The Sasa--Satsuma (complex mKdV II) and the complex sine-Gordon II equation revisited: recursion operators, nonlocal symmetries, and more

Abstract

We found a new symplectic structure and a recursion operator for the Sasa--Satsuma equation widely used in nonlinear optics, $$p_t=p_{xxx}+6pq{p}_x+3p(pq{)}_x,q_t=q_{xxx}+6pq{q}_x+3q(pq{)}_x,$$ along with an integro-differential substitution linking this system to a third-order generalized symmetry of the complex sine-Gordon II system $$u_{xy}=\frac{v{u}_x{u}_y}{uv+c}+(2uv+c)(uv+c)ku,v_{xy}=\frac{u{v}_x{v}_y}{uv+c}+(2uv+c)(uv+c)kv,$$ where $c$ and $k$ are arbitrary constants. Combining these two results yields a highly nonlocal hereditary recursion operator and higher Hamiltonian structures for the complex sine-Gordon II system. We also show that both the Sasa--Satsuma equation and the third order evolutionary symmetry flow for the complex sine-Gordon II system are bihamiltonian systems, and construct several hierarchies of local and nonlocal symmetries for these systems.

arXiv comment16 pages, LaTeX 2e, no figures (in this version the recursion operator for the Sasa--Satsuma equation was written in a somewhat different form, several typos were fixed, and titles of the papers were added in the bibliography)

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Cited by in corpus (7)

• Squared Eigenfunctions for the Sasa-Satsuma Equation
• Anti-dark and Mexican-hat solitons in the Sasa-Satsuma equation on the continuous wave background
• Lenard scheme for two dimensional periodic Volterra chain
• Hamiltonian Flows of Curves in symmetric spaces G/SO(N) and Vector Soliton Equations of mKdV and Sine-Gordon Type
• A new recursion operator for the Viallet equation
• Two-component integrable generalizations of Burgers equations with nondiagonal linearity
• On nonlocal symmetries for the Krichever--Novikov equation