Geometric Flows of Curves, Two-Component Camassa-Holm Equation and Generalized Heisenberg Ferromagnet Equation

Taishiyeva, Aigul; Myrzakul, Tolkynay; Nugmanova, Gulgassyl; Myrzakul, Shynaray; Yesmakhanova, Kuralay; Myrzakulov, Ratbay

doi:10.48550/arxiv.1910.13281

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2020

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“…This work continues our research of Lax-integrable (i.e., admitting Lax pairs with non-vanishing spectral parameter) generalized Heisenberg ferromagnet type equations in 1+1 dimensions related with Camassa-Holm type equations (see, e.g., [1]- [4] and the references therein). In the theory of integrable systems (soliton theory) an important role plays the so-called gauge and geometrical equivalence between two integrable equations.…”

Section: Introductionmentioning

confidence: 55%

Generalized Heisenberg Ferromagnet type Equation and Modified Camassa-Holm Equation: Geometric Formulation, Soliton Solutions and Equivalence

Umurzakhova,

Myrzakul,

Yesmakhanova

et al. 2020

Preprint

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We study the integrability and equivalence of a generalized Heisenberg ferromagnet-type equation (GHFE). The different forms of this equation as well as its reduction are presented. The Lax representation (LR) of the equation is obtained. We observe that the geometrical and gauge equivalent counterpart of the GHFE is the modified Camassa-Holm equation (mCHE) with an arbitrary parameter κ. Finally, the 1-soliton solution of the GHFE is obtained.

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Section: Introductionmentioning

confidence: 55%