Traveling waves of nonlinear Schr\"{o}dinger equation including higher order dispersions

The solitary wave solution and periodic solutions expressed in terms of elliptic Jacobi's functions are obtained for the nonlinear Schr\"{o}dinger equation governing the propagation of pulses in optical fibers including the effects of second, third and fourth order dispersion. The approach is based on the reduction of the generalized nonlinear Schr\"{o}dinger equation to an ordinary nonlinear differential equation. The periodic solutions obtained form one-parameter family which depend on an integration constant $p$. The solitary wave solution with ${\rm sech}^2$ shape is the limiting case of this family with $p=0$. The solutions obtained describe also a train of soliton-like pulses with ${\rm sech}^2$ shape. It is shown that the bounded solutions arise only for special domains of integration constant.Comment: We consider in this paper also the case with negative parameter $\gamma$ (defocusing nonlinearity

Solitary Waves in Optical Fibers Governed by Higher Order Dispersion

An exact solitary wave solution is presented for the nonlinear Schrodinger equation governing the propagation of pulses in optical fibers including the effects of second, third and fourth order dispersion. The stability of this soliton-like solution with sech2 shape is proven by the sign-definiteness of the operator and an integral of the Sobolev type. The main criteria governing the existence of such stable localized pulses propagating in optical fibers are also formulated. A unique feature of these soliton-like optical pulses propagating in a fiber with higher order dispersion is that their parameters satisfy efficient scaling relations. The main soliton solution term given by perturbation theory is also presented when absorption or gain is included in the nonlinear Schrodinger equation. We anticipate that this type of stable localized pulses could find practical applications in communications, slow-light devices and ultrafast lasers.Comment: 4 pages 3 Figure

Extended Korteweg-de Vries equation for long gravity waves in incompressible fluid without strong limitation to surface deviation

We have derived the extended Korteweg-de Vries equation describing the long gravity waves without limitation to surface deviation. The only restriction to the surface deviation is connected with the stability condition for appropriate solutions. The derivation of extended KdV equation is based on the Euler equations for inviscid irrotational and incompressible fluid. It is shown that the extended KdV equation reduces to standard KdV equation for small amplitude of the waves. We have also generalized the extended KdV equation for describing the decaying effect of the waves. Quasi-periodic and solitary wave solutions for extended KdV equation with decaying effect are found as well. We also demonstrate that the fundamental approach based on the inverse scattering method is applicable for solving the extended KdV equation in the case when decaying effect is negligibly small. Such case always occur for restricted propagation distances of the waves.Comment: 15 pages, 2 figure

Propagation of coupled quartic and dipole multi-solitons in optical fibers medium with higher-order dispersions

We present the discovery of two types of multiple-hump soliton modes in a highly dispersive optical fiber with a Kerr nonlinearity. We show that multi-hump optical solitons of quartic or dipole types are possible in the fiber system in the presence of higher-order dispersion. Such nonlinear wave packets are very well described by an extended nonlinear Schrodinger equation involving both cubic and quartic dispersion terms. It is found that the third- and fourth-order dispersion effects in the fiber material may lead to the coupling of quartic or dipole solitons into double-, triple-, and multi-humped solitons. We provide the initial conditions for the formation of coupled multi-hump quartic and dipole solitons in the fiber. Numerical results illustrate that propagating multi-quartic and multi-dipole solitons in highly dispersive optical fibers councide with a high accuracy to our analytical multi-soliton solutions. It is important for applications that described multiple-hump soliton modes are stable to small noise perturbation that was confirmed by numerical simulations. These numerical results confirm that the newly found multi-soliton pulses can be potentially utilized for transmission in optical fibers medium with higher-order dispersions.Comment: 6 pages, 6 figure

Wave-speed management of dipole, bright and W-shaped solitons in optical metamaterials

Wave-speed management of soliton pulses in a nonlinear metamaterial exhibiting a rich variety of physical effects that are important in a wide range of practical applications, is studied both theoretically and numerically. Ultrashort electromagnetic pulse transmission in such inhomogeneous system is described by a generalized nonlinear Schreodinger equation with space-modulated higher-order dispersive and nonlinear effects of different nature. We present the discovery of three types of periodic wave solutions that are composed by the product of Jacobi elliptic functions in the presence of all physical processes. Envelope solitons of the dipole, bright and W-shaped types are also identified, thus illustrating the potentially rich set of localized pulses in the system. We develop an effective similarity transformation method to investigate the soliton dynamics in the presence of the inhomogeneities of media. The application of developed method to control the wave speed of the presented solitons is discussed. The results show that the wave speed of dipole, bright and W-shaped solitons can be effectively controlled through spatial modulation of the metamaterial parameters. In particular, the soliton pulses can be decelerated and accelerated by suitable variations of the distributed dispersion parameters.Comment: 14 pages, 5 figure