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 $M$ 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

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

Stable periodic waves in coupled Kuramoto-Sivashinsky - Korteweg-de Vries equations

Periodic waves are investigated in a system composed of a Kuramoto-Sivashinsky - Korteweg-de Vries (KS-KdV) equation, which is linearly coupled to an extra linear dissipative equation. The model describes, e.g., a two-layer liquid film flowing down an inclined plane. It has been recently shown that the system supports stable solitary pulses. We demonstrate that a perturbation analysis, based on the balance equation for the field momentum, predicts the existence of stable cnoidal waves (CnWs) in the same system. It is found that the mean value U of the wave field u in the main subsystem, but not the mean value of the extra field, affects the stability of the periodic waves. Three different areas can be distinguished inside the stability region in the parameter plane (L,U), where L is the wave's period. In these areas, stable are, respectively, CnWs with positive velocity, constant solutions, and CnWs with negative velocity. Multistability, i.e., the coexistence of several attractors, including the waves with several maxima per period, appears at large value of L. The analytical predictions are completely confirmed by direct simulations. Stable waves are also found numerically in the limit of vanishing dispersion, when the KS-KdV equation goes over into the KS one.Comment: a latex text file and 16 eps files with figures. Journal of the Physical Society of Japan, in pres