Modified Simple Equation Method and its Applications to Nonlinear Partial Differential Equations

Abstract

Abstract:In this paper, the modified simple equation method is used to construct exact periodic and soliton solutions of some nonlinear partial differential equations. Exact solutions of the nonlinear Schrödinger equation, the Hamiltonian amplitude equation, Klein-Gordon equation in 1+2 dimension, the coupled Klein-Gordon equation, the (2 + 1)-dimensional long-wave-short-wave resonance interaction equation, the modified KdV-KP equation and the modified Benjamin-Bona-Mahony equation are successfully obtained. These solutions may be important of significance for the explanation of some practical physical problems.

New Exact Solutions of the New Hamiltonian Amplitude-Equation and Fokas Lenells Equation

“…[24], the solution Equation (22) in [27], the solution Equation (23) in [31], the solution Equation (53) in [32] and the solution Equation (53) in [33] in this study as the solution Equation (40). Besides, we have found the similar solution with the solution Equation (59) in [24] and the solution Equation (54) in [33] in this study as the solution Equation (42). To our knowledge, other solutions of Equation (1) …”

“…Many authors have tackled numerous methods to find exact solutions of the new Hamiltonian amplitude equation such as the general projective Riccati equation method [24], the sinh-Gordon expansion method [25], the extended F-expansion method [26], the first integral method [27], the G / G ′ -expansion method [28], the functional variable method [29], Lie symmetry method [30], the simplest equation method [31], He's semi-inverse variational principle method and Ansatz method [32], modified simplest equation method [33].…”

“…Firstly, we consider the following new Hamiltonian amplitude equation [22][23][24][25][26][27][28][29][30][31][32][33]:…”

Novel Behaviors to the Nonlinear Evolution Equation Describing the Dynamics of Ionic Currents along Microtubules

“…Nonlinear evolution equations arise in the various fields of nonlinear sciences such as optical fibers, chemistry, biology, fluid mechanics etc, and their solutions play a vital roles in our real life situations. Various powerful techniques have been developed for the study of the behavior to the different nonlinear evolution equations such as the (G /G)-expansion method [1], the new extended (G /G)-expansion method [2], the sine-cosine method [3], the sine-Gordon expansion method [4,5], the tanh method [6,7], the extended tanh method [8], the Hirota bilinear method [9], the modified simple equation method [10], the modified tanh-coth method [11], the Riccati-Bernoulli sub-ODE method [12] and so on. In general, various powerful techniques have been developed by various researchers to tackle different class of nonlinear evolution equations [13][14][15][16][17].…”

Implementations of Conservation Theorem and the Exponential Rational Function Method to Viscous Burgers Equation

“…In recent years, a variety of powerful methods for obtaining explicit travelling and solitary wave solutions of NPDEs have been proposed. For example, Hirota bilinear transformation [10], the inverse scattering transformation [11], Bäcklund and Darboux transformation [12], the transformed rational function method [13], the tanh function method [14], the exp-function method [15], the sine-cosine method [16], the modified simple equation method [17], the extended trial equation method [18], the ansatz approach [19] and so on.…”