinary differential euation, dξ 2 + i(2αk c g dφ dξ + (ω αk2 φ + φ 3 = 0. (4 Separating the real part imaginary part yields if we suppose then euation (4 is rewritten as c g = 2αk, (5 ω αk 2 = γ, (6 dξ 2 γφ + φ3 = 0. (7 In order to solve E. (7, we introduce the following crucial ansatz, n φ(ξ = f i (ξa i f(ξ + B i g(ξ + A 0, i= A 2 n + B 2 n 0, (8 where n can be determined by balancing the highest order derivative term with the high degree nonlinear term in E. (7. And f g are solutions to the well-known projective Riccati euations, 4 6 f (ξ = pf(ξg(ξ, g (ξ = + pg 2 (ξ rf(ξ, (9 where p 0 is a real constant, r are two real constants. When p = =, euations (9 reduce to the coupled euations given in Refs. 4 5, when p 0, euations (9 reduce to the coupled euations given in Ref. 6. There is a relation between f g, g 2 = 2rf + r2 + δ f 2, (0 p The project supported by National Natural Science Foundation of China under Grant Nos Correspondence author, fuzt@pku.edu.cn

2 90 FU Zun-Tao, LIU Shi-Da, LIU Shi-Kuo Vol. 4 where δ. Applying the above expansion method, if we take the expansion order of φ as O(φ = n considering the relations (9, then O(dφ/dξ = n +, so partial balance between the highest degree nonlinear term the highest order derivative term leads to n =. Obviously, the formal solution can be written as φ = A 0 + A f(ξ + B g(ξ, A 2 + B 2 0. ( Considering the relation (0, from E. ( one can have φ 3 = ( A p A 0B 2 + 3A 0 A 2 + 6r ( + 3A 2 0B ( p B3 g + 3A 2 0A 3 p A B 2 + 6r ( p A 0B 2 f + 6A 0 A B + 2r p B3 p A B 2 3(r2 + δ A 0 B 2 f 2 + 3A 2 p B (r2 + δ B 3 f 2 g + A 3 3(r2 + δ A B 2 p p fg f 3 (2 d 2 φ dξ 2 = pa f + prb fg + 3prA f 2 2p(r2 + δ B f 2 g 2p(r2 + δ A f 3. (3 Substituting Es (, (2, (3 into E. (7 results in the following algebraic euations, ( γa 0 + A p A 0B 2 = 0, ( γa + 3 p A B 2 + 6r p A 0B 2 + 3A 2 0A αpa = 0, γb + ( p B3 + 3A 2 0B = 0, ( 2r p B3 + 6A 0 A B + αprb = 0, 6r p A B 2 3(r2 + δ p 3A 2 B (r2 + δ B 3 p A 3 3(r2 + δ A B 2 p A 0 B 2 + 3A 0 A 2 + 3αprA = 0, 2αp(r2 + δ B = 0, 2αp(r2 + δ A = 0, (4 for the arbitrariness of the argument ξ, from which the parameters can be determined. For example, for δ =, there are the following solutions: Case If A = 0, A 0 = 0, r = 0, then B 2αp2, p = γ 2α. (5 Obviously, there is the constraint α < 0. Case 2 If A = 0, A 0 = 0, r 0, then B αp2 2, p = 2γ α, r. (6 Obviously, there is the constraint α < 0, too. Case 3 If B = 0, A 0 = 0, then 2α A 2 p 2, p = γ α, r = 0. (7 Case 4 If B = 0, A 0 0, then α A 2 p 2 (r γr, A 0 2 (r 2 + 2, p = 2(r2 γ α(r (8 There is the constraint r 0 r 2.

3 No. 2 New Exact Solutions to NLS Euation Coupled NLS Euations 9 Case 5 If A 0 = 0, A 0, B 0, then A α 2 p 2 (r 2, 4 B αp2 2, p = 2γ α (9 with the constraint r 2. For δ =, there are the following solutions: Case If A = 0, A 0 = 0, r = 0, then B Obviously, there is the constraint that α < 0. Case 2 If B = 0, A 0 = 0, then A 2αp2, p = γ 2α. (20 2α2 p 2, p = γ α, r = 0. (2 Case 3 If B = 0, A 0 0, then A α 2 p 2 (r 2 2 γr, A 0 (r 2 2, p = 2(r2 + γ α(r 2 2. (22 There is the constraint r 0 r 2 2. Case 4 If A 0 = 0, A 0, B 0, then α A 2 p 2 (r 2 +, B αp2 4 2, p = 2γ α. (23 For the projective Riccati euations (9, when p < 0 δ =, its solution is f = r + sinh( pξ, (24 p cosh( p ξ g = p r + sinh( p ξ, (25 when p < 0 δ =, its solution is f 2 = r + cosh( p ξ, (26 p sinh( p ξ g 2 = p r + cosh( p ξ. (27 When p > 0 δ =, its solutions are f 3 = r + sin( p ξ, (28 p cos( p ξ g 3 = p r + sin( p ξ, (29 f 4 = r + cos( p ξ, (30 p sin( p ξ g 4 = p r + cos( p ξ. (3 Combining Es. (3, (, the results from E. (5 (3, we can derive various envelope travelling solutions including envelope solitary wave solutions to NLS euation (, for example, Type For δ =, if α < 0 αγ < 0, then the solution to NLS euation ( is ( γ u = B g 2 e i(kx ωt = tanh γ 2α ξ e i(kx ωt. (32

4 92 FU Zun-Tao, LIU Shi-Da, LIU Shi-Kuo Vol. 4 Type 2 Type 3 Type 4 Type 5 For δ =, if α < 0 αγ > 0, then the solution to NLS euation ( is u 2 = B g 3 e i(kx ωt = γ ( γ cot 2α ξ e i(kx ωt, (33 u 3 = B g 4 e i(kx ωt γ ( γ tan 2α ξ e i(kx ωt. (34 For δ =, if α < 0 αγ < 0, then the solution to NLS euation ( is γ u 4 = B g 2 e i(kx ωt sinh( 2γ/α ξ = cosh( 2γ/α ξ ± e i(kx ωt. (35 For δ =, if α < 0 αγ > 0, then the solution to NLS euation ( is u 5 = B g 3 e i(kx ωt = γ cos( 2γ/α ξ sin( 2γ/α ξ ± e i(kx ωt, (36 u 6 = B g 4 e i(kx ωt γ sin( 2γ/α ξ cos( 2γ/α ξ ± e i(kx ωt. (37 For δ =, if > 0 αγ > 0, then the solution to NLS euation ( is ( 2γ γ u 7 = A f 2 e i(kx ωt = sech α ξ e i(kx ωt. (38 Type 6 Type 7 For δ =, if > 0 αγ < 0, then the solution to NLS euation ( is 2γ u 8 = A f 3 e i(kx ωt ( csc γα ξ e i(kx ωt, (39 2γ u 9 = A f 4 e i(kx ωt ( sec γα ξ e i(kx ωt. (40 For δ =, if > 0 (r 2 αγ < 0, then the solution to NLS euation ( is u 0 = (A 0 + A f 2 e i(kx ωt (r ± 2(r2 γ pα(r α 2 p 2 (r cosh( 2( r 2 γ/α(r ξ + r with the constraint that r 0 r 2. Type 8 For δ =, if > 0 (r 2 αγ > 0, then the solution to NLS euation ( is u = (A 0 + A f 3 e i(kx ωt (r ± 2(r2 γ pα(r u 2 = (A 0 + A f 4 e i(kx ωt (r ± 2(r2 γ pα(r α 2 p 2 (r α 2 p 2 (r sin( 2(r 2 γ/α(r ξ + r cos( 2(r 2 γ/α(r ξ + r with the constraint that r 0 r 2. Type 9 For δ =, if α < 0, αγ < 0, r 2 >, then the solution to NLS euation ( is u 3 = (A f 2 + B g 2 e i(kx ωt 2γ α 2 p 2 (r 2 αp 4 cosh( 2γ/α ξ + r γ sinh( 2γ/α ξ cosh( 2γ/α ξ + r e i(kx ωt (4 e i(kx ωt, (42 e i(kx ωt (43 e i(kx ωt (44

5 No. 2 New Exact Solutions to NLS Euation Coupled NLS Euations 93 with the constraint r 2. Type 0 For δ =, if α < 0, αγ > 0 r 2 <, then the solution to NLS euation ( is u 4 = (A f 3 + B g 3 e i(kx ωt 2γ α 2 p 2 (r 2 αp 4 sin( 2γ/α ξ + r γ cos( 2γ/α ξ sin( e i(kx ωt (45 2γ/α ξ + r u 5 = (A f 4 + B g 4 e i(kx ωt = ± 2γ α 2 p 2 (r 2 αp 4 cos( 2γ/α ξ + r ± γ with the constraint r 2. Type For δ =, if α < 0 αγ < 0, then the solution to NLS euation ( is Type 2 Type 3 ( γ u 6 = B g e i(kx ωt = coth γ 2α ξ sin( 2γ/α ξ cos( 2γ/α ξ + r e i(kx ωt (46 e i(kx ωt. (47 For δ =, if < 0 αγ > 0, then the solution to NLS euation ( is u 7 = A f e i(kx ωt = 2γ ( γα csch ξ e i(kx ωt. (48 For δ =, if (r 2 2 > 0 (r 2 2αγ < 0, then the solution to NLS euation ( is u 8 = (A 0 + A f e i(kx ωt (r 2 2 ± 2(r2 + γ pα(r 2 2 α 2 p 2 (r 2 2 sinh( 2( + r 2 γ/α(2 r 2 ξ + r with the constraint that r 0 r 2 2. Type 4 For δ =, if α < 0 αγ < 0, then the solution to NLS euation ( is u 9 = (A f + B g e i(kx ωt 2γ α 2 p 2 (r 2 + αp 4 sinh( 2γ/αξ + r γ cosh( 2γ/α ξ sinh( 2γ/α ξ + r e i(kx ωt (49 e i(kx ωt. (50 Obviously, the solutions u, u 2, u 3, u 7, u 8, u 9, u 6, u 7 are general envelope solitary wave solutions periodic solutions expressed by sine-cosine functions, which can be found in the usual expansion methods, such as the function transformation method, 7,8 the hyperbolic function expansion method, 9,0 the Jacobi elliptic function expansion method,,2 the sine-cosine method. 3 But the solutions u 4, u 5, u 6, u 0, u, u 2, u 3, u 4, u 5, u 8, u 9 cannot be obtained in these expansion methods. These solutions are new type envelope solitary wave solutions or new type envelope periodic solutions expressed by sine-cosine functions, some of which have not been found before. 3 Exact Solutions to Coupled NLS Euation In order to solve E. (2, we take the following transformation, u = φ(ξ e i(kx ωt, v = ψ(ξ e i(kx ωt, ξ = x c g t. (5 If euations (5 (6 are considered, then euation (2 can be rewritten as If φ = ψ is taken, then dξ 2 γφ + ( φ φψ 2 = 0, α d2 ψ dξ 2 γψ + ( ψ φ 2 ψ = 0. (52 dξ 2 γφ + ( + 2 φ 3 = 0 (53 can be obtained from Es. (52. Comparing E. (53 with E. (7, one can see that the difference between these two euations is that in E. (7 is replaced by + 2 in E. (53, so the solutions to E. (2 can be easily obtained from the solutions to E. (, here we omit these details.

6 94 FU Zun-Tao, LIU Shi-Da, LIU Shi-Kuo Vol. 4 4 Conclusion In this paper, we introduce an intermediate transformation from solutions to the projective Riccati euations apply it to solve the NLS euation the coupled NLS euations. Many solutions are obtained for these nonlinear systems, such as envelope solitary wave solutions constructed in terms of hyperbolic functions envelope periodic solutions expressed in terms of sine cosine functions. Some of which are not given in literatures to our knowledge. Of course, this transformation can be also applied to other nonlinear wave euations. References P.K. Shukla, N.N. Rao, M.Y. Yu, et al., Phys. Rep. 38 ( M.J. Ablowitz H. Segur, Solitons the Inverse Scattering Tranforms, Philadelphia, SIAM (98. 3 Fu Zun-Tao, Liu Shi-Kuo, Liu Shi-Da, Zhao Qiang, Commun. Theor. Phys. (Beijing, China 40 ( R. Conte M. Musette, J. Phys. A: Math. Gen. 25 ( G.X. Zhang, Z.B. Li, Y.S. Duan, Science in China A30 ( Z.Y. Yan, Chaos, Solitons, Fractals 6 ( C.L. Bai, Phys. Lett. A288 ( Z.T. Fu, S.K. Liu, S.D. Liu, Phys. Lett. A299 ( E.G. Fan, Phys. Lett. A277 ( E.J. Parkes B.R. Duffy, Phys. Lett. A229 ( Z.T. Fu, S.K. Liu, S.D. Liu, Q. Zhao, Phys. Lett. A290 ( S.K. Liu, Z.T. Fu, S.D. Liu, Q. Zhao, Phys. Lett. A289 ( C.T. Yan, Phys. Lett. A224 (