Derivation of conserved quantities for one dimensional nonlinear damped wave equation

Abstract

Nonlinear wave equation is a typical nonlinear evolution equation with important theoretical significance and application value. The physical meaning of damping is the attenuation of force, or the energy dissipation of objects in motion. It is to stop the object from continuing to move. When an object vibrates under the action of an external force, a reaction force will be generated to attenuate the external force, which is called damping force. This paper studies the conservation laws for one dimensional damped nonlinear wave equation with constant and non-constant damping coefficient. Multipliers and corresponding conserved quantities are derived for wave equation with simple self-interaction term and constant damping coefficient using method of multipliers. The results are generalized in presence of more complicated source term and non-constant damping coefficient. Certain type of equations conserving all energy-momentum related quantities is found. For each individual conserved quantity, the corresponding multiplier and the corresponding type of equations are found.