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.     

Existence of solitary waves for the discrete Schrödinger equation coupled to a nonlinear oscillator

The discrete Schrödinger equation with a nonlinearity concentrated at a single point is an interesting and important model to study the long-time behavior of solutions, including the asymptotic stability of solitary waves and properties of global attractors. In this note, the global well-posedness of this equation and the existence of solitary waves is proved and the properties of these waves are studied.     

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.     

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.     

Symmetry and asymmetry rogue waves in two-component coupled nonlinear Schrdinger equations

By means of the modified Darboux transformation we obtain some types of rogue waves in two-coupled nonlinear Schr ¨odinger equations.Our results show that the two components admits the symmetry and asymmetry rogue wave solutions,which arises from the joint action of self-phase,cross-phase modulation,and coherent coupling term.We also obtain the analytical transformation from the initial seed solution to unique rogue waves with the bountiful pair structure.In a special case,the asymmetry rogue wave can own the spatial and temporal symmetry gradually,which is controlled by one parameter.It is worth pointing out that the rogue wave of two components can share the temporal inversion symmetry.     

Three-dimensional solitary waves and vortices in a discrete nonlinear Schrödinger lattice

In a benchmark dynamical-lattice model in three dimensions, the discrete nonlinear Schr?dinger equation, we find discrete vortex solitons with various values of the topological charge S. Stability regions for the vortices with S=0,1,3 are investigated. The S=2 vortex is unstable and may spontaneously rearranging into a stable one with S=3. In a two-component extension of the model, we find a novel class of stable structures, consisting of vortices in the different components, perpendicularly oriented to each other. Self-localized states of the proposed types can be observed experimentally in Bose-Einstein condensates trapped in optical lattices and in photonic crystals built of microresonators.     

On the characterization of vector rogue waves in two-dimensional two coupled nonlinear Schrödinger equations with distributed coefficients

We construct vector rogue wave solutions of the two-dimensional two coupled nonlinear Schrödinger equations with distributed coefficients, namely diffraction, nonlinearity and gain parameters through similarity transformation technique. We transform the two-dimensional two coupled variable coefficients nonlinear Schrödinger equations into Manakov equation with a constraint that connects diffraction and gain parameters with nonlinearity parameter. We investigate the characteristics of the constructed vector rogue wave solutions with four different forms of diffraction parameters. We report some interesting patterns that occur in the rogue wave structures. Further, we construct vector dark rogue wave solutions of the two-dimensional two coupled nonlinear Schrödinger equations with distributed coefficients and report some novel characteristics that we observe in the vector dark rogue wave solutions.     

Localized waves in three-component coupled nonlinear Schrdinger equation

We study the generalized Darboux transformation to the three-component coupled nonlinear Schr ¨odinger equation.First-and second-order localized waves are obtained by this technique.In first-order localized wave,we get the interactional solutions between first-order rogue wave and one-dark,one-bright soliton respectively.Meanwhile,the interactional solutions between one-breather and first-order rogue wave are also given.In second-order localized wave,one-dark-one-bright soliton together with second-order rogue wave is presented in the first component,and two-bright soliton together with second-order rogue wave are gained respectively in the other two components.Besides,we observe second-order rogue wave together with one-breather in three components.Moreover,by increasing the absolute values of two free parameters,the nonlinear waves merge with each other distinctly.These results further reveal the interesting dynamic structures of localized waves in the three-component coupled system.     

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     

Dark–bright optical solitary waves and modulation instability analysis with (2 + 1)-dimensional cubic-quintic nonlinear Schrödinger equation

This paper addresses the (2+1)-dimensional cubic-quintic nonlinear Schrödinger equation (CQNLS) that serves as the model to study the light propagation through nonlinear optical media and non-Kerr crystals. A dark–bright optical solitary wave solution of this equation is retrieved by adopting the complex envelope function ansatz. This type of solitary wave describes the properties of bright and dark optical solitary waves in the same expression. The integration naturally lead to a constraint condition placed on the solitary wave parameters which must hold for the solitary waves to exist. Additionally, the modulation instability (MI) analysis of the model is studied based on the standard linear stability analysis and the MI gain spectrum is got. Numerical simulation and physical interpretations of the obtained results are demonstrated. It is hoped that the results reported in this paper can enrich the nonlinear dynamical behaviors of the CQNLS.     

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.     

Analytical treatments of the space–time fractional coupled nonlinear Schrödinger equations

Under investigation in this work is a (\(2+1\))-dimensional the space–time fractional coupled nonlinear Schrödinger equations, which describes the amplitudes of circularly-polarized waves in a nonlinear optical fiber. With the aid of conformable fractional derivative and the fractional wave transformation, we derive the analytical soliton solutions in the form of rational soliton, periodic soliton, hyperbolic soliton solutions by four integration method, namely, the extended trial equation method, the \(\exp (-\,\Omega (\eta ))\)-expansion method and the improved \(\tan (\phi (\eta )/2)\)-expansion method and semi-inverse variational principle method. Based on the the extended trial equation method, we derive the several types of solutions including singular, kink-singular, bright, solitary wave, compacton and elliptic function solutions. Under certain condition, the 1-soliton, bright, singular solutions are driven by semi-inverse variational principle method. Based on the analytical methods, we find that the solutions give birth to the dark solitons, the bright solitons, combine dark-singular, kink, kink-singular solutions with fractional order for nonlinear fractional partial differential equations arise in nonlinear optics.     

Radiationless traveling waves in saturable nonlinear Schrödinger lattices

The long-standing problem of moving discrete solitary waves in nonlinear Schr?dinger lattices is revisited. The context is photorefractive crystal lattices with saturable nonlinearity whose grand-canonical energy barrier vanishes for isolated coupling strength values. Genuinely localized traveling waves are computed as a function of the system parameters for the first time. The relevant solutions exist only for finite velocities.     

Coupled nonlinear Schrödinger equations in optic fibers theory

In this paper a detailed derivation and numerical solutions of Coupled Nonlinear Schrödinger Equations for pulses of polarized electromagnetic waves in cylindrical fibers has been reviewed. Our recent work has been compared with some previous ones and the advantage of our new approach over other methods has been assessed. The novelty of our approach lies is an attempt to proceed without loss of information within the frame of basic approximations. In our work we focused on the multimode The eigen mode definition is based on complete linearized Maxwell equations and Hondros-Debye boundary conditions, which depend on the geometry of the dielectric waveguide. We proved both stability and convergence in the L ₂ space of an explicit finite-difference scheme for the Coupled Nonlinear Schrödinger Equations and those estimations are used for an implicit scheme. To test our hypothesis we compare numerical solutions for Manakov system with known analytical solitonic solutions. We also consider an important example of the general system - an evolution of two pulses with different group velocity which can serve as a model of pulses interaction in multimode optic fibers. Last case, a nonlinear dispersion of rectangular pulse, exhibits an asymptotic behavior similar to Nonlinear Schrödinger Equation solution asymptotics for the rectangular initial condition. Finally, we compared theoretical results with specially arranged experiments employing a photonic crystal fiber.     

On the integrability aspects of nonparaxial nonlinear Schrödinger equation and the dynamics of solitary waves

The integrability nature of a nonparaxial nonlinear Schrödinger (NNLS) equation, describing the propagation of ultra-broad nonparaxial beams in a planar optical waveguide, is studied by employing the Painlevé singularity structure analysis. Our study shows that the NNLS equation fails to satisfy the Painlevé test. Nevertheless, we construct one bright solitary wave solution for the NNLS equation by using the Hirota's direct method. Also, we numerically demonstrate the stable propagation of the obtained bright solitary waves even in the presence of an external perturbation in a form of white noise. We then numerically investigate the coherent interaction dynamics of two and three bright solitary waves. Our study reveals interesting energy switching among the colliding solitary waves due to the nonparaxiality.     

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.     

Nondegenerate soliton solutions in certain coupled nonlinear Schrödinger systems

In this paper, we report a more general class of nondegenerate soliton solutions, associated with two distinct wave numbers in different modes, for a certain class of physically important integrable two component nonlinear Schrödinger type equations through bilinearization procedure. In particular, we consider coupled nonlinear Schrödinger (CNLS) equations (both focusing as well as mixed type nonlinearities), coherently coupled nonlinear Schrödinger (CCNLS) equations and long-wave-short-wave resonance interaction (LSRI) system. We point out that the obtained general form of soliton solutions exhibit novel profile structures than the previously known degenerate soliton solutions corresponding to identical wave numbers in both the modes. We show that such degenerate soliton solutions can be recovered from the newly derived nondegenerate soliton solutions as limiting cases.     

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