Weakly nonlinear extension of d'Alembert's formula

TL;DR

This paper introduces an explicit weakly nonlinear extension of d'Alembert's formula for the Boussinesq equation, using special functions from Korteweg-de Vries equations, validated through analytical and numerical comparisons.

Contribution

It provides a novel explicit formula for weakly nonlinear solutions of the Boussinesq equation, extending classical linear wave solutions with special functions from KdV equations.

Findings

Explicit solutions match numerical simulations

Formula works for exactly solvable initial conditions

Extension captures weak nonlinearity effects

Abstract

We consider a weakly nonlinear solution of the Cauchy problem for the regularised Boussinesq equation, which constitutes an extension of the classical d'Alembert's formula for the linear wave equation. The solution is given by a simple and explicit formula, expressed in terms of two special functions solving the initial-value problems for two Korteweg-de Vries equations. We test the formula by considering several examples with `exactly solvable initial conditions' and their perturbations. Explicit analytical solutions are compared with the results of direct numerical simulations.