Cornelia Schiebold

Nonlinear equations in soliton physics and operator ideals

An operator-theoretic method for the investigation of nonlinear equations in soliton physics is discussed comprehensively. Originating from pioneering work of Marchenko, our operator-method is based on new insights into the theory of traces and determinants on operator ideals. Therefore, we give a systematic and concise approach to some recent developments in this direction which are important in the context of this paper. Our method is widely applicable. We carry out the corresponding arguments in detail for the Kadomtsev-Petviashvili equation and summarize the results concerning the Korteweg-de Vries and the modified Korteweg-de Vries equation as well as for the sine-Gordon equation. Exactly the same formalism works in the discrete case, as the treatment of the Toda lattice, the Langmuir and the Wadati lattice shows. AMS classification scheme numbers: 35C05, 35Q51, 35Q53, 35Q58, 47D50, 47N20

Noncommutative Korteweg-de Vries and modified Korteweg-de Vries hierarchies via recursion methods

Here, noncommutative hierarchies of nonlinear equations are studied. They represent a generalization to the operator level of corresponding hierarchies of scalar equations, which can be obtained from the operator ones via a suitable projection. A key tool is the application of Backlund transformations to relate different operator-valued hierarchies. Indeed, in the case when hierarchies in 1+1-dimensions are considered, a “Backlund chart” depicts links relating, in particular, the Korteweg–de Vries (KdV) to the modified KdV (mKdV) hierarchy. Notably, analogous links connect the hierarchies of operator equations. The main result is the construction of an operator soliton solution depending on an infinite-dimensional parameter. To start with, the potential KdV hierarchy is considered. Then Backlund transformations are exploited to derive solution formulas in the case of KdV and mKdV hierarchies. It is remarked that hierarchies of matrix equations, of any dimension, are also incorporated in the present framework.

An operator theoretic approach to the Toda lattice equation

Abstract We treat the Toda lattice equation with operator methods and derive an explicit solution formula in terms of determinants. As an application, we investigate solutions which are given by special settings. In the finite-dimensional case matrices in Jordan canonical form give rise to a new class of solutions. Within this class the well-known N-soliton solutions can be recovered by the special choice of diagonal matrices. Moreover, using diagonal operators we get solutions depending on an infinite number of parameters. We comprehensively discuss the case involving diagonal operators and show that it can be reduced to a very particular situation.

Explicit solution formulas for the matrix-KP

We study a non-commutative version of the Kadomtsev-Petviashvili equations and construct a family of solutions generalizing naturally the soliton to the non-commutative setting. From this we derive explicit solution formulas as well for the scalar as for the matrix-Kadomtsev-Petviashvili equation which still depend on operator parameters.

The noncommutative AKNS system: projection to matrix systems, countable superposition and soliton-like solutions

Starting from the recent work on noncommutative AKNS systems for functions with values in the bounded operators on a Banach space, it is shown how their formal 1-soliton solution (depending on operator parameters) can be mapped to solutions of matrix AKNS systems. The main result is rather general solution formulas for matrix AKNS systems. The most important applications are the countable superposition of matrix solitons and explicit expressions for the soliton-like solutions of the classical AKNS system.

Structural properties of the noncommutative KdV recursion operator

The present work studies structural properties of the recursion operator of the noncommutative KdV equation. As the main result, it is proved that this operator is hereditary. The notion of hereditary operators was introduced by Fuchssteiner for infinite-dimensional integrable systems, building on classical concepts from differential topology. As an illustration for the consequences of this property, it is deduced that the flows of the noncommutative KdV hierarchy mutually commute.

Matrix Korteweg-de Vries and modified Korteweg-de Vries hierarchies: Noncommutative soliton solutions

The present work continues work on KdV-type hierarchies presented by S. Carillo and C. Schiebold [“Noncommutative Korteweg-de Vries and modified Korteweg-de Vries hierarchies via recursion methods,” J. Math. Phys. 50, 073510 (2009)]10.1063/1.3155080. General solution formulas for the KdV and mKdV hierarchies are derived by means of Banach space techniques both in the scalar and matrix case. A detailed analysis is given of solitons, breathers, their countable superpositions as well as of multisoliton solutions for the matrix hierarchies.

Bäcklund transformations and non-abelian nonlinear evolution equations : A novel bäcklund chart

Classes of third order non-Abelian evolution equations linked to that of Korteweg-de Vries-type are investigated and their connections represented in a non-commutative Backlund chart, generalizing ...

ON THE RECURSION OPERATOR FOR THE NONCOMMUTATIVE BURGERS HIERARCHY

The noncommutative Burgers recursion operator is constructed via the Cole–Hopf transformation, and its structural properties are studied. In particular, a direct proof of its hereditary property is given.

FROM THE NON-ABELIAN TO THE SCALAR TWO-DIMENSIONAL TODA LATTICE

We extend a solution method used for the one-dimensional Toda lattice in [1], [2] to the two-dimensional Toda lattice. The idea is to study the lattice not with values in