1,124 research outputs found
Application of the generalized Kudryashov method to the Eckhaus equation
In this paper, the generalized Kudryashov method is presented to seek exact solutions of the Eckhaus equation. From these solutions, we can derive solitary wave solutions as a special case. The proposed method is direct, effective and convenient and can be applied to many nonlinear evolution equations in mathematical physics
Nonlinear surface waves in left-handed materials
We study both linear and nonlinear surface waves localized at the interface
separating a left-handed medium (i.e. the medium with both negative dielectric
permittivity and negative magnetic permeability) and a conventional (or
right-handed) dielectric medium. We demonstrate that the interface can support
both TE- and TM-polarized surface waves - surface polaritons, and we study
their properties. We describe the intensity-dependent properties of nonlinear
surface waves in three different cases, i.e. when both the LH and RH media are
nonlinear and when either of the media is nonlinear. In the case when both
media are nonlinear, we find two types of nonlinear surface waves, one with the
maximum amplitude at the interface, and the other one with two humps. In the
case when one medium is nonlinear, only one type of surface wave exists, which
has the maximum electric field at the interface, unlike waves in right-handed
materials where the surface-wave maximum is usually shifted into a
self-focussing nonlinear medium. We discus the possibility of tuning the wave
group velocity in both the linear and nonlinear cases, and show that
group-velocity dispersion, which leads to pulse broadening, can be balanced by
the nonlinearity of the media, so resulting in soliton propagation.Comment: 9 pages, 10 figure
Two-component generalizations of the Camassa-Holm equation
A classification of integrable two-component systems of non-evolutionary partial differential equations that are analogous to the Camassa-Holm equation is carried out via the perturbative symmetry approach. Independently, a classification of compatible pairs of Hamiltonian operators is carried out, which leads to bi-Hamiltonian structures for the same systems of equations. Some exact solutions and Lax pairs are also constructed for the systems considered
Singularites in the Bousseneq equation and in the generalized KdV equation
In this paper, two kinds of the exact singular solutions are obtained by the
improved homogeneous balance (HB) method and a nonlinear transformation. The
two exact solutions show that special singular wave patterns exists in the
classical model of some nonlinear wave problems
The connection of the Degasperis-Procesi equation with the Vakhnenko equation
Travelling-wave solutions of the Degasperis-Procesi equation (DPE) are investigated. The solutions are characterized by two parameters. Hump-like, loop-like and coshoidal periodicwave solutions are found; hump-like, loop-like and peakon solitary-wave solutions are obtained as well. Hone and Wang showed a connection between the DPE and the Vakhnenko equation (VE). Comparing the solutions of the DPE and the VE, we observe that, for both equations at interaction of waves, there are three kinds of phaseshift that depend on the ratio of wave amplitudes. In particular, there is a case when two interacted waves have phaseshifts in the positive direction
Solitary wave solutions of the Vakhnenko–Parkes equation
In this paper, two solitary wave solutions are obtained for the Vakhnenko–Parkes equation with power law nonlinearity by the ansatz method. Both topological as well as non-topological solitary wave solutions are obtained. The parameter regimes, for the existence of solitary waves, are identified during the derivation of the solution
Exact Travelling Wave Solutions of Some Nonlinear Nonlocal Evolutionary Equations
Direct algebraic method of obtaining exact solutions to nonlinear PDE's is
applied to certain set of nonlinear nonlocal evolutionary equations, including
nonlinear telegraph equation, hyperbolic generalization of Burgers equation and
some spatially nonlocal hydrodynamic-type model. Special attention is paid to
the construction of the kink-like and soliton-like solutions.Comment: 13 pages, LaTeX2
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