A family of integrable systems of Liouville and lax representation, Darboux transformations for its constrained flows

2002 ◽  
Vol 23 (1) ◽  
pp. 26-34 ◽  
Author(s):  
Zhang Yu-feng ◽  
Zhang Hong-qing
Keyword(s):  
Integrable Systems ◽  
Lax Representation ◽  
Darboux Transformations

scholarly journals Darboux transformations with tetrahedral reduction group and related integrable systems

2016 ◽  
Vol 57 (9) ◽  
pp. 092701 ◽  
Author(s):  
George Berkeley ◽  
Alexander V. Mikhailov ◽  
Pavlos Xenitidis
Keyword(s):  
Integrable Systems ◽  
Darboux Transformations ◽  
Reduction Group

TWO ENLARGED LOOP ALGEBRAS FOR OBTAINING NEW INTEGRABLE HIERARCHIES

2011 ◽  
Vol 25 (19) ◽  
pp. 2637-2656
Author(s):  
YUFENG ZHANG ◽  
HONWAH TAM ◽  
WEI JIANG
Keyword(s):  
Integrable Systems ◽  
Integrable Hierarchies ◽  
Soliton Solutions ◽  
Loop Algebra ◽  
Darboux Transformations ◽  
Soliton Equation ◽  
Hamiltonian Structures ◽  
Soliton Theory ◽  
Loop Algebras ◽  
Soliton Hierarchy

Taking a loop algebra [Formula: see text] we obtain an integrable soliton hierarchy which is similar to the well-known Kaup–Newell (KN) hierarchy, but it is not. We call it a modified KN (mKN) hierarchy. Then two new enlarged loop algebras of the loop algebra [Formula: see text] are established, respectively, which are used to establish isospectral problems. Thus, two various types of integrable soliton-equation hierarchies along with multi-component potential functions are obtained. Their Hamiltonian structures are also obtained by the variational identity. The second hierarchy is integrable couplings of the mKN hierarchy. This paper provides a clue for generating loop algebras, specially, gives an approach for producing new integrable systems. If we obtain a new soliton hierarchy, we could deduce its symmetries, conserved laws, Darboux transformations, soliton solutions and so on. Hence, the way presented in the paper is an important aspect to obtain new integrable systems in soliton theory.

scholarly journals EQUIVALENCE OF THE HUNTER-SAXON EQUATION AND THE GENERALIZED HEISENBERG FERROMAGNET EQUATION

2021 ◽  
Vol 2 (336) ◽  
pp. 33-38
Author(s):  
Zh. R. Myrzakulova ◽  
K. R. Yesmakhanova ◽  
Zh. S. Zhubayeva
Keyword(s):  
Integrable Systems ◽  
Heisenberg Ferromagnet ◽  
Lax Representation ◽  
Director Field ◽  
Weakly Nonlinear ◽  
Scattering Transform ◽  
The Family ◽  
The Matrix ◽  
Special Solutions ◽  
Modern Mathematics

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.

scholarly journals Lax Representation and Darboux Solutions of the Classical Painlevé Second Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Irfan Mahmood ◽  
Muhammad Waseem
Keyword(s):  
Linear System ◽  
Darboux Transformation ◽  
Integrable Systems ◽  
Lax Pair ◽  
Lax Representation ◽  
Gauge Transformations ◽  
Additive Form

In this article, we present Darboux solutions of the classical Painlevé second equation. We reexpress the classical Painlevé second Lax pair in new setting introducing gauge transformations to yield its Darboux expression in additive form. The new linear system of that equation carries similar structure as other integrable systems possess in the AKNS scheme. Finally, we generalize the Darboux transformation of the classical Painlevé second equation to the N -th form in terms of Wranskian.

scholarly journals Differential Galois theory and Darboux transformations for Integrable Systems

2017 ◽  
Vol 115 ◽  
pp. 75-88
Author(s):  
Sonia Jiménez ◽  
Juan J. Morales-Ruiz ◽  
Raquel Sánchez-Cauce ◽  
María-Angeles Zurro
Keyword(s):  
Integrable Systems ◽  
Galois Theory ◽  
Differential Galois Theory ◽  
Darboux Transformations
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