应用Riccati-Bernoulli辅助方程求解广义非线性Schrodinger方程和(2+1)维非线性Ginzburg-Landau方程
doi: 10.21656/1000-0887.400271
Solution of Generalized Nonlinear Schrodinger Equations and (2+1)-Dimensional Nonlinear Ginzburg-Landau Equations With a Riccati-Bernoulli Auxiliary Equation Method
-
摘要: 研究了Riccati-Bernoulli辅助方程法,并应用这种方法得到广义非线性Schrodinger方程和(2+1)维非线性Ginzburg-Landau方程的精确行波解.这些解包括有理函数、三角函数、双曲函数和指数函数.应用这种方法求解过程简洁有效.该研究对于数学物理方程领域诸多非线性偏微分方程精确解的探究具有重要的意义.Abstract: The Riccati-Bernoulli auxiliary equation method was proposed to seek the exact travelling solutions to the generalized nonlinear Schrodinger equation and the (2+1)-dimensional nonlinear Ginzburg-Landau equation. The solutions can be expressed with the rational functions, the trigonometric functions, the hyperbolic functions and the exponential functions. Being effective and concise, the method is important to obtain the exact travelling solutions for more nonlinear partial differential equations in the field of mathematics and physics.
-
[1] ABDOU M A. The extended F -expansion method and its application for a class of nonlinear evolution equations[J]. Chaos, Solitons and Fractals,2007,31(1): 95-104. [2] ABDOU M A. An improved generalized F -expansion method and its applications[J]. Journal of Computational and Applied Mathematics,2008,214(1): 202-208. [3] WANG M L, ZHANG J L, LI X Z. Application of the (G′/G)-expansion to travelling wave solutions of the Broer-Kaup and the approximate long water wave equations[J]. Applied Mathematics and Computation,2008,206(1): 321-326. [4] WANG M L, LI X Z, ZHANG J L. The (G′/G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics[J]. Physics Letters A,2008,372(4): 417-423. [5] LI L, DUAN C N, YU F J. An improved Hirota bilinear method and new application for a nonlocal integrable complex modified Korteweg-de Vries (MKdV) equation[J]. Physics Letters A,2019,383(14): 1578-1582. [6] GUO D, TIAN S F, ZHANG T T. Integrability, soliton solutions and modulation instability analysis of a (2+1)-dimensional nonlinear Heisenberg ferromagnetic spin chain equation[J]. Computers and Mathematics With Applications,2019,77(3): 770-778. [7] CHENG J P, HE J S. Miura and auto-Baecklund transformations for the discrete KP and mKP hierarchies and their constrained cases[J]. Communications in Nonlinear Science and Numerical Simulation,2019,69: 187-197. [8] LIU X Z, YU J, LOU Z M. New Bcklund transformations of the (2+1)-dimensional Bogoyavlenskii equation via localization of residual symmetries[J].Computers and Mathematics With Applications,2018,76(7): 1669-1679. [9] MA L Y, ZHAO H Q, SHEN S F, et al. Abundant exact solutions to the discrete complex mKdV equation by Darboux transformation[J]. Communications in Nonlinear Science and Numerical Simulation,2019,68: 31-40. [10] WANG X, WANG L. Darboux transformation and nonautonomous solitons for a modified Kadomtsev-Petviashvili equation with variable coefficients[J]. 〖Computers and Mathematics With Applications,2018,75: 4201-4213. [11] PARKES E J, DUFFY B R, ABBOTT P C. The Jacobi elliptic-function method for finding periodic-wave solutions to nonlinear evolution equations[J]. Physics Letters A,2002,295(5/6): 280-286. [12] ZAYED E M E, ALURRFI K A E. A new Jacobi elliptic function expansion method for solving a nonlinear PDE describing the nonlinear low-pass electrical lines[J]. Chaos, Solitons and Fractals,2015,78: 148-155. [13] 石兰芳, 莫嘉琪. 一类强非线性非自治方程的奇摄动Robin边值问题[J]. 应用数学, 2017,30(2): 247-251.(SHI Lanfang, MO Jiaqi. A class of singular perturbation solutions to strong nonlinear equation Robin problems[J]. Mathematica Applicata,2017,30(2): 247-251.(in Chinese)) [14] 石兰芳, 周先春. 一类扰动 Burgers 方程的孤子同伦映射解[J]. 物理学报, 2010,59(5): 2915-2918.(SHI Lanfang, ZHOU Xianchun. Homotopic mapping solution of soliton for a class of disturbed Burgers equation[J]. Acta Physica Sinica,2010,59(5): 2915-2918.(in Chinese)) [15] 欧阳成, 石兰芳, 汪维刚, 等. 非线性强迫扰动Klein-Gordon方程的孤波渐进解法[J]. 数学年刊, 2017,38(A): 43-52.(OUYANG Cheng, SHI Lanfang, WANG Weigang, et al. The asymptotic solving method of solitary wave for the nonlinear forced disturbed Klein-Gordon equation[J]. Chinese Annals of Mathematics,2017,38(A): 43-52.(in Chinese)) [16] PENG Y Z, SHEN M, WANG Z J. Exact solutions to the higher order nonlinear Schr?dinger equation[J]. Mathematica Applicata,2007,20(3): 505-511. [17] 石兰芳, 聂子文. 应用全新G′/(G+G′)展开方法求解广义非线性Schr?dinger方程和耦合非线性Schr?dinger方程组[J]. 应用数学和力学, 2017,38(5): 539-552.(SHI Lanfang, NIE Ziwen. Solutions to the nonlinear Schr?dinger equation and coupled nonlinear Schr?dinger equations with a new G′/(G+G′)-expansion method[J]. Applied Mathematics and Mechanics,2017,38(5): 539-552.(in Chinese)) [18] ANKIEWICZ A, SOTO-CRESPO J M, AKHMEDIEV N. Rogue waves and rational solutions of the Hirota equation[J]. Physical Review E,2010,81: 046602. [19] PORSEZIAN K, LAKSHMANAN M. On the dynamics of the radially symmetric Heisenberg ferromagnetic spin system[J]. Journal of Mathematical Physics,1991,32(10): 2923-2928. [20] 施业琼. (2+1)维Ginzburg-Landau方程的精确波解[J]. 数学的实践与认识, 2009,39(16): 247-251.(SHI Yeqiong. The exact wave solutions for 2+1 dimensional cubic-quintic Ginzburg-Landau equation[J]. Mathematics in Practice and Theory,2009,39(16): 247-251.(in Chinese)) [21] SHI Y, DAI Z, LI D. Application of exp-function method for 2D cubic-quintic Ginzburg-Landau equation[J]. Applied Mathematics and Computation,2009,210(1): 269-275. [22] ZAYED E M E, ALURRFI K A E. The (G′/G,1/G) -expansion method and its applications to two nonlinear Schr?dinger equations describing the propagation of femtosecond pulses in nonlinear optical fibers[J]. Optic,2016,127(4): 1581-1589.
点击查看大图
计量
- 文章访问数: 919
- HTML全文浏览量: 168
- PDF下载量: 287
- 被引次数: 0