Application of (G/G') -expansion method to the compound Kdv-burgers type equations

In this Letter, the (G'/G)-expansion method is proposed to seek exact solutions of nonlinear evolution equations. For illustrative examples, we choose the compound KdV-Burgers equation, the compound KdV equation, the KdV-Burgers equation, the mKdV equation. The power of the employed method is confirmed

Solitary-wave solutions of the Degasperis-Procesi equation by means of the homotopy analysis method

The homotopy analysis method is applied to the Degasperis-Procesi equation in order to find analytic approximations to the known exact solitary-wave solutions for the solitary peakon wave and the family of solitary smooth-hump waves. It is demonstrated that the approximate solutions agree well with the exact solutions. This provides further evidence that the homotopy analysis method is a powerful tool for finding excellent approximations to nonlinear solitary waves

On asymptotically equivalent shallow water wave equations

The integrable 3rd-order Korteweg-de Vries (KdV) equation emerges uniquely at linear order in the asymptotic expansion for unidirectional shallow water waves. However, at quadratic order, this asymptotic expansion produces an entire {\it family} of shallow water wave equations that are asymptotically equivalent to each other, under a group of nonlinear, nonlocal, normal-form transformations introduced by Kodama in combination with the application of the Helmholtz-operator. These Kodama-Helmholtz transformations are used to present connections between shallow water waves, the integrable 5th-order Korteweg-de Vries equation, and a generalization of the Camassa-Holm (CH) equation that contains an additional integrable case. The dispersion relation of the full water wave problem and any equation in this family agree to 5th order. The travelling wave solutions of the CH equation are shown to agree to 5th order with the exact solution

Well-posedness of KdV type equations

In this work, we study the initial value problems associated to some linear perturbations of the KdV equations. Our focus is in the well-posedness issues for the initial data given in the $L^2$-based Sobolev spaces. We develop a method that allows us to treat the problem in the Bourgain's space associated to the KdV equation. With this method, we can use the multilinear estimates developed in the KdV context thereby getting analogous well-posedness results for the linearly perturbed equations.Fundação para a Ciência e a Tecnologia (FCT