This paper presents an existence theory for small-amplitude solitary-wave solutions to the classical water-wave problem in the absence of surface tension and with an arbitrary distribution of vorticity. The hydrodynamic problem is formulated as an infinite-dimensional Hamiltonian system in which the horizontal spatial direction is the time-like variable. A centre-manifold reduction technique is...

(1) { ut + f(u)x + uxxx = 0 for x ∈ R, t > 0, u(x, 0) = u0(x) for x ∈ R, where f(u) = |u|p−1u/p (3 ≤ p < 5). I will show that if the speed of the solitary waves are sufficiently close at the initial time, the wave going ahead becomes larger and the wave going behind becomes smaller and the distance between two solitary waves becomes larger as t→∞. This gives an example of multi-pulse solution o...

In this paper, the modified exp-function method is used to seek generalized wave solutions of Klein-Gordon equation. As a result, some new types of exact traveling wave solutions are obtained which include kink wave solutions, periodic wave solution, and solitary wave solutions. Obtained results clearly indicate the reliability and efficiency of the proposed modified exp-function method.

and Applied Analysis 3 25 , the traveling wave solution of 1.2 is a damped oscillatory solution which has a bell profile head. However, they did not present any analytic solution or approximate solution for 1.2 . Xiong 25 obtained a kink profile solitary wave solution for KdV-Burgers equation. In 26 , S. D. Liu and S. D. Liu obtained an approximate damped oscillatory solution to a saddle-focus ...

By considering the long-wavelength limit of the regularized long wave ~RLW! equation, we study its multiple-time higher-order evolution equations. As a first result, the equations of the Korteweg–de Vries hierarchy are shown to play a crucial role in providing a secularity-free perturbation theory in the specific case of a solitary-wave solution. Then, as a consequence, we show that the related...

Via the semi-inverse method, variational principles are established for the Drinfeld-Sokolov-Wilson (DSW) equation. Based on the obtained variational formulations, exact solitary solution and exact singular periodic wave solution can be obtained using the variational approach.

We use a spectral method to solve numerically two nonlocal, nonlinear, dispersive, integrable wave equations, the Benjamin-Ono and the Intermediate Long Wave equations. The proposed numerical method is able to capture well the dynamics of the solutions; we use it to investigate the behaviour of solitary wave solutions of the equations with special attention to those, among the properties usuall...