99 research outputs found
Nonlinear Modulation of Multi-Dimensional Lattice Waves
The equations governing weakly nonlinear modulations of -dimensional
lattices are considered using a quasi-discrete multiple-scale approach. It is
found that the evolution of a short wave packet for a lattice system with cubic
and quartic interatomic potentials is governed by generalized Davey-Stewartson
(GDS) equations, which include mean motion induced by the oscillatory wave
packet through cubic interatomic interaction. The GDS equations derived here
are more general than those known in the theory of water waves because of the
anisotropy inherent in lattices. Generalized Kadomtsev-Petviashvili equations
describing the evolution of long wavelength acoustic modes in two and three
dimensional lattices are also presented. Then the modulational instability of a
-dimensional Stokes lattice wave is discussed based on the -dimensional
GDS equations obtained. Finally, the one- and two-soliton solutions of
two-dimensional GDS equations are provided by means of Hirota's bilinear
transformation method.Comment: Submitted to PR
Multicomponent Burgers and KP Hierarchies, and Solutions from a Matrix Linear System
Via a Cole-Hopf transformation, the multicomponent linear heat hierarchy
leads to a multicomponent Burgers hierarchy. We show in particular that any
solution of the latter also solves a corresponding multicomponent (potential)
KP hierarchy. A generalization of the Cole-Hopf transformation leads to a more
general relation between the multicomponent linear heat hierarchy and the
multicomponent KP hierarchy. From this results a construction of exact
solutions of the latter via a matrix linear system.Comment: 18 pages, 4 figure
Multidimensional Localized Solitons
Recently it has been discovered that some nonlinear evolution equations in
2+1 dimensions, which are integrable by the use of the Spectral Transform,
admit localized (in the space) soliton solutions. This article briefly reviews
some of the main results obtained in the last five years thanks to the renewed
interest in soliton theory due to this discovery. The theoretical tools needed
to understand the unexpected richness of behaviour of multidimensional
localized solitons during their mutual scattering are furnished. Analogies and
especially discrepancies with the unidimensional case are stressed
Integrable (2+1)-Dimensional Spin Models with Self-Consistent Potentials
Integrable spin systems possess interesting geometrical and gauge invariance
properties and have important applications in applied magnetism and
nanophysics. They are also intimately connected to the nonlinear Schr\"odinger
family of equations. In this paper, we identify three different integrable spin
systems in (2 + 1) dimensions by introducing the interaction of the spin field
with more than one scalar potential, or vector potential, or both. We also
obtain the associated Lax pairs. We discuss various interesting reductions in
(2 + 1) and (1 + 1) dimensions. We also deduce the equivalent nonlinear
Schr\"odinger family of equations, including the (2 + 1)-dimensional version of
nonlinear Schr\"odinger--Hirota--Maxwell--Bloch equations, along with their Lax
pairs.Comment: 21 page
New extended generalized Kudryashov method for solving three nonlinear partial differential equations
New extended generalized Kudryashov method is proposed in this paper for the first time. Many solitons and other solutions of three nonlinear partial differential equations (PDEs), namely, the (1+1)-dimensional improved perturbed nonlinear Schrödinger equation with anti-cubic nonlinearity, the (2+1)-dimensional Davey–Sterwatson (DS) equation and the (3+1)-dimensional modified Zakharov–Kuznetsov (mZK) equation of ion-acoustic waves in a magnetized plasma have been presented. Comparing our new results with the well-known results are given. Our results in this article emphasize that the used method gives a vast applicability for handling other nonlinear partial differential equations in mathematical physics
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