[Paper Review] Integrable inhomogeneous Lakshmanan-Myrzakulov equation

K. R. Esmakhanova, Gulgassyl Nugmanova+2 more|ArXiv.org|Apr 17, 2006

Differential Equations and Numerical Methods1 references3 citations

TL;DR

This paper constructs an integrable inhomogeneous extension of the (2+1)-dimensional Lakshmanan-Myrzakulov equation (LME) using prolongation structure theory, establishing its L-equivalence to a generalized (2+1)-dimensional nonlinear Schrödinger equation (NLSE) with variable coefficients. The key result is a new integrable system that generalizes both the inhomogeneous Myrzakulov-I equation and the standard LME, preserving integrability through a Lax pair and gauge equivalence.

ABSTRACT

The integrable inhomogeneous extension of the Lakshmanan-Myrzakulov equation is constructed by using the prolongation structure theory. The corresponding L-equivalent counterpart is also given, which is the (2+1)-dimensional generalized NLSE.

Motivation & Objective

• To extend the integrable Lakshmanan-Myrzakulov equation to an inhomogeneous form with spatially varying coefficients.
• To determine the conditions under which the inhomogeneous LME remains integrable using the prolongation structure theory.
• To establish the L-equivalence between the inhomogeneous LME and a generalized (2+1)-dimensional nonlinear Schrödinger equation (NLSE).
• To generalize known inhomogeneous extensions, such as the inhomogeneous Myrzakulov-I equation, to a broader class of (2+1)-dimensional integrable systems.

Proposed method

• The inhomogeneous LME is formulated with variable coefficients $ f_1(x,t) $, $ f_2(x,t) $, and constant $ \beta $, preserving the original structure with additional inhomogeneous terms.
• The prolongation structure theory is applied to the LME in the case $ \mathbf{S}_t = 0 $, introducing differential forms and prolongation variables to analyze integrability conditions.
• A closed ideal of differential forms is constructed, and the prolongation condition $ d\Omega^k \subset \{I, \Omega^k\} $ leads to a system of PDEs for the connection forms $ F^k $ and $ G^k $.
• Solving the PDEs yields a Lax pair representation, establishing the L-equivalence of the inhomogeneous LME to a generalized (2+1)-dimensional NLSE with variable coefficients.
• The gauge equivalence between the LME and the NLSE is confirmed via the transformation $ \Psi = g^{-1}\Phi $, where $ g = \Phi|_{\lambda=0} $.
• The system is shown to reduce to known cases, such as the inhomogeneous (1+1)-dimensional NLSE and inhomogeneous (2+1)-dimensional NLSE, under specific parameter choices.

Experimental results

Research questions

• RQ1 Can the Lakshmanan-Myrzakulov equation be extended to an integrable inhomogeneous form with spatially varying coefficients?
• RQ2 What conditions on the coefficients $ f_1(x,t) $, $ f_2(x,t) $, and constant $ \beta $ ensure the integrability of the inhomogeneous LME?
• RQ3 Is there a L-equivalent (2+1)-dimensional generalized NLSE corresponding to the inhomogeneous LME?
• RQ4 How does the inhomogeneous LME relate to known integrable extensions such as the Myrzakulov-I and Ishimori equations?
• RQ5 Can the gauge equivalence between the inhomogeneous LME and a generalized NLSE be explicitly constructed?

Key findings

• The inhomogeneous LME is constructed as $ \mathbf{S}_t = \{ \mathbf{S} \times (f_1 \mathbf{S}_x + \beta \mathbf{S}_y) + u\mathbf{S} \}_x + f_2 \mathbf{S}_x $, with $ u_x = -\beta \mathbf{S} \cdot (\mathbf{S}_x \times \mathbf{S}_y) $, generalizing the standard LME and inhomogeneous Myrzakulov-I equation.
• The system admits a Lax pair representation via the prolongation structure, confirming its integrability for arbitrary $ f_1(x,t) $, $ f_2(x,t) $, and constant $ \beta $.
• The L-equivalent counterpart is the generalized (2+1)-dimensional NLSE: $ iq_t - (f_1 q)_{xx} - \beta q_{xy} - i(f_2 q)_x - v q = 0 $, with $ v_x = 2[(f_1 pq)_x + \beta (pq)_y] $, where $ p = \varepsilon \bar{q} $.
• The gauge equivalence between the inhomogeneous LME and the generalized NLSE is established through the transformation $ \Psi = g^{-1} \Phi $, with $ g = \Phi|_{\lambda=0} $, confirming the L-equivalence.
• The system reduces to the inhomogeneous (1+1)-dimensional NLSE when $ \beta = 0 $, and to the inhomogeneous (2+1)-dimensional NLSE when $ f_1 = 0 $ and $ \beta = 1 $.
• The curvature and torsion of the space curve associated with the spin vector satisfy a system of PDEs that map to the generalized NLSE via the complex function $ q = \frac{1}{2}\kappa e^{-i\partial_x^{-1} \tau} $.