Conservation laws of nonlinear PDEs arising in elasticity and acoustics in Cartesian, cylindrical, and spherical geometries

Abstract: Conservation laws are computed for various nonlinear partial differential equations that arise in elasticity and acoustics. Using a scaling homogeneity approach, conservation laws are established for two models describing shear wave propagation in a circular cylinder and a cylindrical annulus. Next, using the multiplier method, conservation laws are derived for a parameterized system of constitutive equations in cylindrical coordinates involving a general expression for the Cauchy stress. Conservation laws for the Khokhlov-Zabolotskaya-Kuznetsov equation and Westervelt-type equations in various coordinate systems are also presented.

• The paper presents a unified algorithmic approach using scaling-homogeneity and multiplier methods to derive conservation laws across Cartesian, cylindrical, and spherical systems.
• It utilizes detailed symbolic computation to classify local conservation laws in nonlinear elasticity and acoustic models, including the KZK and Westervelt equations.
• The study reveals finite conservation laws in cylindrical systems contrasted with infinite hierarchies in multi-dimensional settings, guiding structure-preserving numerical integrator design.

Conservation Laws of Nonlinear PDEs in Elasticity and Acoustics across Multiple Geometries

Introduction

This work provides a comprehensive computational analysis of local conservation laws for nonlinear PDEs arising in elasticity and acoustics, formulated in Cartesian, cylindrical, and spherical coordinates. The focus is on PDEs derived from constitutive relations with nonlinear stress-strain responses, as well as on prototypical nonlinear acoustic models, specifically the Khokhlov-Zabolotskaya-Kuznetsov (KZK) and Westervelt equations. The methodological framework is built upon the scaling-homogeneity technique and the multiplier method, both exploited via symbolic computation packages. The scope includes systems with arbitrary nonlinearities, enabling a broad parametrization of constitutive models.

Conservation Laws for Shear Wave Models in Cylindrical Domains

The initial analysis concerns models for shear wave propagation in nonlinearly elastic media, with implicit constitutive relations expressed in cylindrical coordinates. Two canonical cases are considered: the Kambapalli et al. model for a cylindrical annulus, and the Magan et al. model for a circular cylinder.

Both systems involve strain given as a non-invertible function of stress (often a power-law), which leads to the system:

$$\begin{aligned} \sigma_r + \frac{2\sigma}{r} &= \delta u_{tt}, \ u_r - \frac{u}{r} &= \frac{1}{\delta} \sigma (\beta + \sigma^2)^n, \end{aligned}$$

for the annular geometry, and a slightly different set for the circular cylinder. These nonlinear coupled PDEs embody the balance of linear momentum and the constitutive relation.

The conservation law computation via the scaling-homogeneity method identifies densities and fluxes with fixed scaling weights, enforcing uniformity in rank. In explicit symbolic terms, the admissible conserved densities are restricted, with the homogeneity automatically constraining possible terms. For instance, among the conservation laws presented, a non-trivial one is:

$${\mathrm D}_r (r u \sigma_t) + {\mathrm D}_t \Big( -\frac{r}{2(n+1)\delta} [(\beta + \sigma^2)^{n+1} - (n+1)\delta^2(u_t^2 - 2u u_{tt})] \Big) \dot{=} 0,$$

generalized to arbitrary power-law exponents $n$.

The multiplier method, implemented in symbolic software, systematizes the search for conservation laws by solving an overdetermined system for the multipliers (characteristics) $\Lambda^1$ and $\Lambda^2$. This approach reveals that apart from trivial (divergence form of the equations of motion) and scaling-derived conservation laws, only a finite number of low-order local conservation laws exist for these power-law models in cylindrical geometry—contrasting with the infinite hierarchy often present in Cartesian systems for similar equations.

Generalized Constitutive Models and Parameterized PDE Classes

The methodology extends to constitutive relations of the form $\epsilon = F(\sigma)$, where $F$ is an arbitrary nonlinear function. The parameterized system in cylindrical coordinates captures a wide range of material responses, as documented in an extensive tabulation of $F(\sigma)$ choices drawn from the literature. The combined system:

$$\begin{aligned} \sigma_r + \frac{\kappa_1 \sigma}{r} &= \delta u_{tt}, \ u_r + \frac{\kappa_2 u}{r} &= \frac{1}{\delta} F(\sigma), \end{aligned}$$

with corresponding wave equations for the stress, allows for systematic computation of all admissible local conservation laws for arbitrary parameters and functional forms.

Within this general system, the multiplier analysis yields a three-dimensional space of multipliers, each generating a conserved density-flux pair. Explicit formulas for the densities and fluxes in terms of the parameters and $F$ are presented. These formulas encompass the specific Kambapalli and Magan models as special cases.

For the associated wave equations for the stress (obtained by eliminating displacement variables), the set of admissible conserved vectors grows, parameterized by the scaling exponents and arbitrary functions.

Conservation Laws for the KZK and Westervelt Models

The KZK equation and generalizations in Cartesian, cylindrical, and spherical geometries are treated via the multiplier method. The analysis reveals that for the class:

$$p_{zt} - \frac{c_0}{2}\nabla_\perp^2 p - \frac{\delta}{2 c_0^3} p_{ttt} - \frac{\beta}{2 \rho_0 c_0^3}(p^2)_{tt} = 0,$$

the determining equations for multipliers admit infinite-dimensional solution spaces. The multipliers are always linear in time, with spatial coefficients obeying Laplace and Poisson equations (e.g., in Cartesian geometry, $$\phi_{xx}+\phi_{yy} = 2/c_0 \psi_z, \psi_{xx} + \psi_{yy}=0$$). Analogous conditions hold in cylindrical and spherical coordinates, with differential operators dynamically dependent on the geometry.

Consequently, the KZK equation possesses infinitely many independent conservation laws, each generated by pairing harmonic and appropriate Poisson-equation solutions. The explicit conserved densities and fluxes are formulated in terms of these coefficient functions and the physical variables.

A parallel treatment of Westervelt-type equations:

$$F(p)_{tt} - \alpha p_{ttt} - \beta \nabla^2 p_t = c^2 \nabla^2 p,$$

where $F$ is an arbitrary nonlinear function, demonstrates that for multiple spatial dimensions ($n > 1$), the conservation laws are structured similarly: multipliers are time-linear, and both spatial coefficients must satisfy the Laplacian. For $n = 1$, only a bounded set of conservation laws exists, in full agreement with recent classifications. The approach generalizes to equations in spherical and cylindrical geometries, covering both the standard quadratic and arbitrary nonlinearities.

Numerical and Structural Implications

Strong claims proposed:

• For the KZK and multi-dimensional Westervelt equations, when formulated with generic nonlinearities and physically relevant parameter ranges, there exist infinitely many local conservation laws, all characterized by multiplier functions solving geometric Laplace or Poisson-type equations.
• By contrast, for the corresponding one-dimensional Westervelt models and nonlinear elasticity wave systems in cylindrical geometry, only finitely many local conservation laws exist, and these are explicitly classified.

Numerical implications: The explicit construction of conservation laws plays a critical role in the development of geometric numerical integrators—schemes that preserve invariants discretely for nonlinear PDEs. For systems with rare conservation laws, structure-preserving discretization is severely constrained, whereas for systems with infinite conservation laws (e.g., KZK, multi-D Westervelt), there is more room for numerical schemes preserving arbitrary invariants.

Theoretical implications: The existence and structure of conservation laws are essential in the qualitative analysis of nonlinear PDEs, including blow-up, regularity, and global stability properties. The exhaustive classification here enables future developments in exact solution construction, symmetry reduction, and algorithmic detection of integrability.

Outlook

The thorough application of scaling-homogeneity and multiplier methods across nonlinear elasticity and acoustics models in diverse geometries establishes a unified, algorithmic framework for conservation law discovery. The extension to arbitrary nonlinear constitutive functions and multi-parameter systems highlights the versatility of the symbolic techniques employed. Future research directions include:

• Extending the analysis to nonlocal and higher-order conservation laws;
• Application to integrability studies and higher symmetry algebra classification;
• Automated conservation law computation in emerging generalized elasticity and acoustics models, including fractional and nonlocal PDEs;
• Development of structure-preserving numerical methods leveraging identified conservation laws in complex geometries.

Conclusion

This work solidifies the understanding of conservation law structures in a variety of nonlinear PDEs central to elasticity and nonlinear acoustics. By leveraging algorithmic symmetry techniques and symbolic computation, the analysis yields exhaustive characterizations for both classical and newly proposed models. For key classes such as the KZK and Westervelt equations in higher dimensions, the existence of infinite hierarchies of conservation laws is confirmed, significantly impacting both theoretical analysis and computational simulation of nonlinear wave phenomena.