The existence of Bogomolny decomposition by means of strong necessary conditions

Sokalski, K.; Stȩpień, Ł. T.; Sokalska, D

doi:10.1088/0305-4470/35/29/315

Cited by 16 publications

(26 citation statements)

References 11 publications

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“…Which means that each smooth enough F 1 and F 2 generate V (u, v) leading to the system which is integrable in the sense of the Bogomolny decomposition. In the particular case of a =ã = b =b = 0 and c =c = 1 2 we recover the result for the hyperbolic equations [29]:…”

Section: The Bogomolny Equationssupporting

confidence: 79%

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Existence of Dual Equations by Means of Strong Necessary Conditions - Analysis of Integrability of Partial Differential Nonlinear Equations

Sokalski¹,

Wietecha²,

Sokalska³

2005

JNMP

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A concept of strong necessary conditions for extremum of functional has been applied for analysis an existence of dual equations for a system of two nonlinear Partial Differential Equations (PDE) in 1+1 dimensions. We consider two types of the dual equations: the Bäcklund transformations and the Bogomolny equations. A general form of the second order PDE with a derivative-less non-linear term has been considered. In the case of a coupled system of equations the general conditions for the existence of the Bogomolny decomposition are derived. In the case of an uncoupled system of equations the Bogomolny equations become the Bäcklund transformations. It has been found a denumerable classes of coupled systems possessing the Bogomolny relationship. Weaken the method into semi-strong necessary conditions is presented together with an application to the Lax hierarchy. The method basing on both the strong and the semi-strong necessary condition concept reduces the derivation of the dual equations to an algorithm.

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Section: The Bogomolny Equationssupporting

confidence: 79%

“…Like in Section 2 the reduction of the number of independent equations by an appropriate choice of G 1 , G 2 , G 3 plays essential role in this procedure. Only for a very special V (u, v) such an Ansatz exists [29]. In most cases of V (u, v) the system (3.…”

Section: Reduction Of the (225)-(230) To The Bäcklund Transformationsmentioning

confidence: 99%

Existence of Dual Equations by Means of Strong Necessary Conditions - Analysis of Integrability of Partial Differential Nonlinear Equations

Sokalski¹,

Wietecha²,

Sokalska³

2005

JNMP

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“…Now, in order to determining function F , we compare (38) with Hamilton-Jacobi equation, which has the form, [27]…”

Section: −4βmentioning

confidence: 99%

On Bogomolny Decompositions for the Baby Skyrme Models

Stȩpień

2013

Geometric Methods in Physics

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We derive the Bogomolny decompositions (Bogomolny equations) for: full baby Skyrme model and for its restricted version (so called, pure baby Skyrme model), in (2+0) dimensions, by using so called, concept of strong necessary conditions. It turns out that Bogomolny decomposition can be derived for restricted baby Skyrme model for arbitrary form of the potential term, while for full baby Skyrme model, such derivation is possible only for some class of the potentials.

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“…In 1979 the first attempt to combine both the Rund and the Bogomolny methods has been done [11]. During the two last decades these results have been improved and generalized [12][13][14][15][16][17]. The developed formalism named the strong necessary conditions method (SNCM), uniquely treats the Rund and the Bogomolny approaches.…”

Section: Introductionmentioning

confidence: 99%

The Bogomolny Decomposition for Systems of Two Generalized Nonlinear Partial Differential Equations of the Second Order

Stȩpień

Sokalska

Sokalski

2021

JNMP

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Using a concept of strong necessary conditions we derive the Bogomolny decomposition for systems of two generalized elliptic and parabolic nonlinear partial differential equations (NPDE) of the second order. The generalization means that the equation coefficients depend on the field variables. According to the Cinquini-Cibrario criteria [18-20] the first type is characterized to be an elliptic, whereas the second one is a parabolic system. As a result we derive conditions for existence of the Bogomolny relationships.

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The existence of Bogomolny decomposition by means of strong necessary conditions

Cited by 16 publications

References 11 publications

Existence of Dual Equations by Means of Strong Necessary Conditions - Analysis of Integrability of Partial Differential Nonlinear Equations

Existence of Dual Equations by Means of Strong Necessary Conditions - Analysis of Integrability of Partial Differential Nonlinear Equations

On Bogomolny Decompositions for the Baby Skyrme Models

The Bogomolny Decomposition for Systems of Two Generalized Nonlinear Partial Differential Equations of the Second Order

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