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 ...

In this article, the exact solutions of stochastic conformable Broer–Kaup equations with derivatives which describe bidirectional propagation long waves in shallow water are obtained using modified exponential function method and generalized Kudryashov method. These consist hyperbolic, trigonometric, rational solutions, respectively. This shows that proposed methods competent sufficient. additi...

In this paper, the tanh and sine–cosine methods are used to construct exact periodic and soliton solutions of nonlinear evolution equations arising in mathematical physics. Many new families of exact travelling wave solutions of the generalized Hirota–Satsuma coupled KdV system, generalized-Zakharov equations and (2 + 1)-dimensional Broer–Kaup– Kupershmidt system are successfully obtained. The ...

Nonlinear differential equations are known to describe a wide variety of phenomena not only in physics, in which applications extend over magnetofluid dynamics, water surface gravity waves, electromagnetic radiation reactions, and ion acoustic waves in plasmas, just to name a few, but also in biology and chemistry. It is one of the important tasks in the study of nonlinear differential equation...

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...

Abstract The generalized Broer-Kaup-Kupershmidt (generalized BKK) isospectral problem, including the x-derivative of potential, is considered based on Lie algebra A1. The variational trace identity is extended to construct Hamiltonian structure of generalized BKK system. The Lie algebra A1 is extended to the non-semi-simple Lie algebra of 4 × 4 matrix form, from which a hierarchy of soliton equ...

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 the fractional Riccati equation, we propose an extended fractional Riccati sub-equation method for solving fractional partial differential equations. The fractional derivative is defined in the sense of the modified Riemann-Liouville derivative. By a proposed variable transformation, certain fractional partial differential equations are turned into fractional ordinary di...

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...