Periodic wavetrains for systems of coupled nonlinear Schrödinger equations

Exact, periodic wavetrains for systems of coupled nonlinear Schrödinger equations are obtained by the Hirota bilinear method and theta functions identities. Both the bright and dark soliton regimes are treated, and the solutions involve products of elliptic functions. The validity of these solutions is verified independently by a computer algebra software. The long wave limit is studied. Physical implications will be assessed.     

Novel exact solutions of coupled nonlinear Schro¨dinger equations with time–space modulation

We construct various novel exact solutions of two coupled dynamical nonlinear Schrdinger equations.Based on the similarity transformation,we reduce the coupled nonlinear Schrdinger equations with time-and space-dependent potentials,nonlinearities,and gain or loss to the coupled dynamical nonlinear Schrdinger equations.Some special types of non-travelling wave solutions,such as periodic,resonant,and quasiperiodically oscillating solitons,are used to exhibit the wave propagations by choosing some arbitrary functions.Our results show that the number of the localized wave of one component is always twice that of the other one.In addition,the stability analysis of the solutions is discussed numerically.     

Šilnikov manifolds in coupled nonlinear Schrödinger equations

We consider two coupled nonlinear Schrödinger equations with even, periodic boundary conditions, that are damped and quasiperiodically forced. We prove the existence of invariant manifolds with Šilnikov-type dynamics that are homoclinic to a spatially independent invariant torus. Such manifolds appear to induce complex behavior in numerical experiments.     

Self-similar solutions of equations of the nonlinear Schrödinger type

Experimental data are presented for the temperature dependence of the conductivity of Cu: SiO₂ metal-insulator composite films containing 3-nm Cu granules. At low temperatures in the concentration range 17–33 vol % Cu, all of the conductivity curves have a temperature dependence of the form σ ∝ exp{ (T ₀/T)^1/2}, while at higher temperatures a transition is observed to an activational dependence. A numerical simulation of the conduction in a composite material shows that an explanation of the observed temperature dependence must include the Coulomb interaction and the presence of a rather large random potential. The simulation also yields the size dependence and temperature dependence of the mesoscopic scatter of the conductivities of composite conductors. It is shown that a self-selecting percolation channel of current flow is formed in the region of strong mesoscopic scatter.     

Localized waves of the coupled cubic–quintic nonlinear Schrdinger equations in nonlinear optics

We investigate some novel localized waves on the plane wave background in the coupled cubic–quintic nonlinear Schr o¨dinger(CCQNLS) equations through the generalized Darboux transformation(DT). A special vector solution of the Lax pair of the CCQNLS system is elaborately constructed, based on the vector solution, various types of higherorder localized wave solutions of the CCQNLS system are constructed via the generalized DT. These abundant and novel localized waves constructed in the CCQNLS system include higher-order rogue waves, higher-order rogues interacting with multi-soliton or multi-breather separately. The first-and second-order semi-rational localized waves including several free parameters are mainly discussed:(i) the semi-rational solutions degenerate to the first-and second-order vector rogue wave solutions;(ii) hybrid solutions between a first-order rogue wave and a dark or bright soliton, a second-order rogue wave and two dark or bright solitons;(iii) hybrid solutions between a first-order rogue wave and a breather, a second-order rogue wave and two breathers. Some interesting and appealing dynamic properties of these types of localized waves are demonstrated, for example, these nonlinear waves merge with each other markedly by increasing the absolute value of α.These results further uncover some striking dynamic structures in the CCQNLS system.     

Exact solutions to the space–time fractional Schrödinger–Hirota equation and the space–time modified KDV–Zakharov–Kuznetsov equation

In this paper, the first integral method and the functional variable method are used to establish exact traveling wave solutions of the space–time fractional Schrödinger–Hirota equation and the space–time fractional modified KDV–Zakharov–Kuznetsov equation in the sense of conformable fractional derivative. The results obtained confirm that proposed methods are efficient techniques for analytic treatment of a wide variety of the space–time fractional partial differential equations.     

The fractional discrete nonlinear Schrödinger equation

We examine a fractional version of the discrete nonlinear Schrödinger (dnls) equation, where the usual discrete laplacian is replaced by a fractional discrete laplacian. This leads to the replacement of the usual nearest-neighbor interaction to a long-range intersite coupling that decreases asymptotically as a power-law. For the linear case, we compute both, the spectrum of plane waves and the mean square displacement of an initially localized excitation in closed form, in terms of regularized hypergeometric functions, as a function of the fractional exponent. In the nonlinear case, we compute numerically the low-lying nonlinear modes of the system and their stability, as a function of the fractional exponent of the discrete laplacian. The selftrapping transition threshold of an initially localized excitation shifts to lower values as the exponent is decreased and, for a fixed exponent and zero nonlinearity, the trapped fraction remains greater than zero.     

Modulational instability of bright solitary waves in incoherently coupled nonlinear Schrödinger equations

We present a detailed analysis of the modulational instability (MI) of ground-state bright solitary solutions of two incoherently coupled nonlinear Schr?dinger equations. Varying the relative strength of cross-phase and self-phase effects we show the existence and origin of four branches of MI of the two-wave solitary solutions. We give a physical interpretation of our results in terms of the group-velocity-dispersion- (GVD-) induced polarization dynamics of spatial solitary waves. In particular, we show that in media with normal GVD spatial symmetry breaking changes to polarization symmetry breaking when the relative strength of the cross-phase modulation exceeds a certain threshold value. The analytical and numerical stability analyses are fully supported by an extensive series of numerical simulations of the full model.     

On the finite-time behaviour for nonlinear Schrödinger equations

Consider the nonlinear Schrödinger equationu _t –iu=f(u). Forf(u)=±|u|^1+p , ±i|u|^1+p , ±u|u| ^p (p>0), and the Dirichlet boundary or nonlinear boundary (including the Neumann boundary and the Robin boundary) conditions, we establish the local estimates for the timet to the solutions of the initial-boundary value problems. Being based up on these estimates, we investigate the blowing-up properties of the solutions.Research supported in part by the Youth Foundation of Sichuan Education Committee and the Natural Science Foundation of China     

Long range scattering for nonlinear Schrödinger equations in one space dimension

We consider the scattering problem for the nonlinear Schrödinger equation in 1+1 dimensions: where = /x,R{0},R,p>3. We show that modified wave operators for (*) exist on a dense set of a neighborhood of zero in the Lebesgue spaceL ²(R) or in the Sobolev spaceH ¹(R)., The modified wave operators are introduced in order to control the long range nonlinearity |u|² u.Laboratoire associé au Centre National de la Recherche Scientifique     

Dark and anti-dark vector solitons of the coupled modified nonlinear Schrödinger equations from the birefringent optical fibers

Coupled modified nonlinear Schr?dinger (CMNLS) equations describe the pulse propagation in the picosecond or femtosecond regime of the birefringent optical fibers. A new type of the Lax pair and another hierarchy of the infinitely many conservation laws are derived based on the Wadati-Konno-Ichikawa system. By means of the Hirota method, soliton solutions in the normal dispersion regime are obtained. Parametric regions for the existence of dark and anti-dark vector soliton solutions are given. Asymptotic analysis shows that the collision between two solitons (two anti-dark solitons, two dark solitons, or dark and anti-dark solitons) in each polarization direction is elastic. Moreover, there is no energy transfer between two polarization components of each vector soliton, whether dark or anti-dark vector soliton. In addition, dark and anti-dark solitons can coexist on the same background seen from the collision between the dark and anti-dark solitons in one polarization direction. Our graphical analysis shows that the parameters in the CMNLS equations not only determine the regions for the existence of dark and anti-dark soliton solutions but also control the phase and direction of the propagation of the solitons. Finally, through the linear stability analysis, the modulational instability condition is given.     

On the integration of some classes of weakly deformed nonlinear Schrödinger equations

A method is proposed for constructing the solutions of a nonlinear Schrödinger equation with small corrections arising as a result of the introduction of arbitrary functions of the time and coordinates into the operator that dresses the kernel of a local $\bar \partial $ problem.     

Novel method for solution of coupled radial Schrödinger equations

One of the major problems in numerical solution of coupled differential equations is the maintenance of linear independence for different sets of solution vectors. A novel method for solution of radial Schrödinger equations is suggested. It consists of rearrangement of coupled equations in a way that is appropriate to avoid usual numerical instabilities associated with components of the wave function in their classically forbidden regions. Applications of the new method for nuclear structure calculations within the hyperspherical harmonics approach are given.     

The nonlinear Schrödinger limit of the Zakharov equations governing Langmuir turbulence

We consider the initial value problem for the Zakharov equations $$\begin{gathered} \left( Z \right)\frac{1}{{\lambda ^2 }}n_{tt} - \Delta (n + \left| {\rm E} \right|^2 ) = 0n(x,0) = n_0 (x) \hfill \\ n_t (x,0) = n_1 (x) \hfill \\ iE_t + \Delta E - nE = 0E(x,0) = E_0 (x) \hfill \\ \end{gathered} $$ (x∈? ^k ,k=2, 3,t ≧0) which model the propagation of Langmuir waves in plasmas. For suitable initial data solutions are shown to exist for a time interval independent of λ, a parameter proportional to the ion acoustic speed. For such data, solutions of (Z) converge as λ → ∞ to a solution of the cubic nonlinear Schrödinger equation (CSE)iE _t +ΔE+|E|² E=0. We consider both weak and strong solutions. For the case of strong solutions the results are analogous to previous results on the incompressible limit of compressible fluids.     

Interaction of coupled higher order nonlinear Schrödinger equation solitons

The novel inelastic collision properties of two-soliton interaction for an n-component coupled higher order nonlinear Schr?dinger equation are studied. Some interesting features of three soliton interactions, related to the integrability of the n-component coupled higher order nonlinear Schr?dinger equation are also discussed. Received 17 April 2002 Published online 2 October 2002 RID="a" ID="a"e-mail: abhijit@iitg.ernet.in RID="b" ID="b"e-mail: sasanka@iitg.ernet.in RID="c" ID="c"e-mail: sudipta@iitg.ernet.in     

Analytical and numerical solutions of the Schrödinger–KdV equation

The Schrödinger–KdV equation with power-law nonlinearity is studied in this paper. The solitary wave ansatz method is used to carry out the integration of the equation and obtain one-soliton solution. The G′/G method is also used to integrate this equation. Subsequently, the variational iteration method and homotopy perturbation method are also applied to solve this equation. The numerical simulations are also given.     

Smoothness and non-smoothness of the fundamental solution of time dependent Schrödinger equations

The fundamental solutionE(t,s,x,y) of time dependent Schrödinger equationsi?u/?t=?(1/2)Δu+V(t,x)u is studied. It is shown that

?E(t,s,x,y) is smooth and bounded fort≠s if the potential is sub-quadratic in the sense thatV(t,x)=o(|x|²) at infinity;

? in one dimension, ifV(t,x)=V(x) is time independent and super-quadratic in the sense thatV(x)≧C(1+|x|)^2+ε at infinity,C>0 and ε>0, thenE(t,s,x,y) is nowhereC ¹.

The result is explained in terms of the limiting behavior as the energy tends to infinity of the corresponding classical particle.     

Effect of phase shift in shape changing collision of solitons in coupled nonlinear Schrödinger equations

Soliton interactions in systems modelled by coupled nonlinear Schr?dinger (CNLS) equations and encountered in phenomena such as wave propagation in optical fibers and photorefractive media possess unusual features: shape changing intensity redistributions, amplitude dependent phase shifts and relative separation distances. We demonstrate these properties in the case of integrable 2-CNLS equations. As a simple example, we consider the stationary two-soliton solution which is equivalent to the so-called partially coherent soliton (PCS) solution discussed much in the recent literature. Received 1st October 2001 / Received in final form 4 February 2002 Published online 2 October 2002 RID="a" ID="a"e-mail: lakshman@bdu.ernet.in     

On concentration of positive bound states of nonlinear Schrödinger equations

We study the concentration behavior of positive bound states of the nonlinear Schrödinger equation $$ih\frac{{\partial \psi }}{{\partial t}} = \frac{{ - h^2 }}{{2m}}\Delta \psi + V\left( x \right)\psi - \gamma \left| \psi \right|^{p - 1} \psi .$$ Under certain condition ofV, we show that positive ground state solutions must concentrate at global minimum points ofV ash→0⁺; moreover, a point at which a sequence of positive bound states concentrates must be a critical point ofV. In cases thatV is radial, we prove that the positive radial solutions with least energy among all nontrivial radial solutions must concentrate at the origin ash→0⁺.     

Nature of transitions in augmented discrete nonlinear Schrödinger equations

We investigate the nature of the transitions between free and self-trapping states occurring in systems described by augmented forms of the discrete nonlinear Schr?dinger equation. These arise from an interaction between a moving quasiparticle (such as an electron or an exciton) and lattice vibrations, when the effects of nonlinearities in interaction potential and restoring force are included. We derive analytic conditions for the stability of the free state and the crossover between first- and second-order transitions. We demonstrate our results for different types of nonlinearities in the interaction potential and restoring force. We find that, depending on the type of nonlinearity, it is possible to have both first- and second-order transitions. We discuss possible hysteresis effects.