197 research outputs found
On asymptotically equivalent shallow water wave equations
The integrable 3rd-order Korteweg-de Vries (KdV) equation emerges uniquely at
linear order in the asymptotic expansion for unidirectional shallow water
waves. However, at quadratic order, this asymptotic expansion produces an
entire {\it family} of shallow water wave equations that are asymptotically
equivalent to each other, under a group of nonlinear, nonlocal, normal-form
transformations introduced by Kodama in combination with the application of the
Helmholtz-operator. These Kodama-Helmholtz transformations are used to present
connections between shallow water waves, the integrable 5th-order Korteweg-de
Vries equation, and a generalization of the Camassa-Holm (CH) equation that
contains an additional integrable case. The dispersion relation of the full
water wave problem and any equation in this family agree to 5th order. The
travelling wave solutions of the CH equation are shown to agree to 5th order
with the exact solution
Whitham modulation theory for the Kadomtsev-Petviashvili equation
The genus-1 KP-Whitham system is derived for both variants of the
Kadomtsev-Petviashvili (KP) equation (namely, the KPI and KPII equations). The
basic properties of the KP-Whitham system, including symmetries, exact
reductions, and its possible complete integrability, together with the
appropriate generalization of the one-dimensional Riemann problem for the
Korteweg-deVries equation are discussed. Finally, the KP-Whitham system is used
to study the linear stability properties of the genus-1 solutions of the KPI
and KPII equations; it is shown that all genus-1 solutions of KPI are linearly
unstable while all genus-1 solutions of KPII {are linearly stable within the
context of Whitham theory.Comment: Significantly revised versio
A discrete linearizability test based on multiscale analysis
In this paper we consider the classification of dispersive linearizable
partial difference equations defined on a quad-graph by the multiple scale
reduction around their harmonic solution. We show that the A_1, A_2 and A_3
linearizability conditions restrain the number of the parameters which enter
into the equation. A subclass of the equations which pass the A_3
C-integrability conditions can be linearized by a Mobius transformation
Algorithmic Integrability Tests for Nonlinear Differential and Lattice Equations
Three symbolic algorithms for testing the integrability of polynomial systems
of partial differential and differential-difference equations are presented.
The first algorithm is the well-known Painlev\'e test, which is applicable to
polynomial systems of ordinary and partial differential equations. The second
and third algorithms allow one to explicitly compute polynomial conserved
densities and higher-order symmetries of nonlinear evolution and lattice
equations.
The first algorithm is implemented in the symbolic syntax of both Macsyma and
Mathematica. The second and third algorithms are available in Mathematica. The
codes can be used for computer-aided integrability testing of nonlinear
differential and lattice equations as they occur in various branches of the
sciences and engineering. Applied to systems with parameters, the codes can
determine the conditions on the parameters so that the systems pass the
Painlev\'e test, or admit a sequence of conserved densities or higher-order
symmetries.Comment: Submitted to: Computer Physics Communications, Latex, uses the style
files elsart.sty and elsart12.st
- …