In this paper, based on certain variable transformation, we apply the known (G’/G) method to seek exact solutions for three fractional partial differential equations: the space fractional (2+1)-dimensional breaking soliton equations, the space-time fractional Fokas equation, and the spacetime fractional Kaup-Kupershmidt equation. The fractional derivative is defined in the sense of modified Rie...

We show that the complex modified KdV (cmKdV) equation and generalized nonlinear Schrödinger (GNLS) equation belong to the Ablowitz, Kaup, Newell and Segur or so-called AKNS hierarchy. Both equations do not follow from the action principle and are nonintegrable. By introducing some auxiliary fields we obtain the variational principle for them and study their canonical structures. We make use of...

A one-parameter generalization of the hierarchy negative flows is introduced for integrable hierarchies evolution equations, which yields a wider (new) class non-evolutionary nonlinear wave equations. As main results, several integrability properties these generalized flow equation are established, including their symmetry structure, conservation laws, and bi-Hamiltonian formulation. (The resul...

Nonlocal reductions of a nonisospectral (2+1)-dimensional breaking soliton Ablowitz-Kaup-Newell-Segur equation are discussed on the base double Wronskian reduction technique. Various types solutions, including solutions and Jordan-block for resulting nonlocal equations derived. Dynamics these obtained analyzed illustrated.

By using gauge transformations, we manage to obtain new solutions of (2 + 1)dimensional Kadomtsev-Petviashvili(KP), Kaup-Kuperschmidt(KK) and SawadaKotera(SK) equations from non-zero seeds. For each of the preceding equations, a Galilean type transformation between these solutions u2 and the previously known solutions u2 generated from zero seed is given. We present several explicit formulas of...

Complete eigenfunctions of linearized integrable equations expanded around an arbitrary solution are obtained for the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy and the Korteweg-de Vries (KdV) hierarchy. It is shown that the linearization operators and the recursion operator which generates the hierarchy are commutable. Consequently, eigenfunctions of the linearization operators are precisely ...

Scaling similarity solutions of three integrable PDEs, namely the Sawada-Kotera, fifth order KdV and Kaup-Kupershmidt equations, are considered. It is shown that the resulting ODEs may be written as non-autonomous Hamiltonian equations, which are time-dependent generalizations of the well-known integrable Hénon-Heiles systems. The (time-dependent) Hamiltonians are given by logarithmic derivativ...

We use the integrable Kaup-Boussinesq shallow water system, modified by a small viscous term, to model the formation of an undular bore with a steady profile. The description is made in terms of the corresponding integrable Whitham system, also appropriately modified by friction. This is derived in Riemann variables using a modified finite-gap integration technique for the AKNS scheme. The Whit...

In this paper, based on certain variable transformation, we apply the known (G’/G) method to seek exact solutions for three fractional partial differential equations: the space fractional (2+1)-dimensional breaking soliton equations, the space-time fractional Fokas equation, and the spacetime fractional Kaup-Kupershmidt equation. The fractional derivative is defined in the sense of modified Rie...

We consider the solution of spectral problems with elliptic coefficients in the framework of the Hermite ansatz. We show that the search for exactly solvable potentials and their spectral characteristics is reduced to a system of polynomial equations solvable by the Gröbner bases method and others. New potentials and corresponding solutions of the Sawada–Kotera, Kaup– Kupershmidt, Boussinesq eq...