Soliton-cnoidal Wave Interaction Solutions Research Articles

Nonlocal Symmetry and Interaction Solutions of a Generalized Kadomtsev—Petviashvili Equation**Supported by the Global Change Research Program of China under Grant No. 2015CB953904, National Natural Science Foundation of under Grant Nos. 11275072 and 11435005, Doctoral Program of Higher Education of China under Grant No. 20120076110024, the Network Information Physics Calculation of Basic Research Innovation Research Group of China under Grant No.

A generalized Kadomtsev—Petviashvili equation is studied by nonlocal symmetry method and consistent Riccati expansion (CRE) method in this paper. Applying the truncated Painlevé analysis to the generalized Kadomtsev—Petviashvili equation, some Bäcklund transformations (BTs) including auto-BT and non-auto-BT are obtained. The auto-BT leads to a nonlocal symmetry which corresponds to the residual of the truncated Painlevé expansion. Then the nonlocal symmetry is localized to the corresponding nonlocal group by introducing two new variables. Further, by applying the Lie point symmetry method to the prolonged system, a new type of finite symmetry transformation is derived. In addition, the generalized Kadomtsev—Petviashvili equation is proved consistent Riccati expansion (CRE) solvable. As a result, the soliton-cnoidal wave interaction solutions of the equation are explicitly given, which are difficult to be found by other traditional methods. Moreover, figures are given out to show the properties of the explicit analytic interaction solutions.

Nonlocal symmetry and exact solutions of the (2+1)-dimensional Gardner equation

For the (2+1)-dimensional Gardner equation, the truncated Painlevé method is developed to obtain the nonlocal residual symmetry and Bäcklund transformation. And, the symmetry group transformation can be computed from the extended system. Moreover, (2+1)-dimensional Gardner equation is proved to be consistent Riccati expansion (CRE) solvable. With the help of the Riccati equation and the CRE method, we obtain the soliton–cnoidal wave interaction solution of the equation.

Nonlocal symmetry and exact solutions of the (2+1)-dimensional modified Bogoyavlenskii–Schiff equation**Project supported by the Global Change Research Program of China (Grant No. 2015CB953904), the National Natural Science Foundation of China (Grant Nos. 11275072 and 11435005), the Doctoral Program of Higher Education of China (Grant No. 20120076110024), the Network Information Physics Calculation of Basic Research Innovation Research Group of China (Grant

In this paper, the truncated Painlevé analysis, nonlocal symmetry, Bäcklund transformation of the (2+1)-dimensional modified Bogoyavlenskii–Schiff equation are presented. Then the nonlocal symmetry is localized to the corresponding nonlocal group by the prolonged system. In addition, the (2+1)-dimensional modified Bogoyavlenskii–Schiff is proved consistent Riccati expansion (CRE) solvable. As a result, the soliton–cnoidal wave interaction solutions of the equation are explicitly given, which are difficult to find by other traditional methods. Moreover figures are given out to show the properties of the explicit analytic interaction solutions.

Residual Symmetry and Explicit Soliton–Cnoidal Wave Interaction Solutions of the (2+1)-Dimensional KdV–mKdV Equation

Abstract The truncated Painlevé method is developed to obtain the nonlocal residual symmetry and the Bäcklund transformation for the (2+1)-dimensional KdV–mKdV equation. The residual symmetry is localised after embedding the (2+1)-dimensional KdV–mKdV equation to an enlarged one. The symmetry group transformation of the enlarged system is computed. Furthermore, the (2+1)-dimensional KdV–mKdV equation is proved to be consistent Riccati expansion (CRE) solvable. The soliton–cnoidal wave interaction solution in terms of the Jacobi elliptic functions and the third type of incomplete elliptic integral is obtained by using the consistent tanh expansion (CTE) method, which is a special form of CRE.

CRE Solvability, Exact Soliton-Cnoidal Wave Interaction Solutions, and Nonlocal Symmetry for the Modified Boussinesq Equation

It is proved that the modified Boussinesq equation is consistent Riccati expansion (CRE) solvable; two types of special soliton-cnoidal wave interaction solution of the equation are explicitly given, which is difficult to be found by other traditional methods. Moreover, the nonlocal symmetry related to the consistent tanh expansion (CTE) and the residual symmetry from the truncated Painlevé expansion, as well as the relationship between them, are obtained. The residual symmetry is localized after embedding the original system in an enlarged one. The symmetry group transformation of the enlarged system is derived by applying the Lie point symmetry approach.

Consistent Riccati expansion and exact solutions of the Kuramoto–Sivashinsky equation

A consistent Riccati expansion (CRE) method is developed for a special Kuramoto–Sivashinsky (KS) equation and we prove the general KS equation is non-CRE solvable. Furthermore, we obtain the soliton–cnoidal wave interaction solution of the special KS equation.

Construction of Soliton-Cnoidal Wave Interaction Solution for the (2+1)-Dimensional Breaking Soliton Equation* *Supported by National Natural Science Foundation of China under Grant Nos. 11271211, 11275072, 11435005, and K.C. Wong Magna Fund in Ningbo University

In this paper, the truncated Painlevé analysis and the consistent tanh expansion (CTE) method are developed for the (2+1)-dimensional breaking soliton equation. As a result, the soliton-cnoidal wave interaction solution of the equation is explicitly given, which is difficult to be found by other traditional methods. When the value of the Jacobi elliptic function modulus m = 1, the soliton-cnoidal wave interaction solution reduces back to the two-soliton solution. The method can also be extended to other types of nonlinear evolution equations in mathematical physics.

The consistent Riccati expansion and new interaction solution for a Boussinesq-type coupled system**Project supported by the National Natural Science Foundation of China (Grant No. 11275129).

Starting from the Davey–Stewartson equation, a Boussinesq-type coupled equation system is obtained by using a variable separation approach. For the Boussinesq-type coupled equation system, its consistent Riccati expansion (CRE) solvability is studied with the help of a Riccati equation. It is significant that the soliton–cnoidal wave interaction solution, expressed explicitly by Jacobi elliptic functions and the third type of incomplete elliptic integral, of the system is also given.

Interactions Between Solitons and Cnoidal Periodic Waves of the (2+1)-Dimensional Konopelchenko—Dubrovsky Equation

The (2+1)-dimensional Konopelchenko—Dubrovsky equation is an important prototypic model in nonlinear physics, which can be applied to many fields. Various nonlinear excitations of the (2+1)-dimensional Konopelchenko—Dubrovsky equation have been found by many methods. However, it is very difficult to find interaction solutions among different types of nonlinear excitations. In this paper, with the help of the Riccati equation, the (2+1)-dimensional Konopelchenko—Dubrovsky equation is solved by the consistent Riccati expansion (CRE). Furthermore, we obtain the soliton-cnoidal wave interaction solution of the (2+1)-dimensional Konopelchenko—Dubrovsky equation.

Interactions between Solitons and Cnoidal Periodic Waves of the Gardner Equation

The Gardner equation is one of the most important prototypic models in nonlinear physics. Many scholars pay much attention to the Gardner equation and various nonlinear excitations of the Gardner equation have been found by many methods. However, it is very difficult to find interaction solutions among different types of nonlinear excitations. In this work, with the help of the Riccati equation, the Gardner equation is solved by the consistent Riccati expansion. Furthermore, we obtain the soliton-cnoidal wave interaction solutions of the Gardner equation.

Interactions between solitons and other nonlinear Schrödinger waves.

The nonlinear Schrödinger (NLS) equation is widely used in natural science. Various nonlinear excitations of the NLS equation have been found by many methods. However, except for the soliton-soliton interactions, it is very difficult to find interaction solutions between different types of nonlinear excitations. In this paper, the symmetry reduction method is further developed to find interaction solutions between solitons and other types of NLS waves. Especially, the soliton-cnoidal wave interaction solutions are explicitly studied in terms of the Jacobi elliptic functions and the third type of incomplete elliptic integrals. Some special concrete interaction solutions and their asymptotic behaviors are discussed both in analytical and graphical ways.

Interactions among Periodic Waves and Solitary Waves of the (2+1)-Dimensional Konopelchenko—Dubrovsky Equation

By using the truncated Painlevé analysis and the generalized tanh function expansion approaches, many interaction solutions among solitons and other types of nonlinear excitations of the Konopelchenko—Dubrovsky (KD) equation can be obtained. Particularly, the soliton-cnoidal wave interaction solutions are studied by means of the Jacobi elliptic functions and the third type of incomplete elliptic integrals.