Travelling waves and conservation laws for highly nonlinear wave equations modelling Hertz chains

Abstract

A highly nonlinear, fourth-order wave equation that models the continuum theory of long wavelength pulses in weakly compressed, discrete, homogeneous chains with a general power-law contact interaction is studied. For this wave equation, all solitary wave solutions and all nonlinear periodic wave solutions, along with all conservation laws, are derived. The solutions are explicitly parameterized in terms of the asymptotic value of the wave amplitude in the case of solitary waves and the peak of the wave amplitude in the case of nonlinear periodic waves. All cases in which the solution expressions can be stated in an explicit analytic form using elementary functions are worked out. In these cases, explicit expressions for the total energy and total momentum for all solutions are obtained as well. The derivation of the solutions uses the conservation laws combined with an energy analysis argument to reduce the wave equation directly to a separable first-order differential equation which determines the wave amplitude in terms of the travelling wave variable. This method can be applied more generally to other highly nonlinear wave equations.

References in corpus (2)

• Solitary wave trains in granular chains: Experiments, theory and simulations
• Highly Nonlinear Solitary Waves in Periodic Dimer Granular Chains

Cited by in corpus (5)

• Symmetry multi-reduction method for partial differential equations with conservation laws
• An analytic study on the properties of solitary waves traveling on tensegrity-like lattices
• Conserved norms and related conservation laws for multi-peakon equations
• Travelling wave solutions on a non-zero background for the generalized Korteweg-de Vries equation
• Long wavelength solitary waves in Hertzian chains and their properties in different nonlinearity regimes