Higher-order Dispersive Equation Research Articles

Massera’s Theorems for a Higher Order Dispersive System

This work is devoted to present Massera-type theorems for the Kawahara system, a higher order dispersive equation, posed in a bounded domain. Precisely, thanks to some properties of the semigroup and the decays of the solutions of this equation, we are able to prove its solutions are periodic, quasi-periodic and almost periodic.

Pseudolocalized three-dimensional solitary waves as quasi-particles

A higher-order dispersive equation is introduced as a candidate for the governing equation of a field theory. A new class of solutions of the three-dimensional field equation are considered, which are not localized functions in the sense of the integrability of the square of the profile over an infinite domain. For this new class of solutions, the gradient and/or the Hessian/Laplacian are square integrable. In the linear limiting case, an analytical expression for the pseudolocalized solution is found and the method of variational approximation is applied to find the dynamics of the centers of the quasi-particles (QPs) corresponding to these solutions. A discrete Lagrangian can be derived due to the localization of the gradient and the Laplacian of the profile. The equations of motion of the QPs are derived from the discrete Lagrangian. The pseudomass (“wave mass”) of a QP is defined as well as the potential of interaction. The most important trait of the new QPs is that, at large distances, the force of attraction is proportional to the inverse square of the distance between the QPs. This can be considered analogous to the gravitational force in classical mechanics.

The Cauchy problem for the seventh-order dispersive equation in Sobolev space

This paper is devoted to the Cauchy problem for the higher-order dispersive equation , . The local well-posedness of the associated Cauchy problem is established in Sobolev space with with the aid of the Fourier restriction norm method. MSC:35K30.