Solitary waves and conservation laws for highly nonlinear wave equations modeling granular chains

Abstract

Nonlinear fourth-order wave equations modeling the long wavelength behavior of compression pulses in one-dimensional homogeneous granular chains with weak and strong initial pre-compression are considered. As a main result, the exact solitary wave solution of the weak-compression wave equation is derived, which has been an open mathematical problem for many years. The derivation relies on modifying standard solution techniques. First, conservation laws for energy, momentum, and net displacement speed are found through a multiplier method, without using Noether's theorem. This provides a direct way to reduce the wave equation to a nonlinear first-order ordinary differential equation (ODE). Second, an asymptotic analysis is used to determine the physically appropriate values for the integration constants that appear in this ODE, which allows the solitary wave solution to be obtained by a single quadrature. Third, explicit expressions for energy and momentum of the solitary wave are obtained using only the ODE and the integration constants. These steps are introduced by first applying them to the strong-compression wave equation, giving a new, simple derivation of the known strong-compression solitary wave. The general method is applicable to finding exact solutions for a wide class of nonlinear wave equations and dynamical systems.