Multiple soliton solutions for the sixth-order Ramani equation and a coupled Ramani equation

In this work, the completely integrable sixth-order nonlinear Ramani equation and a coupled Ramani equation are studied. Multiple soliton solutions and multiple singular soliton solutions are formally derived for these two equations. The Hirota’s bilinear method is used to determine the two distinct structures of solutions. The resonance relations for the three cases are investigated.     

Bilinear Bäcklund transformation and Lax pair for a coupled Ramani equation

In this paper, a coupled Ramani equation is proposed. The bilinear Bäcklund transformation and Lax pair for this equation are derived starting from its bilinear form. Multisoliton solutions to the system can also be obtained.     

Soliton solutions for a negative order non-isospectral AKNS equation

In this paper, the bilinear form of a negative order non-isospectral AKNS equation is given. The soliton solutions are obtained through Hirota’s direct method and the Wronskian technique, respectively.     

Bilinear form and new interaction solutions for the sixth-order Ramani equation

By using the Hirota’s bilinear method, the bilinear form of the sixth-order Ramani equation is succinctly obtained. With the aid of the obtained bilinear form, some new interaction solutions which include interaction solutions between exponential and trigonometric functions, interaction solutions between exponential and hyperbolic functions, and interaction solutions between trigonometric and hyperbolic functions are also presented by employing the three wave method.     

Soliton solutions of Burgers equations and perturbed Burgers equation

This paper carries out the integration of Burgers equation by the aid of tanh method. This leads to the complex solutions for the Burgers equation, KdV-Burgers equation, coupled Burgers equation and the generalized time-delayed Burgers equation. Finally, the perturbed Burgers equation in (1+1) dimensions is integrated by the ansatz method.     

Explicit solutions to a coupled integrable lattice equation

A coupled integrable lattice equation is derived from a 4 × 4 matrix spectral problem, then with the help of a special Darboux matrix, explicit solutions of the aforementioned equation are given by means of gauge transformation between the Lax pair. Finally, the density profiles of these exact solutions are presented to illustrate these solutions.     

Exact solutions of coupled nonlinear Klein-Gordon equation

In this paper, we employ the general integral method for traveling wave solutions of coupled nonlinear Klein-Gordon equations. Based on the idea of the exact Jacobi elliptic function, a simple and efficient method is proposed for obtaining exact solutions of nonlinear evolution equations. The solutions obtained include solitons, periodic solutions and Jacobi elliptic function solutions.     

Solitary wave solutions of coupled boussinesq equation

This article presents the general case‐study of our previous works regarding generalized Boussinesq equations [17, 18, 19], that focus on application of various subordinate methods where are applied to construct more general exact solutions of the coupled Boussinesq equations. In this article, the ‐expansion method is applied on coupled Boussinesq equations. Our work is motivated by the fact that the ‐expansion method provides not only more general forms of solutions but also periodic, solitary waves, and rational solutions. The method appears to be easier and faster by means of a symbolic manipulation program. © 2016 Wiley Periodicals, Inc. Complexity 21: 151–155, 2016     

Soliton solutions to the Novikov equation and a negative flow of the Novikov hierarchy

The Novikov equation and a negative flow of the Novikov hierarchy are related to a negative flow of the Sawada–Kotera hierarchy and the Sawada–Kotera equation by reciprocal transformations, respectively. With the help of the Darboux transformations for the negative flow of the Sawada–Kotera hierarchy, the Sawada–Kotera equation and reciprocal transformations, we obtain a parametric representation for $N$-soliton solutions to the Novikov equation and the negative flow of the Novikov hierarchy.     

Bifurcations of traveling wave solutions in a coupled non-linear wave equation

By using the theory of planar dynamical systems to a coupled non-linear wave equation, the existence of solitary wave solutions and uncountably infinite, many smooth and non-smooth, periodic wave solutions is obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given.     

Soliton solutions and Bäcklund transformation for the complex Ginzburg-Landau equation

Symbolically investigated in this paper is the complex Ginzburg-Landau (CGL) equation. With the Hirota method, both bright and dark soliton solutions for the CGL equation are obtained simultaneously. New Bäcklund transformation in the bilinear form is derived. Relevant properties and features are discussed. Solitons can be compressed (amplified) when the nonlinear (linear) dispersion effect is enhanced. Meanwhile, central frequency of the soliton can be affected by the nonlinear and linear dispersion effects. Besides, directions of the movement for the soliton central frequency can be adjusted. Results of this paper would be of certain value to the studies on the soliton compression and amplification.     

Soliton,kink and antikink solutions of a 2-component of the Degasperis–Procesi equation

In this paper, we employ the bifurcation theory of planar dynamical systems to investigate the traveling wave solutions of a 2-component of the Degasperis–Procesi equation. The expressions for smooth soliton, kink and antikink solutions are obtained.     

Solitons and other solutions to a new coupled nonlinear Schrodinger type equation

In this paper, the first integral method combined with Liu's theorem is applied to integrate a new coupled nonlinear Schrodinger type equation. Using this combination, more new exact traveling wave solutions are obtained for the considered equation using ideas from the theory of commutative algebra. In addition, more solutions are also obtained via the application of semi-inverse variational principle due to Ji-Huan He. The used approaches with the help of symbolic computations via Mathematica 9, may provide a straightforward effective and powerful mathematical tools for solving nonlinear partial differential equations in mathematical physics.     

Soliton solutions for the Fitzhugh–Nagumo equation with the homotopy analysis method

An analytic technique, the homotopy analysis method (HAM), is applied to obtain the soliton solution of the Fitzhugh–Nagumo equation. The homotopy analysis method (HAM) is one of the most effective method to obtain the exact solution and provides us with a new way to obtain series solutions of such problems. HAM contains the auxiliary parameter ?, which provides us with a simple way to adjust and control the convergence region of series solution.     

New soliton solutions for a Kadomtsev-Petviashvili (KP) like equation coupled to a Schrödinger equation

The repeated homogeneous balance is used to construct a new exact traveling wave solution of the Kadomtsev-Petviashvili (KP) like equation coupled to a Schrödinger equation, in which the homogeneous balance method is applied to solve the Riccati equation and the reduced nonlinear ordinary differential equation, respectively. Many new exact traveling wave solutions are successfully obtained, which contain rational and periodic-like solutions. This method is straightforward and concise, and it can be applied to other nonlinear evolution equations.     

Soliton solutions of a few nonlinear wave equations

This paper carries out the integration of a few nonlinear wave equations to obtain topological as well as non-topological soliton solutions. The mathematical techniques used to obtain the soliton solutions are He’s variational iteration method, the tanh method and the ansatz method. The nonlinear wave equations that are studied are coupled mKdV equations, Drinfeld-Sokolov equation and its generalized version. Finally, some numerical simulations are given to support the analytical solutions.     

On soliton solutions for a generalized Hirota–Satsuma coupled KdV equation

The extended homogeneous balance method is used to construct exact traveling wave solutions of a generalized Hirota–Satsuma coupled KdV equation, in which the homogeneous balance method is applied to solve the Riccati equation and the reduced nonlinear ordinary differential equation, respectively. Many exact traveling wave solutions of a generalized Hirota–Satsuma coupled KdV equation are successfully obtained, which contain soliton-like and periodic-like solutions This method is straightforward and concise, and it can also be applied to other nonlinear evolution equations.     

Darboux transformation and explicit solutions for the Satsuma-Hirota coupled equation

A Darboux transformation for the Satsuma-Hirota coupled equation is obtained with the help of the gauge transformation between the Lax pairs. As an application of the Darboux transformation, we give some new explicit solutions, including rational solutions, soliton solutions and periodic solutions and others, of the Satsuma-Hirota coupled equation.     

Abundant new travelling wave solutions for the coupled Ito equation

Abundant new travelling wave solutions of the coupled Ito equation are obtained by the generalized Jacobi elliptic function method.