[Paper Review] Integrable Flows of Curves/Surfaces, Generalized Heisenberg Ferromagnet Equation and Complex Coupled Dispersionless Equation

Guldana Bekova, Kuralay Yesmakhanova+3 more|arXiv (Cornell University)|Dec 5, 2018

Nonlinear Waves and Solitons12 references4 citations

TL;DR

This paper establishes a geometric correspondence between the Myrzakulov-XIII (M-XIII) equation and integrable flows of space curves in R³, demonstrating that the M-XIII equation is geometrically equivalent to the complex coupled dispersionless (CCD) equation via curve motion formalism. The gauge equivalence between the two equations is rigorously proven, linking integrable spin systems to differential geometry of curves and surfaces.

ABSTRACT

In the present paper, we study the Myrzakulov-XIII (M-XIII) equation geometrically. From the geometric point of view, we establish a link of the M-XIII equation with the motion of space curves in the 3-dimensional space $R^{3}$. We also show that the complex coupled dispersionless (CCD) equation can be derived from the geometrical formalism such that their curve flows are formulated. Finally, the gauge equivalence between the M-XIII equation and the CCD equation is established.

Motivation & Objective

• To establish a geometric link between the Myrzakulov-XIII (M-XIII) equation and the motion of space curves in R³.
• To derive the complex coupled dispersionless (CCD) equation as the Lakshmanan (geometrical) equivalent of the M-XIII equation.
• To prove the gauge equivalence between the M-XIII equation and the CCD equation.
• To explore the differential geometry of surfaces associated with the M-XIII equation through position vector formulations.
• To present a 1-soliton solution of the M-XIII equation using the CCD equation's seed solution and Sym-Tafel formula.

Proposed method

• Rewriting the M-XIII equation in vector form using spinor-like matrices and curvature-torsion relations.
• Identifying the unit tangent vector of a space curve with the spin vector A in the M-XIII equation.
• Applying the Frenet-Serret equations to model curve evolution and derive compatibility conditions.
• Using the Lax pair formalism of the M-XIII equation to derive the CCD equation as its geometric counterpart.
• Employing the Sym-Tafel formula to construct the 1-soliton solution of the M-XIII equation from the CCD equation's solution.
• Formulating the position vector of a surface in R³ using matrix transformations and the Lax potential, linking it to the M-XIII equation.

Experimental results

Research questions

• RQ1 How is the Myrzakulov-XIII equation geometrically related to the motion of space curves in R³?
• RQ2 What is the Lakshmanan (geometrical) equivalent of the M-XIII equation, and how is it derived from curve dynamics?
• RQ3 Is there a gauge equivalence between the M-XIII equation and the complex coupled dispersionless (CCD) equation?
• RQ4 How can the M-XIII equation be interpreted in terms of surface geometry in R³?
• RQ5 What is the explicit 1-soliton solution of the M-XIII equation, and how is it constructed from the CCD equation?

Key findings

• The M-XIII equation is geometrically equivalent to the complex coupled dispersionless (CCD) equation through the motion of space curves in R³.
• The gauge equivalence between the M-XIII equation and the CCD equation is rigorously established via their Lax pair representations.
• The 1-soliton solution of the M-XIII equation is explicitly constructed using the seed solution of the CCD equation with q=0 and ρ=1.
• The position vector of the 1-soliton surface is derived in component form as r₁, r₂, r₃, with R and W defined in terms of parameters b, a, y, and s.
• The solution satisfies the M-XIII equation in the form rₛ = ρrᵧ + rᵧ ∧ rᵧₛ, confirming its integrability.
• The surface satisfies the normalization condition rₛ² = rᵧ² = 1, indicating unit-speed parametrization in both variables.