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Exact Jacobi elliptic solutions of the $abcd$-system

Jake Daniels, Nghiem V. Nguyen

Abstract

In this manuscript, consideration is given to the existence of periodic traveling-wave solutions to the $abcd$-system. This system was derived by Bona, Saut, and Chen to describe small amplitude, long wavelength gravity waves on the surface of water. These exact solutions are formulated in terms of the Jacobi elliptic function cnoidal. The existence of explicit traveling-wave solutions is very useful in theoretical investigations such as stability of solutions, as well as other numerical analysis of the system.

Exact Jacobi elliptic solutions of the $abcd$-system

Abstract

In this manuscript, consideration is given to the existence of periodic traveling-wave solutions to the -system. This system was derived by Bona, Saut, and Chen to describe small amplitude, long wavelength gravity waves on the surface of water. These exact solutions are formulated in terms of the Jacobi elliptic function cnoidal. The existence of explicit traveling-wave solutions is very useful in theoretical investigations such as stability of solutions, as well as other numerical analysis of the system.
Paper Structure (10 sections, 2 theorems, 35 equations, 2 figures)

This paper contains 10 sections, 2 theorems, 35 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Existence of Periodic Traveling-Wave Solutions
  4. The case when $c \ne 0$
  5. The case when $c = 0$
  6. Exact Jacobi Elliptic Solutions
  7. The case when $c \ne 0$
  8. The case when $c=0$
  9. Semi-trivial Solutions
  10. Conclusion

Key Result

Theorem 3.1

Suppose $c \ne 0$, then the periodic traveling-wave ansatz eqn:eta-w will take on one of the following forms depending on the values of $a$ and $b$:

Figures (2)

  • Figure 5: Graphs of solitary wave solutions of form \ref{['soli-wave-form-1']}
  • Figure 6: Graphs of solitary wave solutions of form \ref{['soli-wave-form-2']} and \ref{['soli-wave-form-3']}

Theorems & Definitions (4)

  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof