In this paper we find a explicit moving frame along curves of Lagrangian planes invariant under the action of the symplectic group. We use the moving frame to find a family of independent and generating differential invariants. We then construct geometric Hamiltonian structures in the space of differential invariants and prove that, if we restrict them to a certain Poisson submanifold, they bec...

A bridge going from Wronskian solutions to generalized Wronskian solutions of the Korteweg-de Vries equation is built. It is then shown that generalized Wronskian solutions can be viewed as Wronskian solutions. The idea is used to generate positons, negatons and their interaction solutions to the Korteweg-de Vries equation. Moreover, general positons and negatons are constructed through the Wro...

In this paper, a generalized KdV equation with time-dependent coefficients will be studied. The K(m,n) equation with generalized evolution will also be examined. The Riccati equation mapping method will be used to obtain some new exact solutions for both equations. These solutions include solitary wave solutions and periodic wave solutions. The results presented in this paper improve the previo...

In this paper we study a shallow water equation derivable using the Boussinesq approximation, which includes as two special cases, one equation discussed by Ablowitz et al [Stud. Appl. Math., 53 (1974) 249–315] and one by Hirota and Satsuma [J. Phys. Soc. Japan, 40 (1976) 611–612]. A catalogue of classical and nonclassical symmetry reductions, and a Painlevé analysis, are given. Of particular i...

We prove local existence and uniqueness of solutions for the one-dimensional nonlinear Schrödinger (NLS) equations iut + uxx ± |u| 2 u = 0 in classes of smooth functions that admit an asymptotic expansion at infinity in decreasing powers of x. We show that an asymptotic solution differs from a genuine solution by a Schwartz class function which solves a generalized version of the NLS equation. ...

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 ...