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Nonlinearly dispersive KP equations with new compacton solutions

Stephen C. Anco, Maria Gandarias

TL;DR

This work addresses compactly supported traveling waves in a nonlinearly dispersive KP-type equation, introducing the KP$_N(m,n)$ generalization and aiming for a complete classification of line and plane compactons in $N$ dimensions. It develops a symmetry-guided multi-reduction approach that yields conservation laws and reduces the travelling-wave problem to quadratures, enabling explicit compacton constructions across multiple parameter regimes. The authors present comprehensive families of explicit profiles, including cosine, Jacobi cn/sn, and algebraic forms, along with antisymmetric variants, and establish precise existence conditions via endpoint asymptotics. The results provide a unified framework linking kinematics (speed and direction) to profile type, extend naturally to higher dimensions, and have potential applications in nonlinear dispersive media and related physical systems.

Abstract

A complete classification of compacton solutions is carried out for a generalization of the Kadomtsev-Petviashvili (KP) equation involving nonlinear dispersion in two and higher spatial dimensions. In particular, precise conditions are given on the nonlinearity powers in this equation under which a travelling wave can be cut off to obtain a compacton. Numerous explicit examples having various profiles are derived, including a quadratic function, powers of a cosine, and powers of Jacobi $\cn$ functions, all of which are symmetric. The cosine and $\cn$ symmetric compactons have an anti-symmetric counterpart. In comparison, explicit solitary waves of the generalized KP equation are found to have profiles given by a power of a sech and a reciprocal quadratic function. Kinematic properties of all of the different types of compactons and solitary waves are discussed, along with conservation laws of the generalized KP equation.

Nonlinearly dispersive KP equations with new compacton solutions

TL;DR

This work addresses compactly supported traveling waves in a nonlinearly dispersive KP-type equation, introducing the KP generalization and aiming for a complete classification of line and plane compactons in dimensions. It develops a symmetry-guided multi-reduction approach that yields conservation laws and reduces the travelling-wave problem to quadratures, enabling explicit compacton constructions across multiple parameter regimes. The authors present comprehensive families of explicit profiles, including cosine, Jacobi cn/sn, and algebraic forms, along with antisymmetric variants, and establish precise existence conditions via endpoint asymptotics. The results provide a unified framework linking kinematics (speed and direction) to profile type, extend naturally to higher dimensions, and have potential applications in nonlinear dispersive media and related physical systems.

Abstract

A complete classification of compacton solutions is carried out for a generalization of the Kadomtsev-Petviashvili (KP) equation involving nonlinear dispersion in two and higher spatial dimensions. In particular, precise conditions are given on the nonlinearity powers in this equation under which a travelling wave can be cut off to obtain a compacton. Numerous explicit examples having various profiles are derived, including a quadratic function, powers of a cosine, and powers of Jacobi functions, all of which are symmetric. The cosine and symmetric compactons have an anti-symmetric counterpart. In comparison, explicit solitary waves of the generalized KP equation are found to have profiles given by a power of a sech and a reciprocal quadratic function. Kinematic properties of all of the different types of compactons and solitary waves are discussed, along with conservation laws of the generalized KP equation.

Paper Structure

This paper contains 15 sections, 4 theorems, 101 equations, 12 figures, 1 table.

Table of Contents

  1. Introduction
  2. Travelling wave ODE
  3. Multi-reduction
  4. Solitary waves
  5. Conditions for existence of compacton solutions
  6. Compactons in a special case
  7. Kinematic features and profiles
  8. Compactons in the general case
  9. Explicit solutions with $m=n$, $k\neq0$
  10. Explicit solutions with $m\neq n$, $k=0$
  11. Explicit solutions with $m\neq n$, $k\neq0$
  12. Antisymmetric compactons
  13. Kinematic features and profiles
  14. Compactons in three (and higher) dimensions
  15. Conclusions

Key Result

Proposition 3.1

For a travelling wave $U(\xi)$ with asymptotic behaviour U.asympt, a compacton profile cutoff.soln gives a classical (strong) solution of the travelling wave ODE U.ODE.recast iff the cut-off power satisfies

Figures (12)

  • Figure 1: Profiles of localized solitary wave \ref{['mlarger1.solitary']} (dotted) and heavy-tail wave \ref{['mlarger1.heavytail']} (solid): $m=\tfrac{7}{5}$, $A=16$, $B=8$.
  • Figure 2: Profiles of solitary wave \ref{['msmaller1.solitary']} (dotted) and heavy-tail wave \ref{['msmaller1.heavytail']} (solid): $m=\tfrac{3}{5}$, $A=-2$, $B=-12$.
  • Figure 3: Cosine profiles \ref{['U.Vlin1.weak']} with $\frac{a}{b}=1$: $n=\tfrac{4}{9}$ solid, $n=\tfrac{1}{10}$ dash. Dots indicate the cut off points.
  • Figure 4: Sine profiles \ref{['U.Vlin2.weak']} with $\frac{a}{b}=1$. Left: $n=\tfrac{2}{5}$ solid, $n=\tfrac{2}{25}$ dash; Right: $n=\tfrac{3}{7}$ solid, $n=\tfrac{1}{15}$ dash. Dots indicate the cut off points.
  • Figure 5: Non-symmetric profiles \ref{['U.Vlin.strong']} with $\frac{a}{b}=1$: $n=\tfrac{1}{5}$ solid, $n=\tfrac{1}{10}$ dash. Left: $\phi=\tfrac{\pi}{3}$, $L=1$; Right: $\phi=\tfrac{2\pi}{5}$, $L=4$. Dots indicate the cut off points.
  • ...and 7 more figures

Theorems & Definitions (9)

  • Remark 3.1
  • Proposition 3.1
  • Proposition 3.2
  • Theorem 4.1
  • Remark 4.1
  • Remark 5.1
  • Theorem 5.1
  • Remark 5.2
  • Remark 5.3