Toda lattice G-Strands

TL;DR

This paper extends the Toda lattice ODEs to PDEs called T-Strands, analyzes their properties, solutions, and integrability, revealing that two- and three-particle T-Strands do not satisfy the zero-curvature relation for integrability.

Contribution

It introduces the T-Strand equations derived from Hamilton's principle, analyzes their structure, solutions, and integrability conditions, expanding the understanding of Toda lattice PDE systems.

Findings

T-Strands form symmetric hyperbolic Lie-Poisson Hamiltonian systems.

Traveling wave solutions are geometrically constructed for two-particle T-Strands.

Two- and three-particle T-Strands do not satisfy the zero-curvature relation for integrability.

Abstract

Hamilton's principle is used to extend for the Toda lattice ODEs to systems of PDEs called the Toda lattice strand equations (T-Strands). The T-Strands in the $n$-particle Toda case comprise $4n-2$ quadratically nonlinear PDEs in one space and one time variable. T-Strands form a symmetric hyperbolic Lie-Poisson Hamiltonian system of quadratically nonlinear PDEs with constant characteristic velocities. The travelling wave solutions for the two-particle T-Strand equations are solved geometrically, and their Lax pair is given to show how nonlinearity affects the solution. The three-particle T-Strands equations are also derived from Hamilton's principle. For both the two-particle and three-particle T-Strand PDEs the determining conditions for the existence of a quadratic zero-curvature relation (ZCR) exactly cancel the nonlinear terms in the PDEs. Thus, the two-particle and three-particle T-Strand PDEs do not pass the ZCR test for integrability.