Numerical approaches in atmospheric and oceanic modeling inevitably introduce diffusion (or dissipation) and dispersion into the approximate solution. From a physical point of view, advection of a passive tracer is the simple transition of a quantity. Therefore, dispersion, the propagation of different spatial scales at different phase speed, and diffusion are processes that are aliens to the p...

Recently, many nonlinear coupled evolution equations, such as the coupled Korteweg–de Vries (KdV) equation, the coupled Boussinesq equation, and the coupled mKdV equation, appear in scientific applications [1 – 13]. The coupled evolution equations attracted a considerable research work in the literature. The aims of these works have been the determination of soliton solutions and the proof of c...

The general variable coefficient cylindrical/spherical KdV equation has been investigated by using the simplified homogeneous balance method. It proven that if its coefficients satisfy certain constraint conditions, then a nonlinear transformation converts solution of quadratic form into equation. admits series solutions expressed exponential functions, therefore one soliton-like and multi can ...

We derive and analyze exact static solutions to the gravitating O(3) σ model with cosmological constant in (2+1) dimensions. Both signs of the gravitational and cosmological constants are considered. Our solutions include geodesically complete spacetimes, and two classes of black holes. e-mail: [email protected] e-mail: [email protected]

We study in a systematic way all static solutions of the Goldstone model in 1+1 dimension with a periodicity condition imposed on the spatial coordinate. The solutions are presented in terms of the standard trigonometric functions and of Jacobi elliptic functions. Their stability analysis is carried out, and the complete list of classically stable quasi-topological solitons is given.

The Darboux–Dressing Transformations are applied to the Lax pair associated to the system of nonlinear equations describing the resonant interaction of three waves in 1+1 dimensions. We display explicit solutions featuring localized waves whose profile vanishes at the spacial boundary |x| = ∞, and which are not pure soliton solutions. These solutions depend on an arbitrary function and allow to...

In this paper, a closed form optical soliton solution is obtained for the nonlinear Schrödinger’s equation with fourth order dispersion in a Kerr law media. The solitary wave ansatze is used to carry out the integration of this equation. Finally, a numerical simulation is given for the closed form soliton solution. Mathematics Subject Classification: 35Q51, 35Q53, 37K10, 78A60 PACS Codes: 02.30...

The derivation of the Demia´nski–Newman solution within the framework of the Ernst complex formalism is considered. We show that this solution naturally arises as a two–soliton specialization of the axisymmet-ric multi–soliton electrovacuum metric, and we work out the full set of the corresponding metrical fields and electromagnetic potentials. Some limits and physical characteristics of the DN...

We show that an interacting double soliton solution to the perturbed mKdV equation (1.1). is close in H2 to a double soliton following an effective dynamics obtained as Hamilton’s equations for the restriction of the mKdV Hamiltonian to the submanifold of solitons. The interplay between algebraic aspects of complete integrability of the unperturbed equation and the analytic ideas related to sol...

I analyze the one-dimensional, cubic Schrödinger equation, with nonlinearity constructed from the current density, rather than, as is usual, from the charge density. A soliton solution is found, where the soliton moves only in one direction. Relation to higher-dimensional Chern–Simons theory is indicated. The theory is quantized and results for the two-body quantum problem agree at weak couplin...