On solving the nonlinear Biswas-Milovic equation with dual-power law nonlinearity using the extended tanh-function method

In this article, we apply the extended tanh-function method to find the exact traveling wave solutions of the nonlinear Biswas-Milovic equation (BME), which describes the propagation of solitons through optical fibers for trans-continental and trans-oceanic distances. This equation is a generalized version of the nonlinear Schrödinger equation with dual-power law nonlinearity. With the aid of computer algebraic system Maple, both constant and time-dependent coefficients of BME are discussed. Comparison between our new results and the well-known results is given. The given method in this article is straightforward, concise and can be applied to other nonlinear partial differential equations (PDEs) in mathematical physics

Some special solutions to the Hyperbolic NLS equation

The Hyperbolic Nonlinear Schrodinger equation (HypNLS) arises as a model for the dynamics of three-dimensional narrowband deep water gravity waves. In this study, the Petviashvili method is exploited to numerically compute bi-periodic time-harmonic solutions of the HypNLS equation. In physical space they represent non-localized standing waves. Non-trivial spatial patterns are revealed and an attempt is made to describe them using symbolic dynamics and the language of substitutions. Finally, the dynamics of a slightly perturbed standing wave is numerically investigated by means a highly acccurate Fourier solver.Comment: 33 pages, 10 figures, 70 references. Other author's papers can be found at http://www.denys-dutykh.com

Resonant Optical Solitons in (3 + 1)-Dimensions Dominated by Kerr Law and Parabolic Law Nonlinearities

This study investigates the optical solitons of of (3+1)-dimensional resonant nonlinear Schrödinger (3D-RNLS) equation with the two laws of nonlinearity. The two forms of nonlinearity are represented by Kerr law and parabolic law. Based on complex transformation, the traveling wave reduction of the governing model is derived. The projective Riccati equations technique is applied to obtain the exact solutions of 3D-RNLS equation. Various types of waves that represent different structures of optical solitons are extracted. These structures include bright, dark, singular, dark-singular and combined singular solitons. Additionally, the obliquity effect on resonant solitons is illustrated graphically and is found to cause dramatic variations in soliton behaviors

Practical use of variational principles for modeling water waves

This paper describes a method for deriving approximate equations for irrotational water waves. The method is based on a 'relaxed' variational principle, i.e., on a Lagrangian involving as many variables as possible. This formulation is particularly suitable for the construction of approximate water wave models, since it allows more freedom while preserving the variational structure. The advantages of this relaxed formulation are illustrated with various examples in shallow and deep waters, as well as arbitrary depths. Using subordinate constraints (e.g., irrotationality or free surface impermeability) in various combinations, several model equations are derived, some being well-known, other being new. The models obtained are studied analytically and exact travelling wave solutions are constructed when possible.Comment: 30 pages, 1 figure, 62 references. Other author's papers can be downloaded at http://www.denys-dutykh.com