Invariant Manifolds of Hyperbolic Integrable Equations and their Applications

Habibullin, I. T.; Khakimova, A. R.

doi:10.1007/s10958-021-05491-3

Cited by 2 publications

(3 citation statements)

References 13 publications

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“…. (1.20) In this article we continue the study begun in our papers [9][10][11]. We conjecture that for any integrable equation of the form (1.13) that has no nontrivial characteristic integrals the sequence of the Laplace transformations associated to its linearization admits a finite field reduction, i.e.…”

Section: Introductionmentioning

confidence: 70%

“…By a direct computation one reduces the order of this Lax pair and arrive at a system of three nonlinear equations from which Dubrovin type equation can be derived [12][13][14], suitable for constructing algebrogeometric solutions (about algebro-geometric solutions see [15]). By passing to new variables in a proper way one can derive from the triple of nonlinear equations the standard Lax pair (see [9][10][11]).…”

Section: Introductionmentioning

confidence: 99%

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Laplace transformations and sine-Gordon type integrable PDE

Habibullin,

Faizulina,

Khakimova

2023

J. Phys. A: Math. Theor.

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It is well known that the Laplace cascade method is an eﬀective tool for constructing solutions to linear equations of hyperbolic type, as well as nonlinear equations of the Liouville type. The connection between the Laplace method and soliton equations of hyperbolic type remains less studied. The article shows that the Laplace cascade also has important applications in the theory of hyperbolic equations of the soliton type. Laplace’s method provides a simple way to construct such fundamental objects related to integrability theory as the recursion operator, the Lax pair and Dubrovin-type equations, allowing one to ﬁnd algebro-geometric solutions. As an application of this approach, previously unknown recursion operators and Lax pairs are found for two nonlinear integrable equations of the sine-Gordon type.

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Section: Introductionmentioning

confidence: 70%

Section: Introductionmentioning

confidence: 99%

Laplace transformations and sine-Gordon type integrable PDE

Habibullin,

Faizulina,

Khakimova

2023

J. Phys. A: Math. Theor.

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“…This symmetry was discovered in [9,18,27], and is characterized as a generating symmetry in [31]. In another paper [10], the authors shows that for a second order hyperbolic equation, in the integrable case, two recursion operators (generating two distinct infinite hierarchies of symmetries) are produced by different parametrizations of the same generalized invariant manifold. This result coincides with the result that we obtain here.…”

Section: Introductionmentioning

confidence: 94%

Symmetry structure of integrable hyperbolic third order equations

Rasin,

Schiff

2023

J. Phys. A: Math. Theor.

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We explore the application of generating symmetries, i.e. symmetries that depend on a parameter, to integrable hyperbolic third order equations, and in particular to consistent pairs of such equations as introduced by Adler and Shabat in [1]. Our main result is that diﬀerent inﬁnite hierarchies of symmetries for these equations can arise from a single generating symmetry by expansion about diﬀerent values of the parameter. We illustrate this, and study in depth the symmetry structure, for two examples. The ﬁrst is an equation related to the potential KdV equation taken from [1]. The second is a more general hyperbolic equation than the kind considered in [1]. Both equations depend on a parameter, and when this parameter vanishes they become part of a consistent pair. When this happens, the nature of the expansions of the generating symmetries needed to derive the hierarchies also changes.

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Invariant Manifolds of Hyperbolic Integrable Equations and their Applications

Cited by 2 publications

References 13 publications

Laplace transformations and sine-Gordon type integrable PDE

Laplace transformations and sine-Gordon type integrable PDE

Symmetry structure of integrable hyperbolic third order equations

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