J. Prada

A Generalization of the Sine-Gordon Equation to 2+1 Dimensions

Abstract The Singular Manifold Method (SMM) is applied to an equation in 2 + 1 dimensions [13] that can be considered as a generalization of the sine-Gordon equation. SMM is useful to prove that the equation has two Painlevé branches and, therefore, it can be considered as the modified version of an equation with just one branch, that is the AKNS equation in 2 + 1 dimensions. The solutions of the former split as linear superposition of two solutions of the second, related by a B¨acklund-gauge transformation. Solutions of both equations are obtained by means of an algorithmic procedure derived from these transformations.

Hodograph transformations for a Camassa–Holm hierarchy in 2 + 1 dimensions

A generalization of the negative Camassa–Holm hierarchy to 2 + 1 dimensions is presented under the name CHH(2+1). Several hodograph transformations are applied in order to transform the hierarchy into a system of coupled CBS (Calogero–Bogoyavlenskii–Schiff) equations in 2 + 1 dimensions that pass the Painleve test. A non-isospectral Lax pair for CHH(2+1) is obtained through the above-mentioned relationship with the CBS spectral problem.

Singular Manifold Method for an Equation in 2+1 Dimensions

Abstract The Singular Manifold Method is presented as an excellent tool to study a 2 + 1 dimensional equation in despite of the fact that the same method presents several problems when applied to 1 + 1 reductions of the same equation. Nevertheless these problems are solved when the number of dimensions of the equation is increased.

On an algorithmic construction of lump solutions in a 2+1 integrable equation

The singular manifold method is used to generate lump solutions of a generalized integrable nonlinear Schrodinger equation in 2 + 1 dimensions. We present several essentially different types of lump solutions. The connection between this method and the Ablowitz–Villarroel scheme is also analysed.

APPELL BASES ON SEQUENCE SPACES

We study conditions for a sequence of Appell polynomials to be a basis on a sequence space.

Integrability properties of a generalized reduction of the KdV6 equation

Abstract We consider the integrability properties of a generalized version of a similarity reduction of the so-called KdV6 equation, an equation that has recently generated much interest. We give a linear problem for this generalized reduction and show that it satisfies the requirements of the Ablowitz–Ramani–Segur algorithm. In addition we give a Backlund transformation to a related equation, giving also an auto-Backlund transformation for this last. Our results mirror those for the Korteweg–de Vries equation itself, which has a similarity reduction to an ordinary differential equation which is related by a Backlund transformation to the second Painleve equation, this last having an auto-Backlund transformation.

On the integrability of the nonlinear Schrödinger equation with randomly dependent linear potential

We show that the Cauchy problem for the nonlinear Schrodinger equation driven by a term of the form where W(t) is a Brownian motion can be integrated by means of the inverse scattering transform. The solution is written in terms of W(t) and the integral of powers , j = 1, 2. A complete set of integrals of the motion is determined. We relate the probability density of the solution to the Greens function of a linear partial differential operator with quadratic coefficients.

1 + 1 spectral problems arising from the Manakov?Santini system

This paper deals with the spectral problem of the Manakov–Santini system. The point Lie symmetries of the Lax pair have been identified. Several similarity reductions arise from these symmetries. An important benefit of our procedure is that the study of the Lax pair instead of the partial differential equations yields the reductions of the eigenfunctions and also the spectral parameter. Therefore, we have obtained five interesting spectral problems in 1 + 1 dimensions.

On Differential Operators on Sequence Spaces

Abstract Two differential operators T 1 and T 2 on a space Λ are said to be equivalent if there is an isomorphism S from Λ onto Λ such that ST 1=T 2 S. The notion was first introduced by Delsarte in 1938 [2] where T 1 and T 2 are differential operators of second order and L a space of functions of one variable defined for x ≥ 0. From them on several authors studied generalizations, applications and related problems [6], [7], [8], [11], [12]. In 1957 Delsarte and Lions [3] proved that if T 1 and T 2 are differential operators of the same order without singularities on the complex plane, Λ being the space of entire functions, then they are equivalent. Using sequence spaces and the fact that the space of entire functions is isomorphic to a power series space of infinite type, we find the same result in a simpler way in our opinion. The same method gives the result for differential operators of the same order with analytic coefficients on the space of holomorphic function on a disc, considering that spaces of holomorphic functions on a disc are isomorphic to a finite power series space. The method can be applied as well to linear differential operators of the same order on other sequence spaces, finding conditions for them to be equivalent. Finally using the fact that the space of all 2π-periodic -functions on ℝ is isomorphic to s, the space of rapidly decreasing sequences, we prove that two linear differential operators of order one with constant coefficients are not equivalent; this result can be extended to linear differential operators of greater order but the proof is essentially the same.

Discrete Spectrum of 2

We consider a natural integrable generalization of nonlinear Schrodinger equation to dimensions. By studying the associated spectral operator we discover a rich discrete spectrum associated with regular rationally decaying solutions, the lumps, which display interesting nontrivial dynamics and scattering. Particular interest is placed in the dynamical evolution of the associated pulses. For all cases under study we find that the relevant dynamics corresponds to a central configuration of a certain -body problem.