300 research outputs found
Two-component Analogue of Two-dimensional Long Wave-Short Wave Resonance Interaction Equations: A Derivation and Solutions
The two-component analogue of two-dimensional long wave-short wave resonance
interaction equations is derived in a physical setting. Wronskian solutions of
the integrable two-component analogue of two-dimensional long wave-short wave
resonance interaction equations are presented.Comment: 16 pages, 9 figures, revised version; The pdf file including all
figures: http://www.math.utpa.edu/kmaruno/yajima.pd
Hydrodynamic chains and a classification of their Poisson brackets
Necessary and sufficient conditions for an existence of the Poisson brackets
significantly simplify in the Liouville coordinates. The corresponding
equations can be integrated. Thus, a description of local Hamiltonian
structures is a first step in a description of integrable hydrodynamic chains.
The concept of Poisson bracket is introduced. Several new Poisson brackets
are presented
Instability and Evolution of Nonlinearly Interacting Water Waves
We consider the modulational instability of nonlinearly interacting
two-dimensional waves in deep water, which are described by a pair of
two-dimensional coupled nonlinear Schroedinger equations. We derive a nonlinear
dispersion relation. The latter is numerically analyzed to obtain the regions
and the associated growth rates of the modulational instability. Furthermore,
we follow the long term evolution of the latter by means of computer
simulations of the governing nonlinear equations and demonstrate the formation
of localized coherent wave envelopes. Our results should be useful for
understanding the formation and nonlinear propagation characteristics of large
amplitude freak waves in deep water.Comment: 4 pages, 4 figures, to appear in Physical Review Letter
A Singular Perturbation Analysis for \\Unstable Systems with Convective Nonlinearity
We use a singular perturbation method to study the interface dynamics of a
non-conserved order parameter (NCOP) system, of the reaction-diffusion type,
for the case where an external bias field or convection is present. We find
that this method, developed by Kawasaki, Yalabik and Gunton for the
time-dependant Ginzburg-Landau equation and used successfully on other NCOP
systems, breaks down for our system when the strength of bias/convection gets
large enough.Comment: 5 pages, PostScript forma
KP solitons in shallow water
The main purpose of the paper is to provide a survey of our recent studies on
soliton solutions of the Kadomtsev-Petviashvili (KP) equation. The
classification is based on the far-field patterns of the solutions which
consist of a finite number of line-solitons. Each soliton solution is then
defined by a point of the totally non-negative Grassmann variety which can be
parametrized by a unique derangement of the symmetric group of permutations.
Our study also includes certain numerical stability problems of those soliton
solutions. Numerical simulations of the initial value problems indicate that
certain class of initial waves asymptotically approach to these exact solutions
of the KP equation. We then discuss an application of our theory to the Mach
reflection problem in shallow water. This problem describes the resonant
interaction of solitary waves appearing in the reflection of an obliquely
incident wave onto a vertical wall, and it predicts an extra-ordinary four-fold
amplification of the wave at the wall. There are several numerical studies
confirming the prediction, but all indicate disagreements with the KP theory.
Contrary to those previous numerical studies, we find that the KP theory
actually provides an excellent model to describe the Mach reflection phenomena
when the higher order corrections are included to the quasi-two dimensional
approximation. We also present laboratory experiments of the Mach reflection
recently carried out by Yeh and his colleagues, and show how precisely the KP
theory predicts this wave behavior.Comment: 50 pages, 25 figure
Weak Transversality and Partially Invariant Solutions
New exact solutions are obtained for several nonlinear physical equations,
namely the Navier-Stokes and Euler systems, an isentropic compressible fluid
system and a vector nonlinear Schroedinger equation. The solution methods make
use of the symmetry group of the system in situations when the standard Lie
method of symmetry reduction is not applicable.Comment: 23 pages, preprint CRM-284
The algebraic and Hamiltonian structure of the dispersionless Benney and Toda hierarchies
The algebraic and Hamiltonian structures of the multicomponent dispersionless
Benney and Toda hierarchies are studied. This is achieved by using a modified
set of variables for which there is a symmetry between the basic fields. This
symmetry enables formulae normally given implicitly in terms of residues, such
as conserved charges and fluxes, to be calculated explicitly. As a corollary of
these results the equivalence of the Benney and Toda hierarchies is
established. It is further shown that such quantities may be expressed in terms
of generalized hypergeometric functions, the simplest example involving
Legendre polynomials. These results are then extended to systems derived from a
rational Lax function and a logarithmic function. Various reductions are also
studied.Comment: 29 pages, LaTe
Evolution of Rogue Waves in Interacting Wave Systems
Large amplitude water waves on deep water has long been known in the sea
faring community, and the cause of great concern for, e.g., oil platform
constructions. The concept of such freak waves is nowadays, thanks to satellite
and radar measurements, well established within the scientific community. There
are a number of important models and approaches for the theoretical description
of such waves. By analyzing the scaling behavior of freak wave formation in a
model of two interacting waves, described by two coupled nonlinear Schroedinger
equations, we show that there are two different dynamical scaling behaviors
above and below a critical angle theta_c of the direction of the interacting
waves below theta_c all wave systems evolve and display statistics similar to a
wave system of non-interacting waves. The results equally apply to other
systems described by the nonlinear Schroedinger equations, and should be of
interest when designing optical wave guides.Comment: 5 pages, 2 figures, to appear in Europhysics Letter
Surface Instability of Icicles
Quantitatively-unexplained stationary waves or ridges often encircle icicles.
Such waves form when roughly 0.1 mm-thick layers of water flow down the icicle.
These waves typically have a wavelength of 1cm approximately independent of
external temperature, icicle thickness, and the volumetric rate of water flow.
In this paper we show that these waves can not be obtained by naive
Mullins-Sekerka instability, but are caused by a quite new surface instability
related to the thermal diffusion and hydrodynamic effect of thin water flow.Comment: 11 pages, 5 figures, Late
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