[Paper Review] Integrable Deformation of Space Curves, Generalized Heisenberg Ferromagnet Equation and Two-Component Modified Camassa-Holm Equation

B.B. Kutum, Gulgassyl Nugmanova+3 more|arXiv (Cornell University)|Aug 4, 2019

Nonlinear Waves and Solitons19 references4 citations

TL;DR

This paper establishes the geometric equivalence between the modified Calogero-Bogoyavlenskii-Schiff (M-CV) equation and the two-component modified Camassa-Holm equation (2-mCHE) through the invariant curve flow in 3D Euclidean space. By deriving the M-CV equation from space curve dynamics and identifying its Lax pair, the authors prove that the M-CV equation is geometrically equivalent to the 2-mCHE, with additional proof of gauge equivalence between the two systems.

ABSTRACT

In this paper, we provide the geometric formulation to the two-component Camassa-Holm equation (2-mCHE). We also study the relation between the 2-mCHE and the M-CV equation. We have shown that these equations arise from the invariant space curve flows in three-dimensional Euclidean geometry. Using this approach we have established the geometrical equivalence between the 2-mCHE and the M-CV equation. The gauge equivalence between these equations is also considered.

Motivation & Objective

• To provide a geometric formulation of the two-component modified Camassa-Holm equation (2-mCHE) using space curve dynamics.
• To investigate the relationship between the M-CV equation and the 2-mCHE through geometric and integrability structures.
• To establish the Lakshmanan (geometrical) equivalence between the M-CV equation and the 2-mCHE.
• To explore the gauge equivalence between the M-CV equation and the 2-mCHE using Lax pair transformations.

Proposed method

• Deriving the motion of space curves in 3D Euclidean space using the Frenet-Serret equations with curvature and torsion as dynamical variables.
• Identifying the spin vector A with the tangent vector e₁ of the space curve to map the M-CV equation to geometric curve evolution.
• Constructing the Lax pair for the M-CV equation using matrix operators involving spectral parameter λ and constant β.
• Deriving the compatibility condition of the Lax pair to obtain the M-CV equation in terms of curvature and torsion components.
• Mapping the geometric evolution equations to the 2-mCHE via substitutions κ₁ = -2ζ, κ₂ = r - q, τ = -i(r + q), and identifying Q = (u - uₓ)(v + vₓ).
• Proving gauge equivalence between M-CV and 2-mCHE by showing Ψ = GΦ transforms one Lax pair into the other.

Experimental results

Research questions

• RQ1 How can the two-component modified Camassa-Holm equation (2-mCHE) be geometrically formulated via space curve dynamics in 3D Euclidean space?
• RQ2 What is the geometric (Lakshmanan) equivalence between the M-CV equation and the 2-mCHE?
• RQ3 Does the M-CV equation arise from invariant space curve flows, and if so, how?
• RQ4 Is there a gauge equivalence between the M-CV equation and the 2-mCHE, and how is it realized through their Lax pairs?
• RQ5 What role do the curvature, torsion, and spin vector dynamics play in connecting the M-CV and 2-mCHE equations?

Key findings

• The M-CV equation is geometrically equivalent to the 2-mCHE, as both arise from the same invariant space curve flow in 3D Euclidean geometry.
• The Lakshmanan equivalent counterpart of the M-CV equation is the 2-mCHE, derived via the identification of the spin vector A with the tangent vector e₁ of the curve.
• The 2-mCHE is recovered from the geometric evolution equations by substituting κ₁ = -2ζ, κ₂ = r - q, and τ = -i(r + q), leading to the system (60)–(63).
• The M-CV equation and the 2-mCHE are gauge equivalent, with the transformation Ψ = GΦ linking their respective Lax pairs.
• When v = u, the 2-mCHE reduces to the standard modified Camassa-Holm equation (4)–(5), confirming consistency with known integrable reductions.
• The compatibility condition of the Lax pair yields the M-CV equation in matrix form, confirming its integrability through the zero-curvature condition.