Solitary-like Solutions Research Articles

Bifurcation and Solitary-Like Solutions for Compound KdV-Burgers-Type Equation

Firstly, based on the improved sub-ODE method and the bifurcation method of dynamical systems, we investigate the bifurcation of solitary waves in the compound KdV-Burgers-type equation. Secondly, numbers of solitary patterns solutions are given for each parameter condition and numerical simulations are used to display the dynamical characteristics. Finally, we obtain twelve solitary patterns solutions under some parameter conditions, such as the trigonometric function solutions and the hyperbolic function solutions.

Stability of steady gravity waves generated by a moving localised pressure disturbance in water of finite depth

We study the stability of the steady waves forced by a moving localised pressure disturbance in water of finite depth. The steady waves take the form of a downstream wavetrain for subcritical flow, but for supercritical flow there is a solution branch for each specified forcing term, which has two localised solitary-like solutions for each Froude number, a small-amplitude and a large-amplitude solution. Our main purpose is to numerically investigate the stability of these steady waves by simulating the unsteady development from appropriate initial conditions. We use a forced Korteweg-de Vries model valid in the transcritical regime for weak forcing, and a fully nonlinear boundary integral simulation. The simulations using the forced Korteweg-de Vries model are in good agreement with the fully nonlinear simulations when the Froude number is near unity for small pressure forcing, as expected. We find that the steady subcritical downstream wavetrain is stable. For supercritical flow, the small-amplitude solitary-like wave which has bifurcated from a uniform flow is stable, whereas the large-amplitude solitary-like wave which has bifurcated from a free solitary wave is unstable. Furthermore, here a difference between the forced Korteweg-de Vries model and the fully nonlinear simulations is revealed. For moderate and large pressure forcing, the forced Korteweg-de Vries model predicts a stable solitary wave moving away from the pressure forcing, while the nonlinear simulation shows that this wave evolves to a breaking wave.

New Jacobi elliptic functions solutions for the variable-coefficient mKdV equation

In this work, a new generalized Jacobi elliptic functions expansion method based upon four new Jacobi elliptic functions is described and abundant new Jacobi-like elliptic functions solutions for the variable-coefficient mKdV equation are obtained by using this method, some of these solutions are degenerated to solitary-like solutions and triangular-like functions solutions in the limit cases when the modulus of the Jacobi elliptic functions m → 1 or 0, which shows that the new method can be also used to solve other nonlinear partial differential equations in mathematical physics.

A note on a class of solitary-like solutions of the Tzitzéica equation generated by a similarity reduction

In this paper, new class of solutions of the Tzitzéica equation are derived by using the classical Lie symmetry analysis. The important aspect of this paper however is the fact that the analysis results in a new class of solitary-like solutions, the so-called cusp-solitary solutions. This special type of solutions are not found in the current literature and represents a necessary contribution for the whole solution manifold. The studied equation was originally found in the field of geometry, otherwise the Tzitzéica equation takes place in many branches of non-linear sciences. Therefore, explicit class of solutions connected by a physical meaning are of great importance. The analysis is restricted to the case of traveling waves represented by a similarity variable describing any wave propagation. A complete characterization of the group properties is given. The classical Lie point symmetries are derived and algebraic properties are determined. Similarity solutions and transformations are given in a most general form and have been derived for the first time in terms of Jacobian elliptic functions. It is worth to mention that the application of known powerful algebraic methods (e.g. special function transform methods) are not appropriate to study the solution manifold. Hence, the present paper is therefore suitable to create a deeper insight into the solution manifold with respect to the traveling wave picture.

The auxiliary equation for constructing the exact solutions of the variable coefficient combined KdV equation with forcible term

Based on the auxiliary equation method, a method of auxiliary equation of elliptic function combined with function transformation is proposed. The method is applied to constructed the Jacobi-like elliptic function exact solutions, degenerated solitary-like solutions and triangle-like function wave solutions of variable coefficient combined KdV equation with forcible term with the help of symbolic computation system Mathematica.

Nonlinear dynamics for magnetic systems with a single-spin potential with variable shapes

Results of computer simulations to investigate the spin dynamics of a classical ferromagnet subject to easy-plane anisotropy and a parameterized Zeeman energy are reported. The variability of the Zeeman term induces a deformable substrate potential in the system. The results show substantial deviations between the conventional sine-Gordon and deformable sine-Gordon description. Solitary-like solutions, shock waves, nanopteron waves and a variety of phenomena, including large easy-plane deviations are also observed. The results display ballistic, diffusive as well as stochastic behaviors. The region of parameters and limits of applicability of deformability effects induced by parameterization of the Zeeman energy are examined.

Surface solitons on elastic surfaces: numerics

In previous work of the authors, the Whitam-Newell technique of study of small-amplitude, modulated, almost monochromatic signals was used to prove the existence of stable surface-wave solitary-like solutions of the envelope type on a structure made of a nonlinear elastic substrate and a superimposed elastic thin film. In the present work we present the results of numerical simulations performed either on the model one-dimensional nonlinear Schrödinger equation, or directly on the original two-dimensional elastic system using analytic solutions of the former as initial conditions in the latter. The numerical experiments confirm the solitary-wave behavior together with exponential decrease of amplitude with depth, as well as a very good solitonic behavior in the original mechanical system in the course of collision of the nonlinear surface waves. Surface elastic shock waves can also be exhibited when compensation between nonlinearity and dispersion is not precisely achieved.