Multiple Soliton Solutions for a New Generalization of the Associated Camassa‐Holm Equation by Exp‐Function Method

Abstract

The Exp-function method is generalized to construct N-soliton solutions of a new generalization of the associated Camassa-Holm equation. As a result, one-soliton, two-soliton, and three-soliton solutions are obtained, from which the uniform formulae of N-soliton solutions are derived. It is shown that the Exp-function method may provide us with a straightforward, effective, and alternative mathematical tool for generating N-soliton solutions of nonlinear evolution equations in mathematical physics.

Investigating One-, Two-, and Triple-Wave Solutions via Multiple Exp-Function Method Arising in Engineering Sciences

“…This section elucidates a systematic explanation of the multiple Exp-function method [60][61][62][63][64] so that it can be further applied to the nonlinear PDEs in order to furnish its exact solutions:…”

“…In this paper, we will study the multiple Exp-function method for determining the multiple soliton solutions. The multiple Exp-function method used by some of the powerful authors for various nonlinear equations include the nonlinear evolution equations [60], the (2 + 1)-dimensional Calogero-Bogoyavlenskii-Schiff equation [61], the generalized (1 + 1)-dimensional and (2 + 1)-dimensional Ito equations [62], a new generalization of the associated Camassa-Holm equation [63], the (3 + 1)-dimensional generalized KP and BKP equations [64], and the new (2 + 1)-dimensional Korteweg-de Vries equation [65]. In [65], Liu et al utilized the multiple Exp-function method for the most well-known equation, namely, the Korteweg-de Vries (KdV) equation, and gained one-soliton-, two-soliton-, and three-soliton-type 2…”

Multiple rogue wave solutions and the linear superposition principle for a (3 + 1)‐dimensional Kadomtsev–Petviashvili–Boussinesq‐like equation arising in energy distributions

“…-expansion method, 7 Hirota's bilinear method, [8][9][10][11][12][13][14][15] He's variational principle, 16,17 binary Darboux transformation, 18 Lie group analysis, 19,20 Bäcklund transformation method, 21 and the multiple exp-function method. [22][23][24][25][26] Moreover, many powerful methods have been employed to investigate the new properties of mathematical models which are symbolizing serious real-world problems. [27][28][29] Originally, the Kadomtsev-Petviashvili (KP) equation introduced by Boris Borisovich Kadomtsev and Vladimir Iosifovich Petviashvili to describe the evolution of the nonlinear and long waves with small and slow dependence on the transverse coordinate [30][31][32] as follows:…”

“…Nonlinear partial differential equations (NPDEs) occur in diverse areas of nonlinear sciences such as hydrodynamics, plasma physics, molecular biology, quantum mechanics, nonlinear optics, and surface water waves. Several analytical and numerical techniques have been formulated for tackling these types of nonlinear models, including the exp‐function method, 1 the homotopy analysis method, 2 the shifted Legendre polynomials method, 3 the natural variational iteration method, 4 the tan ( ϕ /2)‐expansion method, 5,6 the ‐expansion method, 7 Hirota's bilinear method, 8–15 He's variational principle, 16,17 binary Darboux transformation, 18 Lie group analysis, 19,20 Bäcklund transformation method, 21 and the multiple exp‐function method 22–26 . Moreover, many powerful methods have been employed to investigate the new properties of mathematical models which are symbolizing serious real‐world problems 27–29 .…”

A series of the solutions for the Heisenberg ferromagnetic spin chain equation

“…The improved F-expansion method was proposed to find more abundant traveling wave solitions, which is based on the F-expansion and Exp-function method. [27][28][29] The main objective of this work is to seek new exact solutions for Equation 2. A series of abundant exact solutions, namely, periodic and doubly periodic wave solutions, solitary wave solutions, are obtained by using the Jacobi elliptic function method and improved F-expansion method.…”