Exact self-similar solitary waves and collisions in nonlinear optical media

This paper analyses bright and dark spatial self-similar waves propagation and collision in graded-index nonlinear waveguide amplifiers with self-focusing and self-defocusing Kerr nonlinearities. It finds an appropriate transformation for the first time such that the nonlinear Schrodinger equation （NLSE） with varying coefficients transform into standard NLSE. It obtains one-solitonlike, two-solitonlike and multi-solitonlike self-similar wave solutions by using the transformation. Furthermore, it analyses the features of the self-similar waves and their collisions.     

Relativistic effects on the modulational instability of electron plasma waves in quantum plasma

Relativistic effects on the linear and nonlinear properties of electron plasma waves are investigated using the one-dimensional quantum hydrodynamic (QHD) model for a two-component electron?Cion dense quantum plasma. Using standard perturbation technique, a nonlinear Schr?dinger equation (NLSE) containing both relativistic and quantum effects has been derived. This equation has been used to discuss the modulational instability of the wave. Through numerical calculations it is shown that relativistic effects significantly change the linear dispersion character of the wave. Unlike quantum effects, relativistic effects are shown to reduce the instability growth rate of electron plasma waves.     

The nonlinear Schrödinger equation and the propagation of weakly nonlinear waves in optical fibers and on the water surface

The dynamics of waves in weakly nonlinear dispersive media can be described by the nonlinear Schrödinger equation (NLSE). An important feature of the equation is that it can be derived in a number of different physical contexts; therefore, analogies between different fields, such as for example fiber optics, water waves, plasma waves and Bose–Einstein condensates, can be established. Here, we investigate the similarities between wave propagation in optical Kerr media and water waves. In particular, we discuss the modulation instability (MI) in both media. In analogy to the water wave problem, we derive for Kerr-media the Benjamin–Feir index, i.e. a nondimensional parameter related to the probability of formation of rogue waves in incoherent wave trains.     

Self-similar rogue waves in an inhomogeneous generalized nonlinear Schrödinger equation

We present an explicit analytical form of first and second order rogue waves for distributive nonlinear Schrödinger equation (NLSE) by mapping it to standard NLSE through similarity transformation. Upon obtaining the rogue wave solutions, we study the propagation of rogue waves through a periodically distributed system for the two cases when Wronskian of dispersion and nonlinearity is (i) zero, (ii) not equal to zero. For the former case, we discuss a mechanism to control their propagation and for the latter case we depict the interesting features of rogue waves as they propagate through dispersion increasing and decreasing fiber.     

Are rogue waves robust against perturbations?

We study the effect of various perturbations on the fundamental rational solution of the nonlinear Schrödinger equation (NLSE). This solution describes generic nonlinear wave phenomena in the deep ocean, including the notorious rogue waves. It also describes light pulses in optical fibres. We find that the solution can survive at least three types of perturbations that are often used in the physics of nonlinear waves. We show that the rational solution remains rational and localized in each direction, thus representing a modified rogue wave.     

Modulational instability of dust acoustic waves in a dusty plasma with nonthermal electrons and ions

Effects of nonadiabaticity of variable dust charge, dust fluid temperature, trapped electrons as well as nonisothermality of ions on the amplitude modulation of dust acoustic waves in an unmagnetized dusty plasma are investigated. A modified nonlinear Schr?dinger equation (NLSE) is obtained by the standard reductive perturbation technique and is solved numerically by the split-step Fourier method. The modulational instability and the envelope solitary wave structure are found to be modified somewhat by the effects of nonthermally distributed ions and trapped electrons.     

Novel exact self-similar solitary waves in graded-index media with Kerr nonlinearity

A broad class of exact self-similar solitary wave solutions to the nonlinear Schrdinger equation (NLSE) are found by using the extended mapping deformation method. These novel waves can propagate inside planar, graded-index nonlinear waveguide amplifiers with self-focusing and self-defocusing Kerr nonlinearities, respectively. Furthermore, we analyze the features of the self-similar waves and their evolution. Our results in this paper include some in the literature [S.A. Ponomarenko, G.P. Agrawal, Phys. Rev. Lett. 97 (2006) 013901].     

Self-similar cnoidal and solitary wave solutions of the (1+1)-dimensional generalized nonlinear Schrödinger equation

With the help of the similarity transformation and the solvable stationary nonlinear Schrödinger equation (NLSE),we obtain exact chirped and chirp-free self-similar cnoidal wave and solitary wave solutions of the generalized NLSE exhibitingspatial inhomogeneity, inhomogeneous nonlinearity and gain or loss at the same time. As an example, we investigate their propagationdynamics in a nonlinear optical system, and present a series of interesting properties of optical waves.     

Persistence of rogue waves in extended nonlinear Schrödinger equations: Integrable Sasa–Satsuma case

We present the lowest order rogue wave solution of the Sasa–Satsuma equation (SSE) which is one of the integrable extensions of the nonlinear Schrödinger equation (NLSE). In contrast to the Peregrine solution of the NLSE, it is significantly more involved and contains polynomials of fourth order rather than second order in the corresponding expressions. The correct limiting case of the Peregrine solution appears when the extension parameter of the SSE is reduced to zero.     

Modulational instability of dust-ion-acoustic waves in pair-ion plasma having non-thermal non-extensive electrons

The modulational instability (MI) criteria of dust-ion-acoustic (DIA) waves (DIAWs) have been investigated in a four-component pair-ion plasma having inertial pair ions, inertialess non-thermal non-extensive electrons, and immobile negatively charged massive dust grains. A nonlinear Schrödinger equation (NLSE) is derived by using reductive perturbation method. The nonlinear and dispersive coefficients of the NLSE can predict the modulationally stable and unstable parametric regimes of DIAWs and associated first and second-order DIA rogue waves (DIARWs). The MI growth rate and the configuration of the DIARWs are examined, and it is found that the MI growth rate increases (decreases) with increasing the number density of the negatively charged dust grains in the presence (absence) of the negative ions. It is also observed that the amplitude and width of the DIARWs increase (decrease) with the negative (positive) ion mass. The implications of the results to laboratory and space plasmas are briefly discussed.     

Solitary wave solutions for nonlinear Schrödinger equation with non-polynomial nonlinearity

A new kind of non-polynomial nonlinearity is introduced in the nonlinear Schrödinger equation (NLSE) and the conditions are determined for which it admits solitary wave solutions. The study is done for two cases: one in which the nonlinear interaction is of the non-polynomial form and second in which cubic nonlinearity is also included along with the radical nonlinearity. Dark and bright solitary waves solutions are obtained in the respective cases. Further, later case is extended to conditions for which corresponding equation reduces to driven quadratic-cubic NLSE possessing cnoidal solutions with plane wave phase, which reduces to bright soliton for a certain parameter.     

Exact Solutions to Extended Nonlinear Schrödinger Equation in Monomode Optical Fiber

By using the generally projective Riccati equation method, more new exact travelling wave solutions to extended nonlinear Schrödinger equation (NLSE), which describes the femtosecond pulse propagation in monomode optical fiber, are found, which include bright soliton solution, dark soliton solution, new solitary waves, periodic solutions, and rational solutions. The finding of abundant solution structures for extended NLSE helps to study the movement rule of femtosecond pulse propagation in monomode optical fiber.     

简谐势阱中中性原子非线性薛定谔方程的定态解

给出了非线性定态薛定谔方程(NLSE)数值求解的一般方法,并求解了简谐势阱中中性原子NLSE的基态和激发态解．讨论了NLSE的波函数收敛与归一化问题,并对计算精度进行了分析. 关键词：     

Waves that appear from nowhere and disappear without a trace

The title (WANDT) can be applied to two objects: rogue waves in the ocean and rational solutions of the nonlinear Schrödinger equation (NLSE). There is a hierarchy of rational solutions of ‘focussing’ NLSE with increasing order and with progressively increasing amplitude. As the equation can be applied to waves in the deep ocean, the solutions can describe “rogue waves” with virtually infinite amplitude. They can appear from smooth initial conditions that are only slightly perturbed in a special way, and are given by our exact solutions. Thus, a slight perturbation on the ocean surface can dramatically increase the amplitude of the singular wave event that appears as a result.     

Dissipation of Nonlinear Wave in Inhomogeneous Dust Plasma Crystal

A Korteweg-de Vires-type (KdV-type) equation and a modifiedNonlinear Schrödinger equation (NLSE) for the dust lattice wave(DLW) are derived in a weakly inhomogeneous dust plasma crystal. Itseems that the amplitude and the velocity of the dust lattice solitary waves decay exponentially with increasing time in a dust lattice. The modulational instability of this dust lattice envelope waves is investigated as well. It is found that the waves are modulational stable under certain conditions. On the other hand, the waves are modulational unstable if the conditions are not satisfied.     

Nonlinear Schrödinger equation and DNA dynamics

We study a possible solitary wave solution of the nonlinear Schrödinger equation (NLSE). It is shown that the wave can be both modulated and nonmodulated depending on a ratio of the envelope and the carrier wave velocities. We also study the same type of the soliton solution in DNA dynamics. We show that the ratio of these two velocities is a measure of modulation and we conclude that the modulated wave is more stable than the nonmodulated one. Finally, we solved the problem concerning three parameters arising from the applied procedure for the solution of the NLSE.     

Abruptly autofocusing waves

We introduce a new class of (2+1)D spatial and (3+1)D spatiotemporal waves that tend to autofocus in an abrupt fashion. While the maximum intensity of such a radial wave remains almost constant during propagation, it suddenly increases by orders of magnitude right before its focal point. These waves can be generated through the use of radially symmetric Airy waves or by appropriately superimposing Airy wave packets. Possible applications of such abruptly focusing beams are also discussed.     

Nonlinear Waves in an Inhomogeneous Fluid Filled Elastic Tube

In a thin-walled, homogeneous, straight, long, circular, and incompressible fluid filled elastic tube, small but finite long wavelength nonlinear waves can be describe by a KdV (Korteweg de Vries) equation, while the carrier wave modulations are described by a nonlinear Schrödinger equation (NLSE). However if the elastic tube is slowly inhomogeneous, then it is found, in this paper, that the carrier wave modulations are described by an NLSE-like equation. There are soliton-like solutions for them, but the stability and instability regions for this soliton-like waves will change, depending on what kind of inhomogeneity the tube has.     

Electromagnetic waves in dielectrics with memory

We analyse electromagnetic wave propagation in a dielectric with memory of the Maxwell-Hopkinson type. We show that the components of the electric and magnetic fields satisfy two different scalar wave equations and we first look for their harmonic plane wave solutions. Then we prove that dielectrics with memory can also support approximate Courant-Hilbert waves. We discuss the equations to be solved to get all the components of the electromagnetic field from a scalar solution from each wave equation and TE, TM harmonic plane waves are explicitly given.     

Envelope self-similar solutions for the nonautonomous and inhomogeneous nonlinear Schrödinger equation

By means of the similarity transformation connecting with the solvable stationary equation, the self-similar combined Jacobian elliptic function solutions and fractional form solutions of the generalized nonlinear Schrödinger equation (NLSE) are obtained when the dispersion, nonlinearity, and gain or absorption are varied. The propagation dynamics in a periodic distributed amplification system is investigated. Self-similar cnoidal waves and corresponding localized waves including bright and dark similaritons (or solitons) for NLSE and arch and kink similaritons (or solitons) for cubic-quintic NLSE are analyzed. The results show that the intensity and the width of chirped cnoidal waves (or similaritons) change more distinctly than that of chirp-free counterparts (or solitons).