Solitary-wave solutions of the Degasperis-Procesi equation by means of the homotopy analysis method

The homotopy analysis method is applied to the Degasperis-Procesi equation in order to find analytic approximations to the known exact solitary-wave solutions for the solitary peakon wave and the family of solitary smooth-hump waves. It is demonstrated that the approximate solutions agree well with the exact solutions. This provides further evidence that the homotopy analysis method is a powerful tool for finding excellent approximations to nonlinear solitary waves

Initial-Boundary Value Problems for Linear and Soliton PDEs

Evolution PDEs for dispersive waves are considered in both linear and nonlinear integrable cases, and initial-boundary value problems associated with them are formulated in spectral space. A method of solution is presented, which is based on the elimination of the unknown boundary values by proper restrictions of the functional space and of the spectral variable complex domain. Illustrative examples include the linear Schroedinger equation on compact and semicompact n-dimensional domains and the nonlinear Schroedinger equation on the semiline.Comment: 18 pages, LATEX, submitted to the proccedings of NEEDS 2001 - Special Issue, to be published in the Journal of Theoretical and Mathematical Physic

Discrete Reductive Perturbation Technique

We expand a partial difference equation (P$\Delta$E) on multiple lattices and obtain the P$\Delta$E which governs its far field behaviour. The perturbative--reductive approach is here performed on well known nonlinear P$\Delta$Es, both integrable and non integrable. We study the cases of the lattice modified Korteweg--de Vries (mKdV) equation, the Hietarinta equation, the lattice Volterra--Kac--Van Moerbeke (VKVM) equation and a non integrable lattice KdV equation. Such reductions allow us to obtain many new P$\Delta$Es of the nonlinear Schr\"odinger (NLS) type.Comment: 18 pages, 1 figure. submitted to Journal of Mathematical Physic

Perturbative Symmetry Approach

Perturbative Symmetry Approach is formulated in symbolic representation. Easily verifiable integrability conditions of a given equation are constructed in the frame of the approach. Generalisation for the case of non-local and non-evolution equations is disscused. Application of the theory to the Benjamin-Ono and Camassa-Holm type equations is considered.Comment: 16 page

Towards the theory of integrable hyperbolic equations of third order

The examples are considered of integrable hyperbolic equations of third order with two independent variables. In particular, an equation is found which admits as evolutionary symmetries the Krichever--Novikov equation and the modified Landau--Lifshitz system. The problem of choice of dynamical variables for the hyperbolic equations is discussed.Comment: 22

Self-focusing and self-trapping in unbiased centrosymmetric photorefractive media

We predict self-focusing and self-trapping of optical beams propagating in unbiased centrosymmetric photorefractive crystals in the near-transition paraelectric phase, where the nonlinear response is proportional to the square of the diffusion space-charge field

Continuous and discontinuous piecewise linear solutions of the linearly forced inviscid Burgers equation

We study a class of piecewise linear solutions to the inviscid Burgers equation driven by a linear forcing term. Inspired by the analogy with peakons, we think of these solutions as being made up of solitons situated at the breakpoints. We derive and solve ODEs governing the soliton dynamics, first for continuous solutions, and then for more general shock wave solutions with discontinuities. We show that triple collisions of solitons cannot take place for continuous solutions, but give an example of a triple collision in the presence of a shock.Comment: To appear in Journal of Nonlinear Mathematical Physics (proceedings of NEEDS 2007). 16 pages, 3 figures, LaTeX + AMS packages + pstrick