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 A1, A2 and A3 linearizability conditions restrain the number of the parameters which enter into the equation. A subclass of the equations which pass the A3 C-integrability conditions can be linearized by a Möbius transformation

Motion of Curves and Surfaces and Nonlinear Evolution Equations in (2+1) Dimensions

It is shown that a class of important integrable nonlinear evolution equations in (2+1) dimensions can be associated with the motion of space curves endowed with an extra spatial variable or equivalently, moving surfaces. Geometrical invariants then define topological conserved quantities. Underlying evolution equations are shown to be associated with a triad of linear equations. Our examples include Ishimori equation and Myrzakulov equations which are shown to be geometrically equivalent to Davey-Stewartson and Zakharov -Strachan (2+1) dimensional nonlinear Schr\"odinger equations respectively.Comment: 13 pages, RevTeX, to appear in J. Math. Phy

The matrix Kadomtsev--Petviashvili equation as a source of integrable nonlinear equations

A new integrable class of Davey--Stewartson type systems of nonlinear partial differential equations (NPDEs) in 2+1 dimensions is derived from the matrix Kadomtsev--Petviashvili equation by means of an asymptotically exact nonlinear reduction method based on Fourier expansion and spatio-temporal rescaling. The integrability by the inverse scattering method is explicitly demonstrated, by applying the reduction technique also to the Lax pair of the starting matrix equation and thereby obtaining the Lax pair for the new class of systems of equations. The characteristics of the reduction method suggest that the new systems are likely to be of applicative relevance. A reduction to a system of two interacting complex fields is briefly described.Comment: arxiv version is already officia

Multiple-scale analysis of discrete nonlinear partial difference equations: the reduction of the lattice potential KdV

We consider multiple lattices and functions defined on them. We introduce slow varying conditions for functions defined on the lattice and express the variation of a function in terms of an asymptotic expansion with respect to the slow varying lattices. We use these results to perform the multiple--scale reduction of the lattice potential Korteweg--de Vries equation.Comment: 17 pages. 1 figur

Testing Hall-Post Inequalities With Exactly Solvable N-Body Problems

The Hall--Post inequalities provide lower bounds on $N$-body energies in terms of $N'$-body energies with $N'<N$. They are rewritten and generalized to be tested with exactly-solvable models of Calogero-Sutherland type in one and higher dimensions. The bound for $N$ spinless fermions in one dimension is better saturated at large coupling than for noninteracting fermions in an oscillatorComment: 7 pages, Latex2e, 2 .eps figure

Necessary and sufficient conditions for existence of bound states in a central potential

We obtain, using the Birman-Schwinger method, a series of necessary conditions for the existence of at least one bound state applicable to arbitrary central potentials in the context of nonrelativistic quantum mechanics. These conditions yield a monotonic series of lower limits on the "critical" value of the strength of the potential (for which a first bound state appears) which converges to the exact critical strength. We also obtain a sufficient condition for the existence of bound states in a central monotonic potential which yield an upper limit on the critical strength of the potential.Comment: 7 page

G(2)-Calogero-Moser Lax operators from reduction

We construct a Lax operator for the $G_2$-Calogero-Moser model by means of a double reduction procedure. In the first reduction step we reduce the $A_6$-model to a $B_3$-model with the help of an embedding of the $B_3$-root system into the $A_6$-root system together with the specification of certain coupling constants. The $G_2$-Lax operator is obtained thereafter by means of an additional reduction by exploiting the embedding of the $G_2$-system into the $B_3$-system. The degree of algebraically independent and non-vanishing charges is found to be equal to the degrees of the corresponding Lie algebra.Comment: 12 pages, Late

On the Integrability of the Discrete Nonlinear Schroedinger Equation

In this letter we present an analytic evidence of the non-integrability of the discrete nonlinear Schroedinger equation, a well-known discrete evolution equation which has been obtained in various contexts of physics and biology. We use a reductive perturbation technique to show an obstruction to its integrability.Comment: 4 pages, accepted in EP