L. Martínez Alonso

Schrödinger spectral problems with energy–dependent potentials as sources of nonlinear Hamiltonian evolution equations

We develop a method to derive infinite families of completely integrable nonlinear Hamiltonian evolution equations associated with Schrodinger spectral problems whose potential functions depend on the spectral parameter.

Hydrodynamic reductions and solutions of a universal hierarchy

We investigate the diagonal hydrodynamic reductions of a hierarchy of integrable hydrodynamic chains and analyze their compatibility with previously introduced reductions of differential type.

A general setting for Casimir invariants

This paper contains a general description of the theory of invariants under the adjoint action of a given finite‐dimensional complex Lie algebra G, with special emphasis on polynomial and rational invariants. The familiar ’’Casimir’’ invariants are identified with the polynomial invariants in the enveloping algebra U (G). More general structures (quotient fields) are required in order to investigate rational invariants. Some useful criteria for G having only polynomial or rational invariants are given. Moreover, in most of the physically relevant Lie algebras the exact computation of the maximal number of algebraically independent invariants turns out to be very easy. It reduces to finding the rank of a finite matrix. We apply the general method to some typical examples.

Energy-dependent potentials revisited: a universal hierarchy of hydrodynamic type☆

A hierarchy of infinite-dimensional systems of hydrodynamic type is considered and a general scheme for classifying its reductions is provided. Wide families of integrable systems including, in particular, those associated with energy-dependent spectral problems of Schrodinger type, are characterized as reductions of this hierarchy. N-phase type reductions and their corresponding Dubrovin equations are analyzed. A symmetry transformation connecting different classes of reductions is formulated.

On the Prolongation of a Hierarchy of Hydrodynamic Chains

The prolongation of a hierarchy of hydrodynamic chains previously studied by the authors is presented and the properties of the differential reductions of the enlarged hierarchy are derived. Several associated nonlinear integrable models are exhibited. In particular, it is found that the Camassa-Holm equation can be described as a second-order differential reduction of one of the new flows included in the hierarchy.

Dispersionless scalar integrable hierarchies, Whitham hierarchy, and the quasiclassical ∂̄-dressing method

The quasiclassical limit of the scalar nonlocal ∂-problem is derived and a quasiclassical version of the ∂-dressing method is presented. Dispersionless Kadomtsev–Petviashvili (KP), modified KP, and dispersionless two-dimensional Toda lattice (2DTL) hierarchies are discussed as illustrative examples. It is shown that the universal Whitham hierarchy is nothing but the ring of symmetries for the quasiclassical ∂-problem. The reduction problem is discussed and, in particular, the d2DTL equation of B type is derived.

Modified Singular Manifold Expansion: Application to the Boussinesq and Mikhailov-Shabat Systems

We present a modified treatment of the singular manifold method as an improved variant of the Painleve analysis for partial differential equations with two branches in the Painleve expansion. We illustrate the method by fully applying it to the Classical Boussinesq system and to the Mikhailov-Shabat system.

On the Noether map

For a wide class of lagrangian systems we show rigorously that the conventional formulation of Noethers theorem provides a bijective map from the set of equivalence classes of Noethers symmetries onto the set of equivalence classes of conserved currents. We further discuss if Noethers theorem is generalized in a significant way by several formulations proposed in this decade.For a wide class of lagrangian systems we show rigorously that the conventional formulation of Noethers theorem provides a bijective map from the set of equivalence classes of Noethers symmetries onto the set of equivalence classes of conserved currents. We further discuss if Noethers theorem is generalized in a significant way by several formulations proposed in this decade.

Explicit solutions of supersymmetric KP hierarchies: Supersolitons and solitinos

Wide classes of explicit solutions of the Manin‐Radul and Jacobian supersymmetric KP hierarchies are constructed by using line bundles over complex supercurves based on the Riemann sphere. Their construction extends several ideas of the standard KP theory, such as wave functions, ∂‐equations and τ‐functions. Thus, supersymmetric generalizations of N‐soliton solutions, including a new purely odd ‘‘solitino’’ solution, as well as rational solutions, are found and characterized.

Fusion and fission of dromions in the Davey-Stewartson equation

Abstract Multidromion solutions of the Davey-Stewartson equation are considered from the point of view of the bilinear formalism. Their characterization in terms of elements of a Clifford group is used to generate solutions in which the numbers of incoming and outgoing dromions are different. It is shown that processes of fusion and fission of dromions take place.