Soliton structure versus singularity analysis: Third-order completely intergrable nonlinear differential equations in 1 + 1-dimensions

Fuchssteiner, Benno; Carillo, Sandra

doi:10.1016/0378-4371(89)90260-4

Cited by 35 publications

(87 citation statements)

References 27 publications

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“…These connections can be used to derive direct transformations between interacting solitons and solutions of other equations. These aspects have been systematically investigated in joint work with Sandra Carillo (see [15]). But we are far away from understanding all details of these aspects.…”

Section: Discussionmentioning

confidence: 99%

Linear Aspects in the Theory of Solitons and Nonlinear Integrable Equations

Fuchssteiner

1991

J. Phys. Soc. Jpn.

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Section: Discussionmentioning

confidence: 99%

Linear Aspects in the Theory of Solitons and Nonlinear Integrable Equations

Fuchssteiner

1991

J. Phys. Soc. Jpn.

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“…The Korteweg-deVries equation (4.2) is linked [6] is considered. Indeed, such equation enjoys many interesting properties and, in particular, its "general" solution ( [12], [1]) is well known since there is a Bäcklund transformation between this equation and the wave equation ([11], [8]).…”

Section: Connections With Nonlinear Evolution Equationsmentioning

confidence: 99%

Some Remarks on A Class of Ordinary Differential Equations: The Riccati Property

Carillo

Fuchssteiner

1993

Modern Group Analysis: Advanced Analytical and Computational Methods in Mathematical Physics

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Abstract.Here ordinary differential equations of third and higher order are considered; in particular, a class of equations which can be solved by quadratures is exploited.Indeed, crucial to obtain our result is the property of the Riccati equation, according to which, given one particular solution, then its general solution can be determined explicitly.Thus, what we term the "Riccati" Property is introduced to point out that the members of such a class are differential equations which are of a generalized form of Riccati equation. Trivial examples of differential equations which enjoy the Riccati Property are all linear second order ordinary differential equations.Here some further examples of ordinary differential equations which enjoy the same Property are considered. In particular, on the basis of group invariance requirements, a method to construct ordinary differential equations which enjoy the Riccati Property is given.Remarkably, it follows that ordinary differential equations enjoying the Riccati Property are related to nonlinear evolution equations which admit a hereditary recursion operator.Finally, further connections with nonlinear evolution equations are mentioned.

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“…The important role played by Bäcklund transformations to investigate properties enjoyed by nonlinear evolution equations is based on the fact that most of the properties of interest are preserved under Bäcklund transformations [16], [17]. Thus, the construction of a net of Bäcklund transformation to connect nonlinear evolution equations allowed to prove new results both in the case of scalar equations [3]- [5], [18]- [19], as well as in the generalised case of operator equations [7]- [12] 8 A comparison between the scalar (Abelian) and the operator (non-Abelian) cases referring to third order KdV-type equations, Bäcklund transformations connecting them and related properties is comprised in [9]. Notably, Bäcklund transformations indicate a way to construct solutions to nonlinear evolution equations [20] and also to nonlinear ordinary differential equations, [13] and [14].…”

Section: Introductionmentioning

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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Soliton structure versus singularity analysis: Third-order completely intergrable nonlinear differential equations in 1 + 1-dimensions

Cited by 35 publications

References 27 publications

Linear Aspects in the Theory of Solitons and Nonlinear Integrable Equations

Linear Aspects in the Theory of Solitons and Nonlinear Integrable Equations

Some Remarks on A Class of Ordinary Differential Equations: The Riccati Property

Matrix Solitons Solutions of the Modified Korteweg–de Vries Equation

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