S.B. Leble

A wave equation in 2+1: Painleve analysis and solutions

In this paper the nonlinear equation mty=(myxx+mxmy)x is thoroughly analysed. The Painleve test is performed yielding a positive result. The Backlund transformations are found and the Darboux-Moutard-Matveev formalism arises in the context of this analysis. The singular manifold method, based upon the Painleve analysis, is proved to be a useful tool for generating solutions. Some interesting explicit expressions for one and two solitons are obtained and analysed in such a way.

Elementary and binary Darboux transformations at rings

For a given idempotent p and some element σ from a differential associative ring, we introduce gauge transformation λp + σ with the spectral parameter λ that leaves some linear operators form invariant. The explicit form of the σ is derived for the generalized Zakharov-Shabat problem. The maps that factorize Darboux transformations are referred to as elementary ones. Binary transformations that correspond to the iteration of elementary maps with the special choice of solutions of the direct and conjugate problems are introduced and widen the potential reductions set. We show an infinitesimal version of iterated transforms. The spectral operator and soliton theories applications are outlined. The nonabelian N-wave interaction equation modification with additional linear terms, its zero curvature representation, soliton solutions, and their stability with respect to infinitesimal deformations are studied.

Numerical integration of a coupled Korteweg-de Vries system

Abstract We introduce a numerical method for general coupled Korteweg-de Vries systems. The scheme is valid for solving Cauchy problems for an arbitrary number of equations with arbitrary constant coefficients. The numerical scheme takes its legality by proving its stability and convergence, which gives the conditions and the appropriate choice of the grid sizes. The method is applied to the Hirota-Satsuma (HS) system and compared with its known explicit solution investigating the influence of initial conditions and grid sizes on accuracy. We also illustrate the method to show the effects of constants with a transition to nonintegrable cases.

Binary Bell polynomials and Darboux covariant Lax pairs

Hirota representations of soliton equations have proved very useful. They produced many of the known families of multisoliton solutions, and have often lead to a disclosure of the underlying Lax systems and infinite sets of conserved quantities [1,2]. A striking feature is the ease with which direct insight can be gained into the nature of the eigenvalue problem associated with soliton equations derivable from a quadratic Hirota equation (for a single Hirota function), such as the KdV equation, the Boussinesq equation or the Sawada-Kotera equation. A key element is the bilinear Backlund transformation (BT) which can be obtained straight away from the Hirota representation of these equations, through decoupling of a related “ two field condition” by means of an appropriate constraint of minimal weight. Details of this procedure have been reported elsewhere [3,4]. The main point is that bilinear BT’s are obtained systematically, without the need of tricky “exchange formulas” [1]. They arise in the form of “Y-systems”, each equation of which belongs to a linear space spanned by a basis of binary Bell polynomials (Y-polynomials) [5]. An important element is the logarithmic linearizability of Y-systems, which implies that each bilinear BT can be mapped onto a corresponding linear system of Lax type. However, it turns out that these linear systems involve differential operators which, even in the simplest case, do not constitute a Darboux covariant [6] Lax pair. This fact prevents us from obtaining large classes of solutions by direct application of the powerful Darboux machinery to the systems which arise by straightforward linearization of the Y-systems. Here we present a simple scheme by means of which this difficulty can be resolved for a variety of soliton equations which admit a bilinear BT that comprises a constraint of lowest possible weight (weight 2). Darboux covariant Lax pairs for the KdV, Boussinesq and Lax equations are obtained in a unified manner, by exploiting the relations between the coefficients of linear differential operators connected by a classical Darboux transformation. Exponential Bell polynomials [7] and generalized “multipotential” Y-systems are found to be useful for this purpose. The latter reveal deep connections between the 1+1 dimensional equations that are analyzed and the underlying (higher dimensional) KP hierarchy. We start our discussion by recalling the main properties of the Y-polynomials (derived in [5]) and by indicating how the use of the Y-basis can lead systematically from the original NLPDE’s to the associated linear systems. The example of the Lax equation is instructive since this fifth order equation cannot be derived from a single quadratic Hirota equation.

Division of differential operators, intertwine relations and darboux transformations

Abstract The problem of a differential operator left- and right division is solved in terms of generalized Bell polynomials for a nonabelian differential unitary ring. The definition of the polynomials is made by means of recurrent relations. The expressions of classic Bell polynomials via a generalized one is given. Conditions of exact factorization lead to intertwine relations and result in linearizable generalized Burgers equation. An alternative proof of the Matveev theorem is given and transformation formulae for the coefficients of differential operator in terms of differential polynomials follow from the intertwine relation.

Intertwine operators and elementary darboux transforms in differential rings and modules

Abstract This paper presents an algebraic structure allowing to construct an elementary Darboux transformation. The set of necessary propositions is listed and the generic Darboux theorem is formulated for differential rings. The result is given for the generalized Zakharov-Shabat problem. The conditions for the appearance of the Schlesinger transformation and a construction of an analogue of supersymmetry algebra are shown starting from the intertwine relation. Some applications in the theory of nonlinear equations are also discussed.

Piecewise continuous distribution function method in the theory of wave disturbances of inhomogeneous gas

The system of hydrodynamic-type equations for a stratified gas in gravity field is derived from BGK equation by method of piecewise continuous distribution function. The obtained system of the equations generalizes the Navier–Stokes one at arbitrary Knudsen numbers. The problem of a wave disturbance propagation in a rarefied gas is explored. The verification of the model is made for a limiting case of a homogeneous medium. The phase velocity and attenuation coefficient values are in an agreement with former fluid mechanics theories; the attenuation behavior reproduces experiment and kinetics-based results at more wide range of the Knudsen numbers.  2005 Elsevier B.V. All rights reserved.

A KdV Equation in 2 + 1 Dimensions: Painleve Analysis, Solutions and Similarity Reductions

The nonlinear equationmty=(myxx+mxmy)x is throughly analyzed. The Painlevé test yields a positive result. The Bäckhand transformations are found and the Darboux-MoutardMatveev formalism arises in the context of this analysis. Some solutions and their interactions are also analyzed. The singular manifold equations are also used to determine symmetry reductions. This procedure can be related with the direct method of Clarkson and Kruskal.

Nonlinear Waves in Optical Waveguides and Soliton Theory Applications

Starting with dielectric slab as a waveguide, we discuss the formal aspects of a derivation of model (soliton equations) reducing the description of the threedimensional electromagnetic wave. The link to the novel experiments in planar dielectric guides is shown. The derivation we consider as an asymptotic in a small parameter that embed the soliton equation into a general physical model. The resulting system is coupled NS (c NS). Then we go to the nonlinear resonance description; N-wave interactions. Starting from general theory and integration by dressing method arising from Darboux transformation (DT) techniques. Going to linear resonance, we study N-level Maxwell-Bloch (MB) equations with rescaling. Integrability and solutions and perturbation theory of MB equations is treated again via DT approach. The solution of the Manakov system (the cNS with equal nonlinear constants) is integrated by the same Zakharov-Shabat (ZS) problem. The case of non reduced MB equation integrability is discussed in the context of the general quantum Liouville-von Neumann (LvN) evolution equation as associated ZS problem.

Nonlinear von Neumann-type equations: Darboux invariance and spectra

Abstract Generalized Euler-Arnold-von Neumann density matrix equations can be solved by a binary Darboux transformation, given here in a new form: ρ [1]= e P ln( μ / ν ) ρe − P ln( μ / ν ) , where P = P 2 is explicitly constructed in terms of conjugated Lax pairs, and μ , ν are complex. As a result spectra of ρ and ρ [1] are identical. Transformations allowing to shift and rescale spectrum of a solution are introduced, and a class of stationary seed solutions is discussed.