We found a new symplectic structure and a recursion operator for the Sasa–Satsuma (complex mKdV II) equation widely used in nonlinear optics, pt = pxxx + 6pqpx + 3p(pq)x, qt = qxxx + 6pqqx + 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 uxy = vuxuy uv + c + (2uv + c)(uv + c)ku, vxy = uvxvy ...

We derive a new zero-curvature formalism for the sine-Gordon (sG) equation which permits the introduction of a local sine-Gordon hierarchy (in contrast to the traditionally accepted nonlocal higher-order sG equations). In complete analogy to other completely integrable hierarchies of soli-ton equations, such as the KdV, AKNS, and Toda hierarchies, our local sG hierarchy is recursively construct...

In this paper, a type of fully nonlinear Sine-Gordon equations and the approximate Sine-Gordon equation (under the condition: |up|is very small) are studied.Through proper transformation, some initial value problems of many equations with special nonlinear terms are solved by the Adomian decomposition method and some exact solutions: kink solution ,compacton solution, multi-compacton solution, ...

The nonlinear sine-Gordon equation arises in various problems in science and engineering. In this paper, we propose a numerical scheme to solve the two-dimensional damped/undamped sine-Gordon equation. The proposed scheme is based on using collocation points and approximating the solution employing the thin plate splines (TPS) radial basis function (RBF). The new scheme works in a similar fashi...