Gauge Equivalence Between the Г-Spin System and (2+1)-Dimensional Two-Component Nonlinear Schrodinger Equation

Myrzakul, Shynaray; Myrzakulova, Zh. R.

doi:10.32014/2020.2518-1726.22

Cited by 1 publication

(3 citation statements)

References 9 publications

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“…To establish the gauge equivalence between HSE and GFHE (spin system), we need the Lax matrix representation. Since the coefficients of linear systems corresponding to the spin equations are related to the symmetric matrix Lax representations [10]. 7) is the compatibility condition of the (6).…”

Section: Lax Representation Of the Hunter-saxton Equationmentioning

confidence: 99%

“…The HSE (2), (3) with matrix Lax representation (8), ( 9) and the GHFE (13) with Lax representation (16), (17)are gauge equivalent to each other [10,11].…”

Section: A Generalized Heisenberg Ferromagnet Equationmentioning

confidence: 99%

“…Proof.Based on the classical theory of gauge equivalence [10], we begin the proof of the theorem with the transformation…”

Section: A Generalized Heisenberg Ferromagnet Equationmentioning

confidence: 99%

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Equivalence of the Hunter-Saxon Equation and the Generalized Heisenberg Ferromagnet Equation

Myrzakulova

Yesmakhanova

Zhubayeva

2021

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Integrable systems play an important role in modern mathematics, theoretical and mathematical physics. The display of integrable equations with exact solutions and some special solutions can provide important guarantees for the analysis of its various properties. The Hunter-Saxton equation belongs to the family of integrable systems. The extensive and interesting mathematical theory, underlying the Hunter-Saxton equation, creates active mathematical and physical research. The Hunter-Saxton equation (HSE) is a high-frequency limit of the famous Camassa-Holm equation. The physical interpretation of HSE is the propagation of weakly nonlinear orientation waves in a massive nematic liquid crystal director field. In this paper, we propose a matrix form of the Lax representation for HSE in 𝑠𝑢ሺ𝑛 ൅ 1ሻ/𝑠ሺ𝑢ሺ1ሻ ⊕ 𝑢ሺ𝑛ሻሻ - symmetric space for the case 𝑛 ൌ 2. Lax pairs, introduced in 1968 by Peter Lax, are a tool for finding conserved quantities of integrable evolutionary differential equations. The Lax representation expands the possibilities of the equation we are considering. For example, in this paper, we will use the matrix Lax representation for the HSE to establish the gauge equivalence of this equation with the generalized Heisenberg ferromagnet equation (GHFE). The famous Heisenberg Ferromagnet Equation (HFE) is one of the classical equations integrable through the inverse scattering transform. In this paper, we will consider its generalization. Andalso the connection between the decisions of the HSE and the GHFE will be presented.

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Section: Lax Representation Of the Hunter-saxton Equationmentioning

confidence: 99%

“…The HSE (2), (3) with matrix Lax representation (8), ( 9) and the GHFE (13) with Lax representation (16), (17)are gauge equivalent to each other [10,11].…”

Section: A Generalized Heisenberg Ferromagnet Equationmentioning

confidence: 99%