Isotropic extension of first-order wave equations

TL;DR

The work addresses the limited isotropy of classic 1D, first-order-in-time wave equations (e.g., KdV, Burgers, CH/DP) and proposes a principled isotropic extension framework, $T^n\Lambda^m$, that elevates temporal order and tensorizes spatial dependence to preserve original solutions. A key result is the 1D isotropy criterion $p+(q-1)m \in 2\mathbb{N}$, guiding which terms admit isotropic generalization. Applying this, Burgers exhibits $T^{\mathbb{N}_+}\Lambda^{2\mathbb{N}+1}$ extensibility, while KdV admits $T^{2\mathbb{N}_+}\Lambda^{2\mathbb{N}}$ leading to the KdV^2 equation, which retains all KdV solutions and conservation laws and yields linearly stable, higher-order corrections; in 2D, a covariant isotropic $T^2\Lambda^0$ extension emerges as a 2D Boussinesq-like form. CH/DP require $T^{2\mathbb{N}_+}\Lambda^{2\mathbb{N}}$, constraining the dependent variable to scalar or even-order tensors, whereas KdV-Burgers remains inherently anisotropic with no extensibility. Overall, the framework offers a principled path to isotropic, higher-dimensional generalizations and potentially broadens soliton theory beyond 1D.

Abstract

The anisotropy of many one-dimensional and first-order-in-time (T$^1$) scalar wave equations (e.g., Korteweg-de Vries and Camassa-Holm) limits their physical completeness and applicability to bidirectional/high-dimensional systems. We define the T$^nΛ^m$ isotropic extension consisting of temporal order elevation and spatial tensorization, which is the only possible approach to eliminate anisotropy while preserving original solutions. Our analysis finds that the Burgers equation exhibits T$^{\mathbb{N}_+}Λ^{2\mathbb{N}+1}$ extensibility and the Korteweg-de Vries (KdV) equation exhibits the T$^{2\mathbb{N}_+}Λ^{2\mathbb{N}}$ extensibility. The T$^2Λ^0$ extension of the KdV equation leads to the corresponding isotropic T$^2$ equation (KdV$^2$) for shallow water dynamics, which is physically more complete and suitable for 2D generalization. In addition to inheriting all KdV solutions and conservation laws, the KdV$^2$ equation also provides linearly stable corrections to the Boussinesq equation. In contrast, the KdV-Burgers equation is inherently anisotropic as it fails to exhibit any T$^nΛ^m$ extensibility.