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Conservation laws and exact solutions of a nonlinear acoustics equation by classical symmetry reduction

Almudena del Pilar Marquez, Elena Recio, Maria Luz Gandarias

TL;DR

This work analyzes a one-dimensional generalized Westervelt equation $f(p)_{tt}-\beta p_{xxt}=c^2 p_{xx}$, identifying a complete Lie point-symmetry classification under $f''(p)\neq0$ and $\beta\neq0$, and obtaining four low-order conservation laws via the multiplier method with multipliers $Q_1=1$, $Q_2=t$, $Q_3=x$, and $Q_4=tx$. It then constructs two potential systems, introducing first- and second-layer potentials, and derives both inherited and nonlocal symmetries and conservation laws for these extended systems, along with projection relations to the original PDE. Travelling-wave reductions, via translations, yield a reduced ODE that integrates to a separable quadrature, enabling explicit traveling-wave and shock-wave profiles; in particular, for $f(p)=p+\kappa p^n$ with $\kappa>0$, $n>1$, the explicit shock profiles confirm the dissipative nature of the model. Overall, the paper provides a robust symmetry-and-conservation-law framework for invariant solutions, nonlocal structures, and physically relevant shock behavior in nonlinear acoustic waves.

Abstract

Symmetries and conservation laws are studied for a generalized Westervelt equation which is a nonlinear partial differential equation modelling the propagation of sound waves in a compressible medium. This nonlinear wave equation is widely used in nonlinear acoustics and it is especially important in biomedical applications such as ultra-sound imaging in human tissue. Modern methods are applied to uncover point symmetries and conservation laws that can lead to useful developments concerning solutions and their properties. A complete classification of point symmetries is shown for the arbitrary function. Local low-order conservation laws related to net mass of sound waves are obtained by the multiplier method. Two potential systems are derived yielding potential symmetries and nonlocal conservation laws. For the physical case interesting for this equation, travelling wave solutions are studied leading to shock waves.

Conservation laws and exact solutions of a nonlinear acoustics equation by classical symmetry reduction

TL;DR

This work analyzes a one-dimensional generalized Westervelt equation , identifying a complete Lie point-symmetry classification under and , and obtaining four low-order conservation laws via the multiplier method with multipliers , , , and . It then constructs two potential systems, introducing first- and second-layer potentials, and derives both inherited and nonlocal symmetries and conservation laws for these extended systems, along with projection relations to the original PDE. Travelling-wave reductions, via translations, yield a reduced ODE that integrates to a separable quadrature, enabling explicit traveling-wave and shock-wave profiles; in particular, for with , , the explicit shock profiles confirm the dissipative nature of the model. Overall, the paper provides a robust symmetry-and-conservation-law framework for invariant solutions, nonlocal structures, and physically relevant shock behavior in nonlinear acoustic waves.

Abstract

Symmetries and conservation laws are studied for a generalized Westervelt equation which is a nonlinear partial differential equation modelling the propagation of sound waves in a compressible medium. This nonlinear wave equation is widely used in nonlinear acoustics and it is especially important in biomedical applications such as ultra-sound imaging in human tissue. Modern methods are applied to uncover point symmetries and conservation laws that can lead to useful developments concerning solutions and their properties. A complete classification of point symmetries is shown for the arbitrary function. Local low-order conservation laws related to net mass of sound waves are obtained by the multiplier method. Two potential systems are derived yielding potential symmetries and nonlocal conservation laws. For the physical case interesting for this equation, travelling wave solutions are studied leading to shock waves.
Paper Structure (18 sections, 14 theorems, 90 equations, 2 figures)

This paper contains 18 sections, 14 theorems, 90 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Point symmetries
  3. Low-order conservation laws
  4. Conserved quantities
  5. Potential systems, symmetries and conservation laws
  6. First potential system
  7. First-layer potential point symmetries
  8. First-layer potential conservation laws
  9. Second-layer potential point symmetries
  10. Second-layer potential conservation laws
  11. Second potential system
  12. First-layer potential point symmetries
  13. First-layer potential conservation laws
  14. Second-layer potential point symmetries
  15. Second-layer potential conservation laws
  16. ...and 3 more sections

Key Result

Theorem 2.1

The point symmetries of the generalized Westervelt equation pde with $f"(p)\neq 0$ and $\beta\neq0$ are comprised by a time-translation and a space-translation For $f(p)=k(p+p_0)^{1+q}$, an additional scaling in $x$ and $p$ For $f(p)=\tfrac{k}{(p+p_0)^3}$, an additional non-rigid scaling in $x$ and $p$

Figures (2)

  • Figure 1: Shock wave profile \ref{['sol1']} (left) for $c=2$ (solid), $c=3$ (dot), $c=4$ (dash). 3D shock wave space-time \ref{['sol1']} (right) for $c=2$ and $\nu=2$. The parameters used are $U_0=\nu=\beta=\kappa=1$.
  • Figure 2: Shock wave profile \ref{['sol0']} (left) for $c=2$ (solid), $c=3$ (dot), $c=4$ (dash). 3D shock wave space-time \ref{['sol0']} (right) for $c=2$ and $\nu=2$. The parameters used are $n=3$ and $U_0=\nu=\beta=\kappa=1$.

Theorems & Definitions (14)

  • Theorem 2.1
  • Proposition 3.1
  • Theorem 3.2
  • Theorem 4.1
  • Proposition 4.2
  • Theorem 4.3
  • Theorem 4.4
  • Proposition 4.5
  • Theorem 4.6
  • Theorem 4.7
  • ...and 4 more