Existence Of Solitary Wave Solutions Research Articles

Bifurcations of travelling wave solutions for the nonlinear dispersion Drinfel’d–Sokolov ( [formula omitted]) system

By using the bifurcation theory of planar dynamical systems to the nonlinear dispersion Drinfel’d–Sokolov ( D ( m , n ) ) system, the existence of solitary wave solutions, kink and anti-kink wave solutions, compacton solutions and uncountably infinite many smooth and non-smooth periodic wave solutions is proved. Under different regions of parametric spaces, various sufficient conditions to guarantee the existence of above solutions are given. Some exact explicit parametric representations of the above waves are determined.

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.

Bifurcations of travelling wave solutions for a two-component camassa-holm equation

By using the method of dynamical systems to the two-component generalization of the Camassa-Holm equation, the existence of solitary wave solutions, kink and anti-kink wave solutions, and uncountably infinite many breaking wave solutions, 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 exact explicit parametric representations of travelling wave solutions are listed.

Three kinds of nonlinear dispersive waves in elastic rods with finite deformation

On the basis of classical linear theory on longitudinal, torsional and flexural waves in thin elastic rods, and taking finite deformation and dispersive effects into consideration, three kinds of nonlinear evolution equations are derived. Qualitative analysis of three kinds of nonlinear equations are presented. It is shown that these equations have homoclinic or heteroclinic orbits on the phase plane, corresponding to solitary wave or shock wave solutions, respectively. Based on the principle of homogeneous balance, these equations are solved with the Jacobi elliptic function expansion method. Results show that existence of solitary wave solution and shock wave solution is possible under certain conditions. These conclusions are consistent with qualitative analysis.

Bifurcations of travelling wave solutions for the [formula omitted] equations

By using the bifurcation theory of dynamical systems to the K ( n , - n , 2 n ) equations given by Wazwaz, the existence of solitary wave solutions, solitary cusp wave solutions, kink and anti-kink 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 exact explicit parametric representations of the above waves are determined.

Solitary waves of the generalized KdV equation with distributed delays

In this paper, we study an integro-differential equation based on the generalized KdV equation with a convolution term which introduces a time delay in the nonlinearity. Special attention is paid to the existence of solitary wave solutions. Motivated by [M.J. Ablowitz, H. Seger, Soliton and Inverse Scattering Transform, SIAM, Philadelphia, 1981; C.K.R.T. Jones, Geometrical singular perturbation theory, in: R. Johnson (Ed.), Dynamical Systems, in: Lecture Notes in Math., vol. 1609, Springer, New York, 1995; T. Ogawa, Travelling wave solutions to perturbed Korteweg–de Vries equations, Hiroshima Math. J. 24 (1994) 401–422], we prove, using the linear chain trick and geometric singular perturbation analysis, that the solitary wave solutions persist when the average delay is suitably small, for a special convolution kernel.

Travelling wave solutions for a second order wave equation of KdV type

The theory of planar dynamical systems is used to study the dynamical behaviours of travelling wave solutions of a nonlinear wave equations of KdV type.In different regions of the parametric space,sufficient conditions to guarantee the existence of solitary wave solutions,periodic wave solutions,kink and anti-kink wave solutions are given.All possible exact explicit parametric representations are obtained for these waves.

On the Existence of Solitary Wave Solutions to the Shallow-Water Equations on the f Plane

Abstract In a recent paper Li argued that the inviscid shallow-water equations on the f plane exhibit a class of zonally propagating solitary wave solutions, referred to by the author as “geosolitary waves.” These solutions have propagation speeds that are slower than the shallow-water speed gH, and as constructed by Li, have a cusp in the height and zonal velocity field at the origin. The meridional velocity is found to be discontinuous at the origin, though its magnitude is much smaller than the magnitude of the zonal velocity. The authors reconsider the existence of geosolitary waves from the point of view of dynamical system theory and demonstrate that the cusp discontinuity at the origin is an unavoidable property of the solutions. Moreover, it is shown that, in contrast to suggestions made by Li, adding viscosity and diffusivity does not yield smooth, bounded solitary wave solutions. Last, the reason why geosolitary waves cannot be expected to exist in a continuously stratified fluid is discussed.

Bifurcations of travelling wave solutions in a non-linear dispersive equation

By using the theory of bifurcations of dynamical systems to a class of the generalized Benjamin–Bona–Mahony (GBBM) equation, 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.

Bifurcations of travelling wave solutions in a class of generalized KdV equation

By using the bifurcation theory of dynamical systems to a generalized KdV equation given by Cooper et al., the existence of solitary wave solutions, solitary cusp 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 exact explicit parametric representations of the above waves are determined.

Bifurcations of traveling wave solutions for a class of the generalized Benjamin–Bona–Mahony equation

By using the theory of bifurcations of dynamical systems to a class of the generalized Benjamin–Bona–Mahony (GBBM) equation, the existence of solitary wave solutions and uncountably infinite, many smooth and non-smooth periodic wave solutions are obtained. It can be shown that the existence of singular curves in a traveling wave system is the reason why smooth waves converge to cusp waves, finally. When parameters are varied, under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given.

Bifurcations of travelling wave solutions for the Boussinesq equations

By using the theory of bifurcations of dynamical systems to the generalized Boussinesq equations, 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.

Solitary waves in nonlinear microstructured materials

Dispersive effects due to microstructure of materials combined with nonlinearities give rise to solitary waves. In this paper the existence of solitary wave solutions is proved for a rather general hierarchical governing equation which accounts for nonlinearities on both macro- and microscales. Properties of the waves are established. Waves are asymmetric in the case of the nonlinearity in the microscale. Dispersive effects are due to the scale dependence.

The existence of solitary waves of singularly perturbed mKdV–KS equation

We study a sort of nonlinear reaction diffusion equation based on the modified Korteweg–de Vries (mKdV) equation with a higher order singularly perturbing term as the Kuramoto–Sivashinsky (KS) equation, called mKdV–KS equation. Special attention is paid to the question of the existence of solitary wave solutions. Based on the analogue between solitary wave solution and homoclinic orbits of the associated ordinary differential equations, from geometric singular perturbation point of view, we prove that solitary wave persists when the perturbation parameter is suitably small. This argument does not require an explicit expression for the original mKdV solitary wave solution.

Bifurcations of traveling wave solutions in a compound KdV-type equation

By using the theory of planar dynamical systems to a compound KdV-type nonlinear wave equation, the bifurcation boundaries of the system are obtained in this paper. These bifurcation sets divide the parameter space into different regions, which correspond to qualitatively different phase portraits and therefore different types of the solutions may exist in different regions. The parameter conditions for the existence of solitary wave solutions and uncountably infinite, many smooth and non-smooth, periodic wave solutions are therefore obtained.

Bifurcations of traveling wave solutions for two coupled variant Boussinesq equations in shallow water waves

Abstract The bifurcations of traveling wave solutions for two coupled variant Boussinesq equations introduced as a model for water waves are studied in this paper. Transition boundaries have been presented to divide the parameter space into different regions associated with qualitatively different types of solutions. The conditions for the existence of solitary wave solutions and uncountably infinite, smooth, non-smooth and periodic wave solutions are obtained. The explicit exact traveling wave solutions are presented by using an algebraic method.

Bifurcations of traveling wave solutions in a compound KdV-type equation

Abstract By using the theory of planar dynamical systems to a compound KdV-type nonlinear wave equation, the bifurcation boundaries of the system are obtained in this paper. These bifurcation sets divide the parameter space into different regions, which correspond to qualitatively different phase portraits and therefore different types of the solutions may exist in different regions. The parameter conditions for the existence of solitary wave solutions and uncountably infinite, many smooth and non-smooth, periodic wave solutions are therefore obtained.

Bifurcations of travelling wave solutions for the generalized Kadomtsev–Petviashili equation

By using the theory of bifurcations of dynamical systems to the generalized Kadomtsev–Petviashili equation, 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.

Bifurcations of traveling wave solutions in a coupled non-linear wave equation

By using the theory of planar dynamical systems to a coupled non-linear wave equation, 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.

Solitary-wave and multi-pulsed traveling-wave solutions of boussinesq systems

This paper studies traveling-wave solutions of the partial differential equations which model waves in a horizontal water channel traveling in both directions. The existence of solitary-wave solutions for a number of systems is proved. Interesting new multi-pulsed traveling-wave solutions which consist of an arbitrary number of troughs are found numerically. The bifurcation diagrams of N-trough solutions with respect to phase speed and system parameters are presented.