Using a bidifferential graded algebra approach to integrable partial differential or difference equations, a unified treatment of continuous, semi-discrete (Ablowitz-Ladik) and fully discrete matrix NLS systems is presented. These equations originate from a universal equation within this framework, by specifying a representation of the bidifferential graded algebra and imposing a reduction. By application of a general result, corresponding families of exact solutions are obtained that in particular comprise the matrix soliton solutions in the focusing NLS case. The solutions are parametrised in terms of constant matrix data subject to a Sylvester equation (which previously appeared as a rank condition in the integrable systems literature). These data exhibit a certain redundancy, which we diminish to a large extent. More precisely, we first consider more general AKNS-type systems from which two different matrix NLS systems emerge via reductions. In the continuous case, the familiar Hermitian conjugation reduction leads to a continuous matrix (including vector) NLS equation, but it is well-known that this does not work as well in the discrete cases. On the other hand there is a complex conjugation reduction, which apparently has not been studied previously. It leads to square matrix NLS systems, but works in all three cases (continuous, semi- and fully-discrete). A large part of this work is devoted to an exploration of the corresponding solutions, in particular regularity and asymptotic behaviour of matrix soliton solutions.
Using the framework of bidifferential calculi (or bidifferential graded algebras), a unification of integrability aspects and solution generating techniques for a wide class of integrable models has been achieved [1] (see also the references cited therein). In particular, there is a fairly simple and universal method to construct large families of exact solutions from solutions of a linear system of equations. The formalism treats continuous and discrete systems on an equal level. This suggests a corresponding study of the well-known nonlinear Schrödinger (NLS) equation (see e.g. [2,3]), its semi-discretisation, the integrable discrete NLS or Ablowitz-Ladik (AL) equation [4][5][6][7], and a full discretisation [5]. The literature on mathematical aspects of the continuous NLS equation and its applications in science is meanwhile impossible to summarise. A rather incomplete list of references dealing with mathematical aspects of the AL equation is [3,, and more generally for its hierarchy. Physical applications of the AL equation can be found e.g. in [70][71][72][73][74][75][76]. In this work, we consider more generally matrix versions of the (continuous and discrete) NLS equations and present corresponding solutions. Matrix generalisations, including vector versions, of the scalar continuous NLS equation have been studied in particular in [3,. A brief summary of the physical relevance can be found in [3]. In particular, a (symmetric) 2 × 2 matrix NLS equation turned out to be of relevance for the description of a special Bose-Einstein condensate (with atoms in a spin 1 state) [102][103][104][105][106][107][108][109][110][111][112][113][114][115]. Semi-discrete matrix NLS equations appeared in [3,[116][117][118][119][120][121][122][123][124][125][126][127][128][129], a full discretisation has been elaborated in [130] and a dispersionless limit studied in [127].
In section 2 we introduce the general framework. Section 3 then presents bidifferential calculi for the NLS system and its discrete versions. Section 4 derives exact solutions, by application of the method of section 2. The solutions are parametrised by matrix data that have to solve a Sylvester equation. 1 In section 5, these results are translated into a decomposed form, where the NLS systems attain a more familiar form. Section 6 deals with a complex conjugation reduction which applies to the continuous as well as the discrete NLS systems. Quite surprisingly, this has apparently not been studied previously for matrix NLS equations. Our analysis in particular addresses regularity and asymptotic behaviour of matrix soliton solutions. Section 7 treats the more familiar Hermitian conjugation reduction which, however, only works for the continuous NLS system, at least without severe restrictions or introduction of non-locality (cf. appendix D). Section 8 contains some concluding remarks. Some supplementary material has been shifted to appendices. Appendix A briefly treats the considerably simpler example of the matrix KdV equation, in particular to provide the reader with a quick access to the methods used in this work, but also in order to stress the similarities even in details of the calculation. Appendix B establishes a relation between the semi-discrete matrix NLS (AL) system and the matrix modified KdV equation. Appendix C demonstrates that our fully discrete matrix NLS system reduces in the scalar case to the corresponding equation in [5]. Appendix D presents a non-local reduction of the semi-discrete NLS system. Appendix E recovers Lax pairs for the three NLS systems, starting with the respective bidifferential calculus. Finally, Appendix F contains some remarks on the continuum limit of the fully discrete matrix NLS system and the solutions obtained in this work. 2 The framework Definition 2.1. A (complex) graded algebra is an algebra2 Ω that has a direct sum decomposition Ω = r≥0 Ω r
(2.1) into complex vector spaces Ω r , with the property Ω r Ω s ⊆ Ω r+s .
(2.2) Furthermore, A := Ω 0 is a subalgebra and, for r > 0, Ω r is an A-bimodule.
Definition 2.2. A bidifferential graded algebra or bidifferential calculus is a graded algebra Ω equipped with two graded derivations d, d : Ω → Ω of degree one (hence dΩ r ⊆ Ω r+1 , dΩ r ⊆ Ω r+1 ), with the properties
and the graded Leibniz rule
for all χ ∈ Ω r and χ ′ ∈ Ω.
For an algebra A, a corresponding graded algebra is given by
where (C 2 ) denotes the exterior algebra of C 2 . 3 In this case it is sufficient to define the maps d, d on A. They extend in an obvious way to Ω such that the Leibniz rule holds. Elements of (C 2 ) are treated as “constants”. This structure underlies all our examples. In this work, ζ 1 , ζ 2 will denote a basis of 1 (C 2 ). Given a unital algebra B and a bidifferential graded algebra (Ω, d, d) with Ω 0 = B, the actions of d and d extend componentwise to matrices over B. A corresponding example is provided in Appendix A, the matrix KdV equation. By application of result
This content is AI-processed based on open access ArXiv data.
