Generalized Schrödinger equations and Jordan pairs

Svinolupov, S. I.

doi:10.1007/bf02099265

Cited by 55 publications

(39 citation statements)

References 9 publications

(20 reference statements)

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“…As is often the case with one-component soliton systems [7,13,15,16,17,18,19], the Chen-Lee-Liu equation also has plural multi-field generalizations. Using the Lax pairs, the conservation laws and the gauge transformations, we have studied the properties of the two types of the coupled Chen-Lee-Liu equations.…”

Section: Discussionmentioning

confidence: 94%

New integrable systems of derivative nonlinear Schrödinger equations with multiple components

1999

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Section: Discussionmentioning

confidence: 94%

New integrable systems of derivative nonlinear Schrödinger equations with multiple components

1999

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“…If U = k u k e k , where e 1 , . .., e N is a basis in V, then (40) is equivalent to a PDE system of the form (13). We will use the following notation…”

Section: Mkdv Type Systems Related To Pairs Of Compatible Algebraic Smentioning

confidence: 99%

“…Two non-linear terms in the right hand side are defined in terms of the matrix commutator while the others are related to the same triple Jordan system as in (7). Let us write (13) in the algebraic form…”

Section: Introductionmentioning

confidence: 99%

Multi-component generalizations of mKdV equation and nonassociative algebraic structures

Shestakov

Соколов

2020

J. Algebra Appl.

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Relations between triple Jordan systems and integrable multi-component models of the modified Korteveg–de Vries type are established. The most general model is related to a pair consisting of a triple Jordan system and a skew-symmetric bilinear operation. If this operation is a Lie bracket, then we arrive at the Lie–Jordan algebras [Speciality of Lie–Jordan algebras, J. Algebra 237 (2001) 621–636].

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“…; e.g., see [5,11]). For the integrable many-component generalizations of these equations [12,13] related to nonassociative algebras, one can construct integrable boundary conditions. Apparently, these boundary conditions, as well as (2.24), can be concisely ~=:::1 in terms of the correspondir.g .-'.~,nassociative algebras.…”

Section: S)mentioning

confidence: 99%

Integrable boundary conditions for many-component burgers equations

Svinolupov

Habibullin

1996

Math Notes

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ABSTRACT. Infinite series of boundary conditions that are consistent with even-order higher symmetries find ensure the integrability of a Burgers type equation are constructed.KEY WORDS: integrable equations, higher symmetries, boundary conditions, recursion operators.The nonlinear equations of mathematical physics contain the important class of integrable equations that can be either integrated by the inverse problem method or Linearized by differential substitutions (S-and C-integrable equations, respectively, in Calogero's terminology [1]). From the viewpoint of the symmetry approach [2], such systems axe distinguished by the fact that they have infinitely many higher symmetries (commuting flows). From the analytical viewpoint, the integrable equations are characterized by the fact that they admit a wide class of exact and asymptotic solutions. The presence of higher symmetries permits one to reduce the construction of soliton and finite-gap solutions of partial differential equations to solving ordinary differential equations.Sometimes, the solution is subjected to additional boundary conditions. In that connection, the following question arises: What boundary conditions can be satisfied by an exact solution (constructed with the help of higher symmetries). In other words, we must describe boundary conditions consistent with the higher symmetries (the precise definition will be given later). The first examples of nontrivial boundary conditions of this sort were obtained by Moser [3] for the Toda lattice and by Sklyanin [4] for nonlinear Schzfidinger type equations.In [5], an algorithm for constructing integrable boundary condition was suggested. This algorithm involves finding differential reductions for some systems of differential equations that are constructed in a unified way from the higher symmetries of the original equation. In the present paper, we consider the integrable many-component Burgers type equations constructed in [6]. These are systems of N equations of the form u,~ +2ajku u, + bjk,=uJuku '~, i = 1,... ,N, (0.1) where u ~ = u~(t, z), {a~k, b}k,~ } is a collection of constants satisfying certain polynomial identities, and summation from 1 to N ox-er repeated indices is assumed. For these systems, we find an infinite series of boundary conditions consistent with the higher symmetries and having the formwhere U = (u I , ..., u N) and Uk = cgkU/cDz ~ . The boundary conditions from this infinite series can be obtained by applying a differential operator to the simplest of these conditions (by analogy with the recursion operator for symmetries, this operator is naturally referred to as the recursion operator for integrable boundary conditions). Not that so far the authors do not know any similar recursion operators for boundary conditions for other integrable equations. Integrable many-component polynomial systems are closely related to nonassociative algebraic structures; that is why the statement of results and the proofs essentially use the language of nonassociative algebras.

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Generalized Schrödinger equations and Jordan pairs

Cited by 55 publications

References 9 publications

New integrable systems of derivative nonlinear Schrödinger equations with multiple components

New integrable systems of derivative nonlinear Schrödinger equations with multiple components

Multi-component generalizations of mKdV equation and nonassociative algebraic structures

Integrable boundary conditions for many-component burgers equations

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