New exact solution structures and nonlinear dispersion in the coupled nonlinear wave systems

In this Letter, to further understand the role of nonlinear dispersion in coupled nonlinear wave systems in both real and complex fields, we study the coupled Klein–Gordon equations with nonlinear dispersion in real field (called CKG(m,n,k)$\mathrm{CKG}(m,n,k)$ equation) and (2+1)$(2+1)$-dimensional generalization of coupled nonlinear Schrödinger equation with nonlinear dispersion in complex field (called GCNLS(m,n,k)$\mathrm{GCNLS}(m,n,k)$ equation) via some transformations. As a consequence, some types of solutions are obtained, which contain compactons, solitary pattern solutions, envelope compacton solutions, envelope solitary pattern solutions, solitary wave solutions and rational solutions.     

Travelling wave solutions for a class of generalized KdV equation

In this work, the K^∗(l,p) equation is investigated. The sine-cosine method, the tanh method and the extended tanh method are efficiently used for analytic study of this equation. New solitary patterns solutions and compactons solutions are formally derived. The proposed schemes are reliable and manageable.     

Elastic rods, Weierstrass’ theory and special travelling waves solutions with compact support

By an extension of the usual Weierstrass discussion for first-order differential equations, we determine a simple criterion which allows us to determine when compact structures are possible in second-order wave equations. For higher-order wave equations, such as the modified improved Boussinesq equation, this criterion may still be used, but the appearance of an overdetermined system of differential equations which is not trivially satisfied, unlike in the second-order case, renders the possibility of compact structures a happenstance.     

Bifurcations of travelling wave solutions for the generalized KP-BBM equation

By using the bifurcation theory of dynamical systems to the generalized Kadomtsov-Petviashvili-Benjamin-Bona-Mahony equation, the existence of solitary wave solutions, compactons solution, non-smooth periodic cusp wave solutions and uncountably infinite many smooth periodic wave solutions is obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given. Some exact explicit parametric representations of the above waves are determined.     

A repository of equations with cosine/sine compactons

Nonlinear evolution equations with cosine/sine compacton solutions are reviewed, including the Rosenau-Hyman equation and generalizations of Korteweg-de Vries, Camassa-Holm, Boussinesq, Benjamin-Bona-Mahony, Klein-Gordon and other equations. Each equation is generalized to three dimensions and the conditions for its cosine solitary waves to be either a compacton or a soliton are determined. Several equations claimed in the literature to be different among them are found to be equivalent.     

Bifurcations of travelling wave solutions for modified nonlinear dispersive phi-four equation

By using the bifurcation theory of dynamical systems to modified nonlinear dispersive phi-four equation, we analysis all bifurcations and phase portraits in the parametric space, the existence of solitary wave solutions and uncountably infinite many smooth and non-smooth periodic wave solutions is obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given. Some explicit exact solution formulas are acquired for some special cases.     

Qualitative Analysis and Periodic Cusp Waves to a Class of Generalized Short Pulse Equations

In this paper, we qualitatively study periodic cusp waves to a class of generalized short pulse equations, which are of the general form of three special generalized short pulse equations, from the perspective of dynamical systems. We show the existence of smooth periodic waves, periodic cusp wave and compactons, obtain exact expression of periodic cusp wave and illustrate the limiting process of periodic cusp wave from smooth periodic waves.     

Compactons structures for fifth-order KdV like equations in higher dimensions

In this paper we study a variant of the fifth-order KdV equation (fKdV) that exhibits compactons: solitons with finite wave lengths. The work formally shows how to construct compact dispersive structures in higher dimensions. Two sets of general formulas for compactons solutions, that are of substantial interest, are developed for this variant fK(n,n) for all positive integers n, n1.     

Some New Expressions of Compactons in Compressible Elastic Rods

In this paper, we use phase plane analysis to study the compactons of the nonlinear equation. Four new implicit expressions of the compactons are obtained. These new implicit expressions are given by inverse tangent functions. Our work extends previous results. For two sets of the data, the graphs of the implicit functions are drawn and numerical simulations are given to test the correctness of our theoretical results.