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.     

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.     

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.     

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.     

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.     

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.     

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 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.     

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 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.     

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.     

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.     

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.     

A study on two extensions of the Bogoyavlenskii-Schieff equation

In this work, two extensions of the Bogoyavlenskii-Schieff equation are examined. N-soliton solutions are formally determined for each extended equation. We show that the extension terms do not kill the integrability of the typical Bogoyavlenskii-Schieff equation. The simplified Hirota’s method established by Hereman and Nuseir is applied to achieve this goal.     

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.     

Exact soliton solutions for the general fifth Korteweg-de Vries equation

With the aid of computer symbolic computation system such as Maple, the extended hyperbolic function method and the Hirota’s bilinear formalism combined with the simplified Hereman form are applied to determine the soliton solutions for the general fifth-order KdV equation. Several new soliton solutions can be obtained if we taking parameters properly in these solutions. The employed methods are straightforward and concise, and they can also be applied to other nonlinear evolution equations in mathematical physics. The article is published in the original.     

Lump wave and hybrid solutions of a generalized(3+1)-dimensional nonlinear wave equation in liquid with gas bubbles

We investigate a generalized (3 + 1)-dimensional nonlinear wave equation, which can be used to depict many nonlinear phenomena in liquid containing gas bubbles. By employing the Hirota bilinear method, we derive its bilinear formalism and soliton solutions succinctly. Meanwhile, the first-order lump wave solution and second-order lump wave solution are well presented based on the corresponding two-soliton solution and four-soliton solution. Furthermore, two types of hybrid solutions are systematically established by using the long wave limit method. Finally, the graphical analyses of the obtained solutions are represented in order to better understand their dynamical behaviors.     

The multi-soliton solutions of the CH-γ equation

This paper solves the integrable CH-γ equation for analytical multiple soliton solutions with the Darboux transformation method. Some properties of the soliton solutions are different from the CH equation. This work was partially supported by the National Natural Science Foundation of China (Grant No. 10401022) and the Research Grants Council of Hong Kong