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General Classification, Invariance and Conservation Laws Analyses of Nonlinear Fourth Order Wave and Nerve Membrane Equations with Dissipation

Ali Raza, F M Mahomed, F D Zaman, A H Kara

TL;DR

The work delivers a comprehensive Lie symmetry analysis of a nonlinear fourth‑order wave equation with dissipation, including an extension with dispersion and a biomembrane nerve‑model. It develops equivalence transformations, complete symmetry classifications across several forms of the arbitrary function $\Phi(u)$, and builds one‑dimensional optimal systems to obtain invariant reductions and explicit solutions. The authors also implement the multiplier method to derive local conservation laws for both the general fourth‑order and specific nerve‑equation cases, and they present detailed invariant and soliton solutions for second and fourth order nerve models with graphical illustrations. The results provide a rigorous symmetry‑based framework for analyzing nonlinear dispersive systems relevant to biomembranes and highlight potential applications to pulse propagation and soliton dynamics in nerve membranes.

Abstract

We study the nonlinear wave equation for arbitrary function with fourth order dissipation. A special case that is analysed exclusively is the model of nerve membranes; we consider this model, both, in the presence and absence of the fourth order dissipation. The equivalence transformations, Lie symmetries and a complete classification is presented. We also discuss the one dimensional optimal system in each case obtained via classification. The reduction of the partial differential equations (PDEs) is carried out and the forms of invariant solutions are presented. The study also include the construction of conservation laws using the direct method. The invariant solutions and some special type of solutions including solitions are presented with their graphical illustrations.

General Classification, Invariance and Conservation Laws Analyses of Nonlinear Fourth Order Wave and Nerve Membrane Equations with Dissipation

TL;DR

The work delivers a comprehensive Lie symmetry analysis of a nonlinear fourth‑order wave equation with dissipation, including an extension with dispersion and a biomembrane nerve‑model. It develops equivalence transformations, complete symmetry classifications across several forms of the arbitrary function , and builds one‑dimensional optimal systems to obtain invariant reductions and explicit solutions. The authors also implement the multiplier method to derive local conservation laws for both the general fourth‑order and specific nerve‑equation cases, and they present detailed invariant and soliton solutions for second and fourth order nerve models with graphical illustrations. The results provide a rigorous symmetry‑based framework for analyzing nonlinear dispersive systems relevant to biomembranes and highlight potential applications to pulse propagation and soliton dynamics in nerve membranes.

Abstract

We study the nonlinear wave equation for arbitrary function with fourth order dissipation. A special case that is analysed exclusively is the model of nerve membranes; we consider this model, both, in the presence and absence of the fourth order dissipation. The equivalence transformations, Lie symmetries and a complete classification is presented. We also discuss the one dimensional optimal system in each case obtained via classification. The reduction of the partial differential equations (PDEs) is carried out and the forms of invariant solutions are presented. The study also include the construction of conservation laws using the direct method. The invariant solutions and some special type of solutions including solitions are presented with their graphical illustrations.

Paper Structure

This paper contains 60 sections, 6 theorems, 408 equations, 12 figures, 10 tables.

Table of Contents

  1. Introduction
  2. General Classification and Optimal System
  3. $\boldsymbol{\Phi = \epsilon (u+k_1)^{k_2} }$, where $\boldsymbol{\epsilon = \pm 1}$
  4. $\boldsymbol{\Phi = \epsilon \exp (ku) }$, where $\boldsymbol{\epsilon = \pm 1}$
  5. $\boldsymbol{\Phi = \epsilon (u)^{k_1} }$, where $\boldsymbol{\epsilon = \pm 1}$
  6. $\boldsymbol{\Phi = \epsilon \exp (u) }$, where $\boldsymbol{\epsilon = \pm 1}$
  7. $\boldsymbol{\Phi^{\prime} = 0}$
  8. Principal Case
  9. Discussion on Optimal System
  10. Optimal System of Fourth Order Wave
  11. Optimal system: $\boldsymbol{\Phi^{\prime} = 0}$
  12. Optimal system: Principal Case - For Arbitrary $\boldsymbol{\Phi}$
  13. Optimal system: $\boldsymbol{\Phi = \epsilon (u+k_1)^{k_2} }$, where $\boldsymbol{\epsilon = \pm 1}$
  14. Optimal system: $\boldsymbol{\Phi = \epsilon \exp (ku) }$, where $\boldsymbol{\epsilon = \pm 1}$
  15. Optimal system: $\boldsymbol{\Phi = \epsilon (u)^{k_1} }$, where $\boldsymbol{\epsilon = \pm 1}$
  16. ...and 45 more sections

Key Result

Theorem 1

Consider the PDE (lin1op) admits an infinite dimensional algebra spanned by The set of one-dimensional optimal system of sub-algebra, which can not be reduced further under any adjoint action is given by the following two lists. For the finite-dimensional part $\mathfrak{g}=\left\{X_1, X_2, X_3\right\}$ and for the infinite-dimensional part $\mathfrak{h}=\left\{X_\beta\right\}$

Figures (12)

  • Figure 1: Solution $u(x,t)$ : Visualised for $t,x=-2,..,2$ and $t,x=-1,..,1$.
  • Figure 2: Solution $u(x,t)$ : Visualised for $t,x=0,..,2$ and $t,x=0,..,2$.
  • Figure 3: Solution $u(x,t)$ : Visualised for $t,x=0,..,10$ and $t,x=0,..,100$.
  • Figure 4: Similarity solution is depicted along the axes $x = -2\pi \ldots 2\pi, t = 0 \ldots 2\pi$.
  • Figure 5: Solution $u(x,t)$ : Visualised for two sets of values $t,x$
  • ...and 7 more figures

Theorems & Definitions (6)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6