The tanh and the sine-cosine methods for the complex modified K dV and the generalized K dV equations

AbstractThe complex modified K dV (CMK dV) equation and the generalized K dV equation are investigated by using the tanh method and the sine-cosine method. A variety of exact travelling wave solutions with compact and noncompact structures are formally obtained for each equation. The study reveals the power of the two schemes where each method complements the other

Soliton and periodic wave solutions to the osmosis K(2, 2) equation

In this paper, two types of traveling wave solutions to the osmosis K(2, 2) equation are investigated. They are characterized by two parameters. The expresssions for the soliton and periodic wave solutions are obtained.Comment: 14 pages, 16 figure

Compactons and kink-like solutions of BBM-like equations by means of factorization

In this work, we study the Benjamin-Bona-Mahony like equations with a fully nonlinear dispersive term by means of the factorization technique. In this way we find the travelling wave solutions of this equation in terms of the Weierstrass function and its degenerated trigonometric and hyperbolic forms. Then, we obtain the pattern of periodic, solitary, compacton and kink-like solutions. We give also the Lagrangian and the Hamiltonian, which are linked to the factorization, for the nonlinear second order ordinary differential equations associated to the travelling wave equations.Comment: 10 pages, 8 figure

Shock-peakon and shock-compacton solutions for K(p,q) equation by variational iteration method

AbstractBy variational iteration method, we obtain new solitary solutions for non-linear dispersive equations. Particularly, shock-peakon solutions in K(2,2) equation and shock-compacton solutions in K(3,3) equation are found by this simple method. These two types of solutions are new solitary wave solutions which have the shapes of shock solutions and compacton solutions (or peakon solutions)

Solitary Wave and Shock Wave Solutions of the Variants of Boussinesq Equations

This paper obtains the solitary wave as well as the shock wave solutions of the variants of the Boussinesq equations in both (1+1) and (1+2) dimensions. The domain restrictions are also identified in the process

Soliton Solutions of the Kaup-Kupershmidt and Sawada-Kotera Equations

In this paper I seek soliton solutions of two-component generalizations of the Kaup- Kupershmidt and Sawada-Kotera equations, for this purpose I will apply the extended tanh method. The extended tanh method with a computerized symbolic computation, is used for constructing the travelling wave solutions of coupled nonlinear equations arising in physics. The obtained solutions include soliton, kink and plane periodic solutions. KeyWords: Soliton Solutions; Kaup-Kupershmidt Equation; Sawada-Kotera Equatio

Peakon, Cuspon, Compacton, and Loop Solutions of a Three-Dimensional 3DKP(3, 2) Equation with Nonlinear Dispersion

We study peakon, cuspon, compacton, and loop solutions for the three-dimensional Kadomtsev-Petviashvili equation (3DKP(3,2) equation) with nonlinear dispersion. Based on the method of dynamical systems, the 3DKP(3,2) equation is shown to have the parametric representations of the solitary wave solutions such as peakon, cuspon, compacton, and loop solutions. As a result, the conditions under which peakon, cuspon, compacton, and loop solutions appear are also given

PT-symmetry breaking in complex nonlinear wave equations and their deformations

We investigate complex versions of the Korteweg-deVries equations and an Ito type nonlinear system with two coupled nonlinear fields. We systematically construct rational, trigonometric/hyperbolic, elliptic and soliton solutions for these models and focus in particular on physically feasible systems, that is those with real energies. The reality of the energy is usually attributed to different realisations of an antilinear symmetry, as for instance PT-symmetry. It is shown that the symmetry can be spontaneously broken in two alternative ways either by specific choices of the domain or by manipulating the parameters in the solutions of the model, thus leading to complex energies. Surprisingly the reality of the energies can be regained in some cases by a further breaking of the symmetry on the level of the Hamiltonian. In many examples some of the fixed points in the complex solution for the field undergo a Hopf bifurcation in the PT-symmetry breaking process. By employing several different variants of the symmetries we propose many classes of new invariant extensions of these models and study their properties. The reduction of some of these models yields complex quantum mechanical models previously studied.Comment: 50 pages, 39 figures (compressed in order to comply with arXiv policy; higher resolutions maybe obtained from the authors upon request

Bright and Dark Soliton Solutions of the (2 + 1)-Dimensional Evolution Equations

In this paper, we obtained the 1-soliton solutions of the (2+1)-dimensional Boussinesq equation and the Camassa–Holm–KP equation. By using a solitary wave ansatz in the form of sechp function, we obtain exact bright soliton solutions and another wave ansatz in the form of tanhp function we obtain exact dark soliton solutions for these equations. The physical parameters in the soliton solutions are obtained nonlinear equations with constant coefficients