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

Carillo, Sandra; Schiebold, Cornelia

doi:10.1063/1.3155080

Cited by 37 publications

(102 citation statements)

References 30 publications

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“…A very short outline on how this method can be adopted to construct solutions of the matrix mKdV equation is provided. Indeed, the study in [10], [7], [8], is devoted to study nonlinear evolution equations in which the unknown is an operator on a Banach space. The idea of the method can be sketched as follows:…”

Section: Matrix Soliton Solutionsmentioning

confidence: 99%

Matrix Solitons Solutions of the Modified Korteweg–de Vries Equation

Carillo

Schiavo

Schiebold

2020

Nonlinear Dynamics of Structures, Systems and Devices

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Nonlinear non-Abelian Korteweg-de Vries (KdV) and modified Korteweg-de Vries (mKdV) equations and their links via Bäcklund transformations are considered. The focus is on the construction of soliton solutions admitted by matrix modified Korteweg-de Vries equation. Matrix equations can be viewed as a specialisation of operator equations in the finite dimensional case when operators admit a matrix representation. Bäcklund transformations allow to reveal structural properties [10] enjoyed by non-commutative KdV-type equations, such as the existence of a recursion operator. Operator methods combined with Bäcklund transformations, allow to construct explicit solution formulae [11]. The latter are adapted to obtain solutions admitted by the 2 × 2 and 3 × 3 matrix mKdV equation. Some of these matrix solutions are visualised to show the solitonic behaviour they exhibit. A further key tool used to obtain the presented results is an ad hoc construction of computer algebra routines to implement non-commutative computations.

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Section: Matrix Soliton Solutionsmentioning

confidence: 99%

Matrix Solitons Solutions of the Modified Korteweg–de Vries Equation

Carillo

Schiavo

Schiebold

2020

Nonlinear Dynamics of Structures, Systems and Devices

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“…Summary: First we observe that from 1 [X ] = (D 2 ) [X ] = 0 it is immediate that both S 1 + 2 + 3 , 1 = 0 and S 4 , 1 = 0. Thus, on the one hand,…”

Section: Proof Of Stepmentioning

confidence: 96%

“…Finally, starting from C H D −1 C X D 1 2 C H C X which is obtained from (a) by multiplication of both sides with D − 1 from the left, and applying then (b), we get (c).…”

Section: Examplementioning

confidence: 99%

“…The somewhat surprising success of that approach motivates our present goal to find structural properties behind the involved calculations in Ref. 1 From a geometric viewpoint, one may regard (2) as a section of the endomorphism bundle End(T F) of the tangent bundle T F. The fiber of End(T F) over u ∈ F is End(T u F), the space of bounded linear operators on T u F. Sections of End(T F) will be called operator-valued functions or just operators. Before stating our main result, we recall some terminology.…”

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confidence: 99%

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Structural properties of the noncommutative KdV recursion operator

Schiebold

2011

Journal of Mathematical Physics

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Alternative sets of multisoliton solutions of some integrable KdV type equations via direct methods AIP Conf.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. C 2011 American Institute of Physics. [doi:10.1063/1.3656271] I. INTRODUCTIONThe noncommutative Korteweg-de Vries equation (ncKdV) readswhere u = u(x, t) is a function with values in some (possibly noncommutative) Banach algebra A and {u, v} = uv + vu denotes the anticommutator. In the sequel we will consider (1) as an ordinary differential equationwhere u = u(t) is a point varying in an infinite-dimensional algebra F of A-valued functions depending on the space variable x ∈ R, and where K is the vector fieldOur main topic are the structural properties of the noncommutative KdV recursion operator KdV (u) = D 2 + 2A

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“…Thus, the Bäcklund chart in [10,9] is recalled in the subsection devoted to the classical commutative setting. The following subsection is concerned about the Bäcklund chart, constructed in [12] and later extended [15,17]. The last subsection is concerned about pointing out differences and analogies between the two cases.…”

Section: Introductionmentioning

confidence: 99%

Abelian versus non-Abelian Bäcklund charts: Some remarks

Carillo¹,

Schiavo²,

Schiebold³

2019

Evolution Equations &Amp; Control Theory

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Connections via Bäcklund transformations among different nonlinear evolution equations are investigated aiming to compare corresponding Abelian and non Abelian results. Specifically, links, via Bäcklund transformations, connecting Burgers and KdV-type hierarchies of nonlinear evolution equations are studied. Crucial differences as well as notable similarities between Bäcklund charts in the case of the Burgers -heat equation, on one side and KdV -type equations are considered. The Bäcklund charts constructed in [16] and [17], respectively, to connect Burgers and KdV-type hierarchies of operator nonlinear evolution equations show that the structures, in the noncommutative cases, are richer than the corresponding commutative ones.

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

Cited by 37 publications

References 30 publications

Matrix Solitons Solutions of the Modified Korteweg–de Vries Equation

Matrix Solitons Solutions of the Modified Korteweg–de Vries Equation

Structural properties of the noncommutative KdV recursion operator

Abelian versus non-Abelian Bäcklund charts: Some remarks

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