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Lie symmetries, conservation laws and exact solutions of a generalized quasilinear KdV equation with degenerate dispersion

María de los Santos Bruzón, Elena Recio, Tamara María Garrido, Rafael de la Rosa

Abstract

We provide a complete classification of point symmetries and low-order local conservation laws of the generalized quasilinear KdV equation in terms of the arbitrary function. The corresponding interpretation of symmetry transformation groups are given. In addition, a physical description of the conserved quantities is included. Finally, few travelling wave solutions have been obtained.

Lie symmetries, conservation laws and exact solutions of a generalized quasilinear KdV equation with degenerate dispersion

Abstract

We provide a complete classification of point symmetries and low-order local conservation laws of the generalized quasilinear KdV equation in terms of the arbitrary function. The corresponding interpretation of symmetry transformation groups are given. In addition, a physical description of the conserved quantities is included. Finally, few travelling wave solutions have been obtained.
Paper Structure (7 sections, 2 theorems, 45 equations, 3 figures)

This paper contains 7 sections, 2 theorems, 45 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Point Symmetries
  3. Conservation laws
  4. Travelling wave reductions
  5. Integrating factors and first integrals
  6. Some exact solutions
  7. Conclusions

Key Result

Theorem 1

The point symmetries admitted by the generalized quasilinear KdV equation genedp for arbitrary $f(u)$ are generated by: Additional point symmetries are admitted by the generalized quasilinear KdV equation genedp in the following cases: In the above symmetries, $n \neq 0$, $f_0\neq 0$, $f_1$ and $f_2$ are arbitrary constants. On the other hand, symmetries (symm1)-(symm2) represent space and time-

Figures (3)

  • Figure 1: Solution $u(x,t)={\rm sech}^2(x-t)$ of Eq. (\ref{['genedp']})
  • Figure 2: Solution $u(x,t)={\rm sinh}(x-t)$ of Eq. \ref{['genedp']}
  • Figure 3: Solution $u(x,t)={\rm sin}(x-t)$ of Eq. \ref{['genedp']}

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2