Diversity of Rogue Wave Solutions to the (1+1)-Dimensional Boussinesq Equation

A periodically homoclinic solution and some rogue wave solutions of (1+1)-dimensional Boussinesq equation are obtained via the limit behavior of parameters and different polynomial functions. Besides, the mathematics reasons for different spatiotemporal structures of rogue waves are analyzed using the extreme value theory of the two-variables function. The diversity of spatiotemporal structures not only depends on the disturbance parameter u0 but also has a relationship with the other parameters c0, α, β.

Influence of dissipation on solitary wave solution to generalized Boussinesq equation

This paper uses the theory of planar dynamic systems and the knowledge of reaction-diffusion equations,and then studies the bounded traveling wave solution of the generalized Boussinesq equation affected by dissipation and the influence of dissipation on solitary waves.The dynamic system corresponding to the traveling wave solution of the equation is qualitatively analyzed in detail.The influence of the dissipation coefficient on the solution behavior of the bounded traveling wave is studied,and the critical values that can describe the magnitude of the dissipation effect are,respectively,found for the two cases of b_3<0 and b_3>0 in the equation.The results show that,when the dissipation effect is significant(i.e.,r is greater than the critical value in a certain situation),the traveling wave solution to the generalized Boussinesq equation appears as a kink-shaped solitary wave solution;when the dissipation effect is small(i.e.,r is smaller than the critical value in a certain situation),the traveling wave solution to the equation appears as the oscillation attenuation solution.By using the hypothesis undetermined method,all possible solitary wave solutions to the equation when there is no dissipation effect(i.e.,r=0)and the partial kink-shaped solitary wave solution when the dissipation effect is significant are obtained;in particular,when the dissipation effect is small,an approximate solution of the oscillation attenuation solution can be achieved.This paper is further based on the idea of the homogenization principles.By establishing an integral equation reflecting the relationship between the approximate solution of the oscillation attenuation solution and the exact solution obtained in the paper,and by investigating the asymptotic behavior of the solution at infinity,the error estimate between the approximate solution of the oscillation attenuation solution and the exact solution is obtained,which is an infinitesimal amount that decays exponentially.The influence of the dissipation coefficient on the amplitude,frequency,period,and energy of the bounded traveling wave solution of the equation is also discussed.

Dispersive propagation of optical solitions and solitary wave solutions of Kundu-Eckhaus dynamical equation via modified mathematical method

In this research work,we constructed the optical soliton solutions of nonlinear complex Kundu-Eckhaus(KE)equation with the help of modified mathematical method.We obtained the solutions in the form of dark solitons,bright solitons and combined dark-bright solitons,travelling wave and periodic wave solutions with general coefficients.In our knowledge earlier reported results of the KE equation with specific coefficients.These obtained solutions are more useful in the development of optical fibers,dynamics of solitons,dynamics of adiabatic parameters,dynamics of fluid,problems of biomedical,industrial phenomena and many other branches.All calculations show that this technique is more powerful,effective,straightforward,and fruitfulness to study analytically other higher-order nonlinear complex PDEs involves in mathematical physics,quantum physics,Geo physics,fluid mechanics,hydrodynamics,mathematical biology,field of engineering and many other physical sciences.

New Explicit Solitary Wave Solutions and Periodic Wave Solutions for the Generalized Coupled Hirota-Satsuma KdV System

In this paper, we study the generalized coupled Hirota Satsuma KdV system by using the new generalizedtransformation in homogeneous balance method. As a result, many explicit exact solutions, which contain new solitarywave solutions, periodic wave solutions, and the combined formal solitary wave solutions, and periodic wave solutions,are obtained.

Discrete doubly periodic and solitary wave solutions for the semi-discrete coupled mKdV equations

In this paper, the improved Jacobian elliptic function expansion approach is extended and applied to constructing discrete solutions of the semi-discrete coupled modified Korteweg de Vries （mKdV） equations with the aid of the symbolic computation system Maple. Some new discrete Jacobian doubly periodic solutions are obtained. When the modulus m →1, these doubly periodic solutions degenerate into the corresponding solitary wave solutions, including kink-type, bell-type and other types of excitations.

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 methodto 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 cross kink-wave solutions)for the new(2+1)-dimensional KdV equation.These results enrich the variety of the dynamics of higher-dimensional nonlinear wavefield.

Solitary Wave and Periodic Wave Solutions for the Relativistic Toda Lattices

In this work, an adaptation of the tanh/tan-method that is discussed usually in the nonlinear partial differential equations is presented to solve nonlinear polynomial differential-difference equations. As a concrete example,several solitary wave and periodic wave solutions for the chain which is related to the relativistic Toda lattice are derived.Some systems of the differential-difference equations that can be solved using our approach are listed and a discussion is given in conclusion.

Periodic Solitary Wave Solutions of the (2 + 1)-Dimensional Variable-Coefficient Caudrey-Dodd-Gibbon-Kotera-Sawada Equation

In this paper, through symbolic computations, we obtain two exact solitary wave solitons of the (2 + l)-dimensional variable-coefficient Caudrey-Dodd- Gibbon-Kotera-Sawada equation. We study basic properties of l-periodic solitary wave solution and interactional properties of 2-periodic solitary wave solution by using asymptotic analysis.

Rational Form Solitary Wave Solutions and Doubly Periodic Wave Solutions to （1＋1）-Dimensional Dispersive Long Wave Equation

In this work we devise an algebraic method to uniformly construct rational form solitary wave solutions and Jacobi and Weierstrass doubly periodic wave solutions of physical interest for nonlinear evolution equations. With the aid of symbolic computation, we apply the proposed method to solving the (1+1)-dimensional dispersive long wave equation and explicitly construct a series of exact solutions which include the rational form solitary wave solutions and elliptic doubly periodic wave solutions as special cases.

Exact explicit solitary wave and periodic wave solutions and their dynamical behaviors for the Schamel–Korteweg–de Vries equation

The Schamel–Korteweg–de Vries equation is investigated by the approach of dynamics.The existences of solitary wave including ω-shape solitary wave and periodic wave are proved via investigating the dynamical behaviors with phase space analyses.The sufficient conditions to guarantee the existences of the above solutions in different regions of the parametric space are given.All possible exact explicit parametric representations of the waves are also presented.Along with the details of the analyses,the analytical results are numerically simulated lastly.

Exact transverse solitary and periodic wave solutions in a coupled nonlinear inductor-capacitor network

Through two methods, we investigate the solitary and periodic wave solutions of the differential equation describing a nonlinear coupled two-dimensional discrete electrical lattice. The fixed points of our model equation are examined and the bifurcations of phase portraits of this equation for various values of the front wave velocity are presented. Using the sineGordon expansion method and classic integration, we obtain exact transverse solutions including breathers, bright solitons,and periodic solutions.

New Periodic Solitary Wave Solutions for a Variable-Coefficient Gardner Equation from Fluid Dynamics and Plasma Physics

The Gardner equation with a variable-coefficient from fluid dynamics and plasma physics is investigated. Different kinds of solutions including breather-type soliton and two soliton solutions are obtained using bilinear method and extended homoclinic test approach. The proposed method can also be applied to solve other types of higher dimensional integrable and non-integrable systems.

Combined periodic wave and solitary wave solutions in two-component Bose-Einstein condensates

In this paper, the Jacobi elliptic function expansion method provides an effective approach to obtain the exact periodic wave solutions of two-component Bose-Einstein condensates. Exact combined bright-bright and dark-dark soliton wave solutions can be achieved in their limit conditions. We also obtain the different formation regions of combined solitons. Our results show that the intraspecies （interspecies） interaction strengths clearly affect the formation of dar^dark, bright-bright and dark-bright soliton solutions in different regions.

The Periodic Solitary Wave Solutions for the (2 + 1)-Dimensional Fifth-Order KdV Equation

The (2 + 1)-dimensional fifth-order KdV equation is an important higher-dimensional and higher-order extension of the famous KdV equation in fluid dynamics. In this paper, by constructing new test functions, we investigate the periodic solitary wave solutions for the (2 + 1)-dimensional fifth-order KdV equation by virtue of the Hirota bilinear form. Several novel analytic solutions for such a model are obtained and verified with the help of symbolic computation.

Integrability, Multi-Solitary Wave Solutions and Riemann Theta Functions Periodic Wave Solutions of the Newell Equation

This paper systematically studies the complete integrability of the Newell equation. Using generalized Bell polynomials, the corresponding bilinear equation, bilinear B&#228;cklund transformation, Lax pair, and multi-shock wave solutions are successfully obtained. In addition, using the multidimensional Riemann theta functions, the periodic wave solutions of the Newell equation are constructed. On this basis, the asymptotic behavior of the periodic wave solution is given, which is the relationship between the periodic wave solution and the solitary wave solution.

Periodic Wave Solutions and Solitary Wave Solutions of the (2+1)-Dimensional Korteweg-de-Vries Equatio

In this paper, we investigate the periodic wave solutions and solitary wave solutions of a (2+1)-dimensional Korteweg-de Vries (KDV) equation by applying Jacobi elliptic function expansion method. Abundant types of Jacobi elliptic function solutions are obtained by choosing different coefficients p, q and r in the elliptic equation. Then these solutions are coupled into an auxiliary equation and substituted into the (2+1)-dimensional KDV equation. As a result, a large number of complex Jacobi elliptic function solutions are obtained, and many of them have not been found in other documents. As , some complex solitary solutions are also obtained correspondingly. These solutions that we obtained in this paper will be helpful to understand the physics of the (2+1)-dimensional KDV equation.

New exact periodic solutions to (2+1)-dimensional dispersive long wave equations

In this paper, we make use of the auxiliary equation and the expanded mapping methods to find the new exact periodic solutions for （2＋1）-dimensional dispersive long wave equations in mathematical physics, which are expressed by Jacobi elliptic functions, and obtain some new solitary wave solutions （m → 1）. This method can also be used to explore new periodic wave solutions for other nonlinear evolution equations.

Applications of F-expansion to Periodic Wave Solutions for Variant Boussinesq Equations

We present an F-expansion method for finding periodic wave solutions of nonlinear evolution equations in mathematical physics, which can be thought of as a concentration of extended Jacobi elliptic function expansion method proposed recently. By using the F-expansion, without calculating Jacobi elliptic functions, we obtain simultaneously many periodic wave solutions expressed by various Jacobi elliptic functions for the variant Boussinesq equations. When the modulus m approaches 1 and 0, the hyperbolic function solutions (including the solitary wave solutions) and trigonometric solutions are also given respectively.

Riemann theta function periodic wave solutions for the variable-coefficient mKdV equation

In this paper, a variable-coefficient modified Korteweg-de Vries （vc-mKdV） equation is considered. Bilinear forms are presented to explicitly construct periodic wave solutions based on a multidimensional Riemann theta function, then the one and two periodic wave solutions are presented~ and it is also shown that the soliton solutions can be reduced from the periodic wave solutions.

Binary Bell Polynomials,Bilinear Approach to Exact Periodic Wave Solutions of(2+l)-Dimensional Nonlinear Evolution Equations

In the present letter,we get the appropriate bilinear forms of(2 + l)-dimensionaI KdV equation,extended (2+1)-dimensional shallow water wave equation and(2 +1)-dimensional Sawada-Kotera equation in a quick and natural manner,namely by appling the binary Bell polynomials.Then the Hirota direct method and Riemann theta function are combined to construct the periodic wave solutions of the three types nonlinear evolution equations.And the corresponding figures of the periodic wave solutions are given.Furthermore,the asymptotic properties of the periodic wave solutions indicate that the soliton solutions can be derived from the periodic wave solutions.