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.     

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.     

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.     

Negative‐order modified KdV equations: multiple soliton and multiple singular soliton solutions

In this work, we develop the negative‐order modified Korteweg–de Vries (nMKdV) equation. By means of the recursion operator of the modified KdV equation, we derive negative order forms, one for the focusing branch and the other for the defocusing form. Using the Weiss–Tabor–Carnevale method and Kruskal's simplification, we prove the Painlevé integrability of the nMKdV equations. We derive multiple soliton solutions for the first form and multiple singular soliton solutions for the second form. Copyright © 2015 John Wiley & Sons, Ltd.     

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.     

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.     

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.     

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.     

N-soliton solutions for the integrable bidirectional sixth-order Sawada-Kotera equation

In this work, the integrable bidirectional sixth-order Sawada-Kotera equation is examined. The equation considered is a KdV6 equation that was derived from the fifth order Sawada-Kotera equation. Multiple soliton solutions and multiple singular soliton solutions are formally derived for this equation. The Cole-Hopf transformation method combined with the Hirota’s bilinear method are used to determine the two sets of solutions, where each set has a distinct structure.     

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.     

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.     

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.     

Multiple complex soliton solutions for integrable negative-order KdV and integrable negative-order modified KdV equations

In this work we show that the integrable negative-order Korteweg–de Vries (nKdV) and the integrable negative-order modified Korteweg–de Vries (nMKdV) equation admit multiple complex soliton solutions. To achieve this goal, we introduce two complex forms of the simplified Hirota’s method, the first works effectively for the nKdV equation, and the other form is nicely applicable for the nMKdV equation. We believe that the newly proposed complex forms and the obtained findings will shed light on complex solitons of other integrable equations.     

Darboux transformation and Rogue waves of the Kundu–nonlinear Schrödinger equation

In this paper, the Darboux transformation of the Kundu–nonlinear Schrödinger equation is derived and generalized to the matrix of n‐fold Darboux transformation. From known solution Q, the determinant representation of n‐th new solutions of Q^[n] are obtained by the n‐fold Darboux transformation. Then soliton solutions and positon solutions are generated from trivial seed solutions, breather solutions and rogue wave solutions that are obtained from periodic seed solutions. After that, the higher order rogue wave solutions of the Kundu–nonlinear Schrödinger equation are given. We show that free parameters in eigenfunctions can adjust the patterns of the higher order rogue waves. Meanwhile, the third‐order rogue waves are given explicitly. Copyright © 2014 John Wiley & Sons, Ltd.     

A series of complexiton soliton solutions for non-linear Hirota–Satsuma equations using the generalized multiple Riccati equations rational expansion method

In this article, the generalized multiple Riccati equations rational expansion method has been used to construct a series of complexiton soliton solutions for the non-linear Hirota–Satsuma equations. With the help of symbolic computation software as Maple or Mathematica, we obtain many new types of complexiton soliton solutions, i.e. various combination of trigonometric periodic function and hyperbolic function solutions, various combination of trigonometric periodic function and rational function solutions, and various combination of hyperbolic function and rational function solutions.     

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.     

Elliptic function solutions for some nonlinear pdes in mathematical physics

In this work, we have constructed various types of soliton solutions of the generalized regularized long wave and generalized nonlinear Klein-Gordon equations by the using of the extended trial equation method. Some of the obtained exact traveling wave solutions to these nonlinear problems are the rational function, 1-soliton, singular, the elliptic integral functions $F, E, \Pi$ and the Jacobi elliptic function sn solutions. Also, all of the solutions are compared with the exact solutions in literature, and it is seen that some of the solutions computed in this paper are new wave solutions.     

A new symmetry approach to solve space–time‐dependent KdV systems

In this paper, a new method to solve space–time‐dependent non‐linear equations is proposed. After considering the variable coefficient of a non‐linear equation as a new dependent variable, some special types of space–time‐dependent equations can be solved from corresponding space–time‐independent equations by using the general classical Lie approach. The rich soliton solutions of space–time‐dependent KdV equation and mKdV equation are given with the help of the approach. Copyright © 2004 John Wiley & Sons, Ltd.     

Construction of a hierarchy of negative‐order integrable Burgers equations of higher orders

In this work, we employ the recursion operator, the Burgers equation and its inverse operator, for constructing a hierarchy of negative‐order integrable Burgers equations of higher orders. The complete integrability of each established equation emerges by virtue of the correlation between integrability and recursion operators. We use the simplified Hirota's method to obtain multiple kink solutions for some of the derived equations, and in particular, for the generalized negative‐order Burgers equation.