Pseudolocalized Three-Dimensional Solitary Waves as Quasi-Particles

TL;DR

This paper introduces a higher-order dispersive field equation with pseudolocalized 3D solitary wave solutions that behave like quasi-particles, exhibiting inverse-square force interactions similar to gravity.

Contribution

It presents a new class of pseudolocalized solutions for a 3D dispersive equation and derives their dynamics as quasi-particles with gravitational-like interactions.

Findings

Pseudolocalized solutions have square-integrable derivatives but not the profiles themselves.

The dynamics of these quasi-particles are governed by a discrete Lagrangian.

At large distances, the interaction force is proportional to the inverse square of the separation.

Abstract

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.