In this paper, a moving boundary technique is developed to investigate wave runup and rundown with depth-integratedequations. Highly nonlinear and weakly dispersive equations are solved using a high-order finite difference scheme. An eddyviscosity model is adopted for wave breaking so as to investigate breaking wave runup. The moving boundary technique utilizeslinear extrapolation t...

A generalized Fisher equation involving a nonlinear term of any order and timedependent coefficients is investigated. We present exact analytical solutions describing periodic and solitary wave solutions using the modified sine-cosine method. A certain class of exact soliton-like solutions has been found by means of the auxiliary equation method. The conditions of existence and uniqueness of th...

Soliton interactions for the extended Korteweg-de Vries (KdV) equation are examined. It is shown that the extended KdV equation can be transformed (to its order of approximation) to a higher-order member of the KdV hierarchy of integrable equations. This transformation is used to derive the higher-order, two-soliton solution for the extended KdV equation. Hence it follows that the higher-order ...

We study the singularly perturbed (sixth-order) Boussinesq equation recently introduced by Daripa and Hua [Appl. Math. Comput. 101 (1999), 159-207]. This equation describes the bi-directional propagation of small amplitude and long capillary-gravity waves on the surface of shallow water for Bond number less than but very close to 1/3. On the basis of far-field analyses and heuristic arguments, ...

The long-time asymptotics is analyzed for finite energy solutions of the 1D Schrödinger equation coupled to a nonlinear oscillator. The coupled system is invariant with respect to the phase rotation group U(1). For initial states close to a solitary wave, the solution converges to a sum of another solitary wave and dispersive wave which is a solution to the free Schrödinger equation. The proofs...

The Exp-function method can be used as an alternative to obtain analytic and approximate solutions of different types of differential equations. In this work Exp -function method is used to find exact solitary wave solutions for Fitzhugh-Nagumo equation. The method is straightforward and concise and its applications for Fitzhugh-Nagumo equation enable one to find exact traveling wave solutions....

The complete analytical solution of the governing nonlinear vector equations is found which describes periodic and solitary propagating-rotating waves in an initially helical fiber. A detailed description of various types of these waves is given. The solitary wave velocity is found to be proportional to the square root of the amplitude of the internal force, and the effective wave length is sho...

We consider some Boussinesq systems of water wave theory, which are of coupled KdV type. After a brief review of the theory of existence-uniqueness of solutions of the associated initial-value problems, we turn to the numerical solution of their initialand periodic boundary-value problems by unconditionally stable, highly accurate methods that use Galerkin/finite element type schemes with perio...

Axisymmetric disturbances that preserve their form as they move along the vortex lines in uniform Bose-Einstein condensates are obtained numerically by the solution of the Gross-Pitaevskii equation. A continuous family of such solitary waves is shown in the momentum (p)-substitution energy (Epsilon) plane with p-->0.09 rho kappa(3)/c(2), Epsilon-->0.091 rho kappa(3)/c as U-->c, where rho is the...

We find new analytic solitary-wave solutions of the higher order wave equations of Korteweg – De Vries (KdV) type (I), using the auxiliary function method. We study the dynamical properties of the solitary-waves by numerical simulations. It is shown that the solitary-waves are stable for wide ranges of the model coefficients. We study the dynamics of the two solitary-waves by using the analytic...