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We found a new symplectic structure and a recursion operator for theSasa--Satsuma equation widely used in nonlinear optics, $$ p_t=p_{xxx}+6 p qp_x+3 p (p q)_x,\quad q_t=q_{xxx}+6 p q q_x+3 q (p q)_x, $$ along with anintegro-differential substitution linking this system to a third-ordergeneralized symmetry of the complex sine-Gordon II system $$ u_{xy}=\frac{v u_xu_y}{u v + c} + (2 u v + c)(u v + c) k u,\qquad v_{xy}=\frac{u v_x v_y}{u v +c} + (2 u v + c)(u v + c) k v, $$ where $c$ and $k$ are arbitrary constants.  Combining these two results yields a highly nonlocal hereditary recursionoperator and higher Hamiltonian structures for the complex sine-Gordon IIsystem. We also show that both the Sasa--Satsuma equation and the third orderevolutionary symmetry flow for the complex sine-Gordon II system arebihamiltonian systems, and construct several hierarchies of local and nonlocalsymmetries for these systems.Comment: 16 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

journal of mathematical physics

Sasa-Satsuma (complex modified Korteweg–de Vries II) and the complex sine-Gordon II equation revisited: Recursion operators, nonlocal symmetries, and more