The Weierstrass elliptic function expansion method and its applications in nonlinear wave equations

Abstract

Abstract. In this paper, based on the close relationship between the Weierstrass elliptic function ℘(ξ; g 2 , g 3 )(g 2 , g 3 , invariants) and nonlinear ordinary differential equation, a Weierstrass elliptic function expansion method is developed in terms of the Weierstrass elliptic function instead of many Jacobi elliptic functions. The mechanism is constructive and can be carried out in computer with the aid of computer algebra (for example, Maple). Many important nonlinear wave equations arising from nonlinear science are chosen to illustrate this technique such as the new integrable Davey-Stewartsontype equation, the (2+1)-dimensional modified KdV equation, the the generalized Hirota equation in 2+1 dimensions, the Generalized KdV equation, the (2+1)-dimensional modified Novikov-Veselov equations, (2+1)-dimensional generalized system of modified KdV equation, the coupled Klein-Gordon equation, and the (2+1)-dimensional generalization of coupled nonlinear Schrodinger equation. As a consequence, some new doubly periodic solutions are obtained in terms of the Weierstrass elliptic function. Moreover solitary wave solutions and singular solitary wave solutions are also given as simple limits of doubly periodic solutions. These solutions may be useful to explain some physical phenomena. The algorithm is also applied to other many nonlinear wave equations. Moreover we also present the general form of the technique.

Doubly Periodic Meromorphic Solutions of Autonomous Nonlinear Differential Equations

“…В то время как целый ряд методов поиска эллиптических решений, таких как метод эллиптической функции Вейерштрасса [3,11], метод эллиптических функций Якоби [5,6,9], их различные обобщения и модификации, теряют решения, не попадающие в априорно заданное представление, предлагаемый метод позволяет находить любое эллиптическое решение и проводить их классификацию. Еще один метод, лишенный этих недостатков, был предложен Контом и Мюзетт [19,20].…”

“…Автономные нелинейные дифференциальные уравнения, как обыкновенные, так и в частных производных, встречаются при описании многих процессов и явлений в физике, биологии, химии экономике и т. д. В последние годы появилось большое количество работ, посвященных проблеме построения точных решений автономных нелинейных дифференциальных уравнений [1][2][3][4][5][6][7][8][9][10][11][12]. Был предложен целый ряд мето-дов и алгоритмов.…”

New generalized Jacobi elliptic function expansion method

“…In the past decades, there has been significant progression in the development of methods such as the inverse scattering method [1], Hirota's bilinear method [2], similarity transformation method [3,4], homogeneous balance method [5], the sine-cosine method [6], tanh function method [7,8], mapping method [9,10], F-expansion method [11], Riccati equation rational expansion method [12], Jacobi and Weierstrass elliptic function method [13,14] and new generalized Jacobi elliptic function expansion method [15]. In [16][17][18][19], Wang and Chen present a new elliptic function rational expansion method and is more powerful than exiting Jacobi elliptic function method [20] to uniformly construct more new doubly periodic solutions in terms of rational formal Jacobi elliptic function of nonlinear evolution equations.…”

Application of the Weierstrass Elliptic Expansion Method to the Long-Wave and Short-Wave Resonance Interaction System

“…In this paper, enlightened by the idea in [5,6], we attempt to develop an algorithm in terms of the Weierstrass elliptic function ℘(ξ ; g 2 , g 3 ) to seek new types of doubly periodic solutions of the (2+1)-dimensional equation describing the interaction of a long wave with 0932-0784 / 08 / 0500-0273 $ 06.00 c 2008 Verlag der Zeitschrift für Naturforschung, Tübingen · http://znaturforsch.com a packet of short waves. Such process can arise in fluid mechanics.…”

“…There exist some transformations to study Weierstrass elliptic function solutions of nonlinear wave equations. In [5], many important nonlinear wave equations arising from nonlinear science are chosen to illustrate the Weierstrass elliptic function expansion method such as the new integrable Davey-Stewartson-type equation, the (2+1)-dimensional modified Korteweg-de Vries equation, the generalized Hirota equation, the (2+1)-dimensional modified Novikov-Veselov equations, and the coupled Klein-Gordon equation.…”