On the Pseudolocalized Solutions in Multi-dimension of Boussinesq Equation

TL;DR

This paper introduces a new class of pseudolocalized solutions for the three-dimensional Boussinesq equation, characterized by non-localized profiles with square-integrable derivatives, analyzed both analytically and numerically.

Contribution

It presents the first analytical and numerical construction of pseudolocalized solutions for the 3D Boussinesq equation, expanding understanding of non-localized wave profiles.

Findings

Pseudolocalized solutions exist for quadratic and cubic nonlinearities.

Analytical expressions are derived in the linear limit.

Numerical methods effectively handle boundary conditions at the origin and infinity.

Abstract

A new class of solutions of three-dimensional equations from the Boussinesq paradigm are considered. The corresponding profiles are not localized functions in the sense of the integrability of the square over an infinite domain. For the new type of solutions, the gradient and the Hessian/Laplacian are square integrable. In the linear limiting case, analytical expressions for the profiles of the pseudolocalized solutions are found. The nonlinear case is treated numerically with a special approximation of the differential operators with spherical symmetry that allows for automatic acknowledgement of the behavioral conditions at the origin of the coordinate system. The asymptotic boundary conditions stem from the $1/r$ behavior at infinity of the pseudolocalized profile. A special approximation is devised that allows us to obtain the proper behavior for much smaller computational box. The pseudolocalized solutions are obtained for both quadratic and cubic nonlinearity.