Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Weak compactons of nonlinearly dispersive KdV and KP equations

Stephen C. Anco, Maria Gandarias

TL;DR

The paper introduces a rigorous weak formulation for the nonlinear dispersive K(m,n) equation and its KP(m,n) generalization, enabling compacton solutions in a distributional sense. By analyzing travelling-wave reductions and introducing a cutoff construction, it derives explicit weak compacton profiles—including forms that do not correspond to strong solutions—and establishes concrete endpoint conditions and pn > 2-type criteria for existence. It catalogs multiple symmetric profile families (algebraic, cosine, Jacobi cn/sn, and rational cn) with precise parameter regions, and demonstrates numerical weak compactons via an ODE for V(ξ) that respects the weak formulation. The work broadens the understanding of well-posedness for nonlinearly dispersive equations and provides a framework extendable to other generalized KdV/KP-type models and higher-dimensional compacton structures.

Abstract

A weak formulation is devised for the K(m,n) equation which is a nonlinearly dispersive generalization of the gKdV equation having compacton solutions. With this formulation, explicit weak compacton solutions are derived, including ones that do not exist as classical (strong) solutions. Similar results are obtained for a nonlinearly dispersive generalization of the gKP equation in two dimensions, which possesses line compacton solutions.

Weak compactons of nonlinearly dispersive KdV and KP equations

TL;DR

The paper introduces a rigorous weak formulation for the nonlinear dispersive K(m,n) equation and its KP(m,n) generalization, enabling compacton solutions in a distributional sense. By analyzing travelling-wave reductions and introducing a cutoff construction, it derives explicit weak compacton profiles—including forms that do not correspond to strong solutions—and establishes concrete endpoint conditions and pn > 2-type criteria for existence. It catalogs multiple symmetric profile families (algebraic, cosine, Jacobi cn/sn, and rational cn) with precise parameter regions, and demonstrates numerical weak compactons via an ODE for V(ξ) that respects the weak formulation. The work broadens the understanding of well-posedness for nonlinearly dispersive equations and provides a framework extendable to other generalized KdV/KP-type models and higher-dimensional compacton structures.

Abstract

A weak formulation is devised for the K(m,n) equation which is a nonlinearly dispersive generalization of the gKdV equation having compacton solutions. With this formulation, explicit weak compacton solutions are derived, including ones that do not exist as classical (strong) solutions. Similar results are obtained for a nonlinearly dispersive generalization of the gKP equation in two dimensions, which possesses line compacton solutions.
Paper Structure (13 sections, 6 theorems, 59 equations, 5 figures, 1 table)

This paper contains 13 sections, 6 theorems, 59 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Travelling wave ODE and solution profiles
  3. Compacton profiles
  4. Weak formulation for compacton solutions
  5. Weak form of $K(m,n)$ equation
  6. Weak form of ${\it KP}(m,n)$ equation
  7. Derivation of weak compactons via cutoff conditions
  8. Existence conditions for strong solutions
  9. Explicit weak compactons
  10. Solution profiles
  11. Examples when compactons exist only as weak solutions
  12. Examples of numerical weak compactons
  13. Concluding remarks

Key Result

Proposition 3.1

Weak compactons of the $K(m,n)$ equation are travelling waves $u=U(\xi)\in C_c^0({\mathbb R})$ that satisfy the equivalent integral equations weak.Kmn.pde and weak.Kmn.ode.

Figures (5)

  • Figure 1: Parameter regions for $K(m,n)$: (left) algebraic solutions \ref{['zsq1']} [green] and \ref{['zsq2']} [blue]; (right) cosine solutions \ref{['cos1']} [green] and \ref{['cos2']} [blue].
  • Figure 2: Parameter regions for $K(m,n)$: (left) cn & sn solutions \ref{['cn1']} & \ref{['sn1']} [brown] and \ref{['cn2']} & \ref{['sn2']} [blue]; (right) rational cn solutions \ref{['ratcn1']} & \ref{['ratcn2']} [green] and \ref{['ratcn3']} [pink], \ref{['ratcn4']} & \ref{['ratcn5']} [brown] and \ref{['ratcn6']} [blue].
  • Figure 3: Parameter regions for ${\it KP}(m,n)$: (left) algebraic solutions \ref{['zsq1']} [green] and \ref{['zsq2']} [blue]; (right) cosine solutions \ref{['cos1']} [green] and \ref{['cos2']} [blue].
  • Figure 4: Parameter regions for ${\it KP}(m,n)$: (left) cn & sn solutions \ref{['cn1']} & \ref{['sn1']} [brown] and \ref{['cn2']} & \ref{['sn2']} [blue]; (right) rational cn solutions \ref{['ratcn1']} & \ref{['ratcn2']} [green] and \ref{['ratcn3']} [pink], \ref{['ratcn4']} & \ref{['ratcn5']} [brown] and \ref{['ratcn6']} [blue].
  • Figure 5: Profile of symmetric weak compactons: (left) $n=2$, $m=9/4$, $a=b=g=1$; (right) $n=9/10$, $m=1/2$, $a=-b=g=1$

Theorems & Definitions (8)

  • Definition 3.1
  • Proposition 3.1
  • Proposition 3.2
  • Lemma 3.1
  • Proposition 3.3
  • Remark 3.1
  • Theorem 3.1
  • Proposition 3.4