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Conservation laws and variational structure of damped nonlinear wave equations

Stephen C. Anco, Almudena P. Marquez, Tamara M. Garrido, Maria L. Gandarias

Abstract

All low-order conservation laws are found for a general class of nonlinear wave equations in one dimension with linear damping which is allowed to be time-dependent. Such equations arise in numerous physical applications and have attracted much attention in analysis. The conservation laws describe generalized momentum and boost momentum, conformal momentum, generalized energy, dilational energy, and light-cone energies. Both the conformal momentum and dilational energy have no counterparts for nonlinear undamped wave equations in one dimension. All of the conservation laws are obtainable through Noether's theorem, which is applicable because the damping term can be transformed into a time-dependent self-interaction term by a change of dependent variable. For several of the conservation laws, the corresponding variational symmetries have a novel form which is different than any of the well known variation symmetries admitted by nonlinear undamped wave equations in one dimension.

Conservation laws and variational structure of damped nonlinear wave equations

Abstract

All low-order conservation laws are found for a general class of nonlinear wave equations in one dimension with linear damping which is allowed to be time-dependent. Such equations arise in numerous physical applications and have attracted much attention in analysis. The conservation laws describe generalized momentum and boost momentum, conformal momentum, generalized energy, dilational energy, and light-cone energies. Both the conformal momentum and dilational energy have no counterparts for nonlinear undamped wave equations in one dimension. All of the conservation laws are obtainable through Noether's theorem, which is applicable because the damping term can be transformed into a time-dependent self-interaction term by a change of dependent variable. For several of the conservation laws, the corresponding variational symmetries have a novel form which is different than any of the well known variation symmetries admitted by nonlinear undamped wave equations in one dimension.
Paper Structure (12 sections, 7 theorems, 126 equations)

This paper contains 12 sections, 7 theorems, 126 equations.

Table of Contents

  1. Introduction
  2. Multipliers, variational structure, and Noether's theorem
  3. Conservation laws and multipliers
  4. Variational symmetries
  5. Determining equations
  6. Wave equations with removable damping
  7. Conservation laws
  8. Light-cone energies
  9. Conservation laws for intrinsic damping
  10. Conserved quantities
  11. Variational symmetries for intrinsic damping
  12. Concluding remarks

Key Result

Proposition 1

For any damped wave equation waveeqn.gen, there is a one-to-one correspondence between non-trivial conservation laws conslaw (up to local equivalence) and non-zero multipliers Q.relation, where, without loss of generality, $(T,\Phi)$ and $Q$ are functions of $u$, $u_t$, and their $x$-derivatives, in

Theorems & Definitions (7)

  • Proposition 1
  • Proposition 2
  • Theorem 1
  • Proposition 3
  • Proposition 4
  • Theorem 2
  • Proposition 5