Interactions Between Solitons and Cnoidal Periodic Waves of the Boussinesq Equation

The Boussinesq equation is one of important prototypic models in nonlinear physics.Various nonlinear excitations of the Boussinesq equation have been found by many methods.However,it is very difcult to find interaction solutions among diferent types of nonlinear excitations.In this peper,two equivalent very simple methods,the truncated Painlev′e analysis and the generalized tanh function expansion approaches,are developed to find interaction solutions between solitons and any other types of Boussinesq waves.     

Interactions Between Solitons and Cnoidal Periodic Waves of the Boussinesq Equation

The Boussinesq equation is one of important prototypic models in nonlinear physics. Various nonlinear excitations of the Boussinesq equation have been found by many methods. However, it is very difficult to find interaction solutions among different types of nonlinear excitations. In this peper, two equivalent very simple methods, the truncated Painlevé analysis and the generalized tanh function expansion approaches, are developed to find interaction solutions between solitons and any other types of Boussinesq waves.     

Interaction Behaviours Between Solitons and Cnoidal Periodic Waves for (2+1)-Dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada Equation

The consistent tanh expansion (CTE) method is employed to the (2+1)-dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada (CDGKS) equation. The interaction solutions between solitons and the cnoidal periodic waves are explicitly obtained. Concretely, we discuss a special kind of interaction solution in the form of tanh functions and Jacobian elliptic functions in both analytical and graphical ways. The results show that the profiles of the soliton-cnoidal periodic wave interaction solutions can be designed by choosing different values of wave parameters.     

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.     

Soliton molecules and the CRE method in the extended mKdV equation

The soliton molecules of the(1+1)-dimensional extended modified Korteweg–de Vries(mKdV)system are obtained by a new resonance condition, which is called velocity resonance. One soliton molecule and interaction between a soliton molecule and one-soliton are displayed by selecting suitable parameters. The soliton molecules including the bright and bright soliton, the dark and bright soliton, and the dark and dark soliton are exhibited in figures 1–3, respectively.Meanwhile, the nonlocal symmetry of the extended mKdV equation is derived by the truncated Painlevé method. The consistent Riccati expansion(CRE) method is applied to the extended mKdV equation. It demonstrates that the extended mKdV equation is a CRE solvable system. A nonauto-B?cklund theorem and interaction between one-soliton and cnoidal waves are generated by the CRE method.     

Various Kinds Waves and Solitons Interaction Solutions of Boussinesq Equation Describing Ultrashort Pulse in Quadratic Nonlinear Medium

The consistent tanh expansion (CTE) method is applied to the (2+1)-dimensional Boussinesq equation which describes the propagation of ultrashort pulse in quadratic nonlinear medium. The interaction solutions are explicitly given, such as the bright soliton-periodic wave interaction solution, variational amplitude periodic wave solution, and kink-periodic wave interaction solution. We also obtain the bright soliton solution, kind bright soliton solution, double well dark soliton solution and kink-bright soliton interaction solution by using Painlevé truncated expansion method. And we investigate interactive properties of solitons and periodic waves.     

Solitons on a Periodic Wave Background of the Modified KdV-Sine-Gordon Equation

The Bäcklund transformation(BT) of the mKdV-sG equation is constructed by introducing a new transformation. Infinitely many nonlocal symmetries are obtained in terms of its BT. The soliton-periodic wave interaction solutions are explicitly derived by the classical Lie-group reduction method. Particularly, some special concrete soliton and periodic wave interaction solutions and their behaviours are discussed both in analytical and graphical ways.     

An Approach for Solving Short-Wave Models for Camassa-Holm Equation and Degasperis-Procesi Equation

In this paper, to construct exact solution of nonlinear partial differential equation, an easy-to-use approach is proposed. By means of the transformation of the independent variables and the travelling wave transformation, the partial differential equation is reduced to an ordinary differential equation. To solve the ordinary differential equation, we assume the soliton solution in the explicit expression and obtain the travelling wave solution. By the transformation back to the original independent variables, the soliton solution of the original partial differential equation is derived. We investigate the short wave model for the Camassa-Holm equation and the Degasperis-Procesi equation respectively. One-cusp soliton solution of the Camassa-Flolm equation is obtained. One-loop soliton solution of the Degasperis- Procesi equation is also obtained, the approximation of which in a closed form can be obtained firstly by the Adomian decomposition method. The obtained results in a parametric form coincide perfectly with those given in the present reference. This illustrates the efficiency and reliability of our approach.     

An Approach for Solving Short-Wave Models for Camassa-Holm Equation and Degasperis-Procesi Equation

In this paper, to construct exact solution of nonlinear partial differential equation, an easy-to-use approach is proposed. By means of the transformation of the independent variables and the travelling wave transformation, the partial differential equation is reduced to an ordinary differential equation. To solve the ordinary differential equation, we assume the soliton solution in the explicit expression and obtain the travelling wave solution. By the transformation back to the original independent variables, the soliton solution of the original partial differential equation is derived. We investigate the short wave model for the Camassa-Holm equation and the Degasperis-Procesi equation respectively. One-cusp soliton solution of the Camassa-Holm equation is obtained. One-loop soliton solution of the Degasperis-Procesi equation is also obtained, the approximation of which in a closed form can be obtained firstly by the Adomian decomposition method. The obtained results in a parametric form coincide perfectly with those given in the present reference. This illustrates the efficiency and reliability of our approach.     

New Exact Periodic Solitary-Wave Solutions for New (2+1)-Dimensional KdV Equation

The extended homoclinic test function method is a kind of classic, efficient and well-developed method to solve nonlinear evolution equations. In this paper, with the help of this approach, we obtain new exact solutions (including kinky periodic solitary-wave solutions, periodic soliton solutions, and crosskink-wave solutions) for the new (2+1)-dimensional KdV equation. These results enrich the variety of the dynamics of higher-dimensional nonlinear wave field.     

Bifurcation and Solitary Waves of the Combined KdV and KdV Equation

Bifurcation, bistability and solitary waves of the combined KdV and mKdV equation are investigatedsystematically. At first, bifurcation and bistability are analyzed by selecting an integral constant as the bifurcationparameter. Then, different conditions expressed in terms of the bifurcation parameter are obtained for the existence ofbreather-like, algebraic, pulse-like solitary waves, and shock waves. All types of the solitary wave and shock wave solutionsare given by direct integration. Finally, an approximate analytic method by employing the interpolation polynomials iscomplete and the theoretical methods are the simplest hitherto.     

Solitary Solution of Discrete mKdV Equation by Homotopy Analysis Method

In this paper, we apply homotopy analysis method to solve discrete mKdV equation and successfully obtain the bell-shaped solitary solution to mKdV equation. Comparison between our solution and the exact solution shows that homotopy analysis method is effective and validity in solving hybrid nonlinear problems, including solitary solution of difference-differential equation.     

Bifurcation and Solitary Waves of the Combined KdV and mKdV Equation

Bifurcation, bistability and solitary waves of the combined KdV and mKdV equation are investigated systematically. At first, bifurcation and bistability are analyzed by selecting an integral constant as the bifurcation parameter. Then, different conditions expressed in terms of the bifurcation parameter are obtained for the existence of breather-like, algebraic, pulse-like solitary waves, and shock waves. All types of the solitary wave and shock wave solutions are given by direct integration. Finally, an approximate analytic method by employing the interpolation polynomials is utilized to give simpler forms for the pulse-like solitary wave solutions. In view of the references, our results are the most complete and the theoretical methods are the simplest hitherto.     

New Generalized Transformation Method and Its Application in Higher-Dimensional Soliton Equation

A new generalized transformation method is presented to find more exact solutions of nonlinear partial differential equation. As an application of the method, we choose the (3+1)-dimensional breaking soliton equation to illustrate the method. As a result many types of explicit and exact traveling wave solutions, which contain solitary wave solutions, trigonometric function solutions, Jacobian elliptic function solutions, and rational solutions, are obtained. The new method can be extended to other nonlinear partial differential equations in mathematical physics.     

Doubly Periodic Propagating Wave for （2d-1）-Dimensional Breaking Soliton Equation

Using the variable separation approach, we obtain a general exact solution with arbitrary variable separation functions for the （2＋1）-dimensional breaking soliton system. By introducing Jacobi elliptic functions in the seed solution, two families of doubly periodic propagating wave patterns are derived. We investigate these periodic wave solutions with different modulus m selections, many important and interesting properties are revealed. The interaction of Jabcobi elliptic function waves are graphically considered and found to be nonelastic.     

Solitons and Waves in (2+1)-Dimensional Dispersive Long-Wave Equation

For a higher-dimensional integrable nonlinear dynamical system, there are abundant coherent soliton excitations. With the aid of an improved projective Riccati equation approach, the paper obtains several types of exact solutions to the (2 1)-dimensional dispersive long-wave equation, including multiple-soliton solutions, periodic soliton solutions, and Weierstrass function solutions. From these solutions, apart from several multisoliton excitations, we derive some novel features of wave structures by introducing some types of lower-dimensional patterns.     

Solitons and Waves in （2＋l）-Dimensional Dispersive Long-Wave Equation

For a higher-dimensional integrable nonlinear dynamical system, there are abundant coherent soliton excitations. With the aid of an improved projective Riccati equation approach, the paper obtains several types of exact solutions to the （2＋l）-dimenslonal dispersive long-wave equation, including multiple-soliton solutions, periodic soliton solutions, and Weierstrass function solutions. From these solutions, apart from several multisoliton excitations, we derive some novel features of wave structures by introducing some types of lower-dimensional patterns.     

Rational and Periodic Wave Solutions of Two-Dimensional Boussinesq Equation

Two new exact, rational and periodic wave solutions are derived for the two-dimensional Boussinesq equation. For the first solution it is obtained by performing an appropriate limiting procedure on the soliton solutions obtained by Hirota bilinear method. The second one in terms of Riemann theta function is explicitly presented by virtue of Hirota bilinear method and its asymptotic property is also analyzed in detail. Moreover, it is of interest to note that classical soliton solutions can be reduced from the periodic wave solutions.