Darboux and Bäcklund transformations of the bidirectional Sawada-Kotera equation

Darboux and Bäcklund transformations of the bidirectional Sawada-Kotera equation are derived with the help of the resulting Riccati equation. As an application, some explicit solutions of the bidirectional Sawada-Kotera equation are obtained, including rational solutions, periodic solutions, and soliton solutions.     

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.     

Multiple soliton solutions for the Bogoyavlenskii’s generalized breaking soliton equations and its extension form

In this work, two generalized breaking soliton equations, namely, the Bogoyavlenskii’s breaking soliton equation and its extended form, are examined. The complete integrability of these equation are justified. Multiple soliton solutions and multiple singular soliton solutions are formally derived for each equation. The additional terms of these equations do not kill the integrability of the typical breaking soliton equation. The Cole-Hopf transformation method and the simplified Hereman’s method are applied to conduct this analysis.     

The variational iteration method combined with improved generalized tanh-coth method applied to Sawada-Kotera equation

We obtain new exact solutions to generalized Sawada-Kotera equation. Using the variational iteration method combined with the improved generalized tanh-coth method, we construct new traveling wave solutions for the standard Sawada-Kotera equation and, by means of scaling, we obtain new solutions to general Sawada-Kotera equation. Periodic and soliton solutions are formally derived for both models.     

Multiple soliton solutions for (2 + 1)‐dimensional Sawada–Kotera and Caudrey–Dodd–Gibbon equations

Multiple soliton solutions for the (2 + 1)‐dimensional Sawada–Kotera and the Caudrey–Dodd–Gibbon equations are formally derived. Moreover, multiple singular soliton solutions are obtained for each equation. The simplified form of Hirota's bilinear method is employed to conduct this analysis. Copyright © 2011 John Wiley & Sons, Ltd.     

A (3 + 1)-dimensional nonlinear evolution equation with multiple soliton solutions and multiple singular soliton solutions

In this work, a (3 + 1)-dimensional nonlinear evolution equation is investigated. The Hirota’s bilinear method is applied to determine the necessary conditions for the complete integrability of this equation. Multiple soliton solutions are established to confirm the compatibility structure. Multiple singular soliton solutions are also derived. The resonance phenomenon does not exist for this model.     

A variety of distinct kinds of multiple soliton solutions for a ( 3 + 1)‐dimensional nonlinear evolution equation

In this work, a variety of distinct kinds of multiple soliton solutions is derived for a ( 3 + 1)‐dimensional nonlinear evolution equation. The simplified form of the Hirota's method is used to derive this set of distinct kinds of multiple soliton solutions. The coefficients of the spatial variables play a major role in the existence of this variety of multiple soliton solutions for the same equation. The resonance phenomenon is investigated as well. Copyright © 2012 John Wiley & Sons, Ltd.     

Nonautonomous soliton solutions for a nonintegrable Korteweg–de Vries equation with variable coefficients by the variational approach

A nonintegrable Korteweg–de Vries equation with variable coefficients is investigated in this paper. Due to the existence of variable coefficients, the equation becomes nonintegrable, which leads to the invalidity of the traditional analytical methods to obtain soliton solutions. In order to overcome this difficulty, the variational approach is employed in this paper. The variational principle corresponding to this nonintegrable equation is established. Based on that, the first- and second-order nonautonomous soliton solutions are derived. We note that the obtained solutions can be degenerated to the integrable cases. Properties of the nonautonomous solitons and influence of the variable coefficients are discussed.     

Multiple soliton solutions and other exact solutions for a two‐mode KdV equation

In this work, we study the two‐mode Korteweg–de Vries (TKdV) equation, which describes the propagation of two different waves modes simultaneously. We show that the TKdV equation gives multiple soliton solutions for specific values of the nonlinearity and dispersion parameters involved in the equation. We also derive other distinct exact solutions for general values of these parameters. We apply the simplified Hirota's method to study the specific of the parameters, which gives multiple soliton solutions. We also use the tanh/coth method and the tan/cot method to obtain other set of solutions with distinct physical structures. Copyright © 2016 John Wiley & Sons, Ltd.     

A new integrable equation that combines the KdV equation with the negative‐order KdV equation

In this work, we develop a new integrable equation by combining the KdV equation and the negative‐order KdV equation. We use concurrently the KdV recursion operator and the inverse KdV recursion operator to construct this new integrable equation. We show that this equation nicely passes the Painlevé test. As a result, multiple soliton solutions and other soliton and periodic solutions are guaranteed and formally derived.     

The improved Fan sub-equation method and its application to the SK equation

In this paper, many types of exact solutions of a first-order nonlinear ordinary differential equation called Fan sub-equation, is further investigated by using bifurcation method of dynamical systems. As a result, more types of exact solutions to Sawada-Kotera (SK) equation are obtained, which include more general soliton solutions, kink solutions, triangular function solutions, Jacobian elliptic function solutions with double periods and so on.     

A study on a two‐wave mode Kadomtsev–Petviashvili equation: conditions for multiple soliton solutions to exist

We establish a two‐wave mode equation for the integrable Kadomtsev–Petviashvili equation, which describes the propagation of two different wave modes in the same direction simultaneously. We determine the necessary conditions that make multiple soliton solutions exist for this new equation. The simplified Hirota's method will be used to conduct this work. We also use other techniques to obtain other set of periodic and singular solutions for the two‐mode Kadomtsev‐Petviashvili equation. Copyright © 2017 John Wiley & Sons, Ltd.     

Sine–Cosine method for finding the soliton solutions of the generalized fifth-order nonlinear equation

In this paper, we use a modified form of the Sine–Cosine method for obtaining exact soliton solutions of the generalized fifth-order nonlinear evolution equation. Analysis for this method is presented. The present method shows that the solutions involve either sec² or sech² under certain conditions. General forms of those conditions are determined for the first time. Exact solutions for special cases of this problem such as the Sawada-Kotera and Lax equations are determined and found to be compared well with the previous studies.     

Multiple soliton solutions for the new (2+1)-dimensional Korteweg–de Vries equation by multiple exp-function method

The multiple exp-function method is utilized for solving the multiple soliton solutions for the new $(2+1)$-dimensional Korteweg–de Vries equation, which include one-soliton, two-soliton, and three-soliton type solutions. The physical phenomena of these obtained multiple soliton solutions are analyzed and illustrated in figures by selecting appropriate parameters.     

共振长短波方程的孤波解

本文研究了共振长短波方程的孤波解.利用扩展映射法和符号计算,得到许多新的孤波解.这些孤波解能很好地模拟水波,辅助方程用更一般方程代替的扩展映射法能更有效找到这些孤波解.

Abstract：

In this article,soliton solutions of the long-short wave resonance equations are investigated.By the extended mapping method and symbolic computation,many new exact soliton solutions are obtained.These soliton solutions are fascinating in modeling water waves.The extended mapping method,with the auxiliary ordinary equations replaced by more general ones,is more effective to find these soliton solutions.     

A study on the (2 + 1)‐dimensional KdV4 equation derived by using the KdV recursion operator

We derive a new ( 2 + 1)‐dimensional Korteweg–de Vries 4 (KdV4) equation by using the recursion operator of the KdV equation. This study shows that the new KdV4 equation possess multiple soliton solutions the same as the multiple soliton solutions of the KdV hierarchy, but differ only in the dispersion relations. We also derive other traveling wave solutions. Copyright © 2013 John Wiley & Sons, Ltd.     

Four (2 + 1)-dimensional integrable extensions of the Kadomtsev-Petviashvili equation

In this work, four (2 + 1)-dimensional nonlinear extensions of the Kadomtsev-Petviashvili (KP) equation are developed. The complete integrability of these models are investigated. Multiple-soliton solutions and multiple singular soliton solutions are determined to demonstrate the compatibility of these models. The resonance phenomenon does not exist for any of the derived models.     

Four (2 + 1)-dimensional integrable extensions of the KdV equation: Multiple-soliton and multiple singular soliton solutions

In this work, four (2 + 1)-dimensional nonlinear completely integrable equations, generated by extending the KdV equation are developed. The necessary condition for the complete integrability of these equation are formally derived. Multiple-soliton solutions and multiple singular soliton solutions are determined to emphasize the compatability of these models. The dispersion relations of these models are characterized by distinct physical structures. The resonance phenomenon for these equations does not exist for any model.     

Bell polynomial approach to an extended Korteweg–de Vries equation

Under investigation in this paper is an extended Korteweg–de Vries equation. Via Bell polynomial approach and symbolic computation, this equation is transformed into two kinds of bilinear equations by choosing different coefficients, namely KdV–SK‐type equation and KdV–Lax‐type equation. On the one hand, N‐soliton solutions, bilinear Bäcklund transformation, Lax pair, Darboux covariant Lax pair, and infinite conservation laws of the KdV–Lax‐type equation are constructed. On the other hand, on the basis of Hirota bilinear method and Riemann theta function, quasiperiodic wave solution of the KdV–SK‐type equation is also presented, and the exact relation between the one periodic wave solution and the one soliton solution is established. It is rigorously shown that the one periodic wave solution tend to the one soliton solution under a small amplitude limit. Copyright © 2013 John Wiley & Sons, Ltd.     

An extended simplest equation method and its application to several forms of the fifth-order KdV equation

In this paper, an extended simplest equation method is proposed to seek exact travelling wave solutions of nonlinear evolution equations. As applications, many new exact travelling wave solutions for several forms of the fifth-order KdV equation are obtained by using our method. The forms include the Lax, Sawada-Kotera, Sawada-Kotera-Parker-Dye, Caudrey-Dodd-Gibbon, Kaup-Kupershmidt, Kaup-Kupershmidt-Parker-Dye, and the Ito forms.