Discrete Nonlinear Schrodinger Equation Research Articles

Kuznetsov–Ma breather-like solutions in the Salerno model

The Salerno model is a discrete variant of the celebrated nonlinear Schrödinger (NLS) equation interpolating between the discrete NLS (DNLS) equation and completely integrable Ablowitz–Ladik (AL) model by appropriately tuning the relevant homotopy parameter. Although the AL model possesses an explicit time-periodic solution known as the Kuznetsov–Ma (KM) breather, the existence of time-periodic solutions away from the integrable limit has not been studied as of yet. It is thus the purpose of this work to shed light on the existence and stability of time-periodic solutions of the Salerno model. In particular, we vary the homotopy parameter of the model by employing a pseudo-arclength continuation algorithm where time-periodic solutions are identified via fixed-point iterations. We show that the solutions transform into time-periodic patterns featuring small, yet non-decaying far-field oscillations. Remarkably, our numerical results support the existence of previously unknown time-periodic solutions even at the integrable case whose stability is explored by using Floquet theory. A continuation of these patterns towards the DNLS limit is also discussed.

Discrete nonlocal nonlinear Schrödinger systems: Integrability, inverse scattering and solitons

A number of integrable nonlocal discrete nonlinear Schrödinger (NLS) type systems have been recently proposed. They arise from integrable symmetry reductions of the well-known Ablowitz–Ladik scattering problem. The equations include: the classical integrable discrete NLS equation, integrable nonlocal: PT symmetric, reverse space time (RST), and the reverse time (RT) discrete NLS equations. Their mathematical structure is particularly rich. The inverse scattering transforms (IST) for the nonlocal discrete PT symmetric NLS corresponding to decaying boundary conditions was outlined earlier. In this paper, a detailed study of the IST applied to the PT symmetric, RST and RT integrable discrete NLS equations is carried out for rapidly decaying boundary conditions. This includes the direct and inverse scattering problem, symmetries of the eigenfunctions and scattering data. The general linearization method is based on a discrete nonlocal Riemann–Hilbert approach. For each discrete nonlocal NLS equation, an explicit one soliton solution is provided. Interestingly, certain one soliton solutions of the discrete PT symmetric NLS equation satisfy nonlocal discrete analogs of discrete elliptic function/Painlevé-type equations.

Solving a Discrete Nonlinear Schrödinger Equation with a Trap

The wave function and ground state energy for a discrete nonlinear Schrodinger equation (DNLSE) with a trap can be found numerically using an iterative process. However, this approach does not always give convergent results. A functional is proposed that is not an energy functional, the minimum of which always gives the ground state of the system under consideration.

Quantum compacton and quantum peakon in Heisenberg ferromagnetic chains with Dzyaloshinsky–Moriya interaction

The one-dimensional ferromagnet XXZ spin chain with Dzyaloshinsky–Moriya (DM) interaction, recently introduced by Djoufack and coworkers is reexplored, with a particular attention carried on their found discrete nonlinear Schrödinger (DNLS) equation, governing the quantum breathers states behaviors. This DNLS equation admits exact bright compacton and peakon-like solutions, where analytical expressions, the existence and stability criteria are found and used to obtain their quantized energy levels. Using the semi-discrete multiple-scale method, the DNLS equation is reduced to the extended nonlinear Schrödinger equation which consists of the basic NLS equation with additional nonlinear dispersive terms Via the bifurcation diagram and Liapunov exponents, we provide a summary of essential dynamics and show that the equation would admit several forms of solutions among which the localized Hartree n compacton and peakon-like boson quantum breathers states. Furthermore, we notice that on the contrary to the classical outcomes where amplitudes of both solutions are free parameters, the amplitudes for quantum states are not free since the obtained solutions need to be normalized. The stationary localization of both compacton and peakon-like states is confirmed by numerical simulations performed on the DNLS equation.

Derivation of the Tight-Binding Approximation for Time-Dependent Nonlinear Schrödinger Equations

In this paper, we consider the nonlinear one-dimensional time-dependent Schrodinger equation with a periodic potential and a bounded perturbation. In the limit of large periodic potential, the time behavior of the wavefunction can be approximated, with a precise estimate of the remainder term, by means of the solution to the discrete nonlinear Schrodinger equation of the tight-binding model.

Solutions of the Discrete Nonlinear Schrödinger Equation with a Trap

We obtain solutions of the discrete nonlinear Schrodinger equation with an impurity center in two ways. In the first of them, we construct the wave function as a series in a certain parameter. In the second, approximate method, we obtain the wave function in the continuum limit. We compare the obtained solutions with numerical results, and the relative accuracy of the solution in the form of a series does not exceed 10−15in order of magnitude.

Asymptotic Formula for “Transparent Points” for Cubic–Quintic Discrete NLS Equation

A “transparent point” is a particular value of a governing parameter in a nontranslationally invariant system that makes the system “almost” translationally invariant. This concept was introduced recently in the context of the discrete nonlinear Schrodinger (DNLS) equation with saturable nonlinearity — it was discovered that a tuning of the lattice spacing parameter h in this model affects the soliton mobility. In this paper, we study the DNLS equation with competing cubic–quintic nonlinearity that also admits the transparent points with respect to the lattice spacing parameter h. We give a geometrical interpretation of the transparent points in terms of dynamical system theory and present a simple asymptotical formula for them at h → 0. Although the derivation of this formula is heuristic and nonrigorous, it gives the values of transparent points with remarkable accuracy even for quite large values of h.

Dynamic instability in neuronal microtubules

The Hamiltonian describing the interaction between the two neighbouring filaments of neuronal microtubules (nMTs) and in addition of u and ф model, which may be represented as longitudinal and radial displacements. From this Hamiltonian, we get discrete Nonlinear Schrodinger (DNLS) like equations by using the Rotating wave approximation (RWA). In order to analyze the instability nature of the system, we perform modulational instability (MI) analysis for the DNLS equations of motion which attribute for the energy localization phenomenon and it paves way for bio-energy transport in intracellular communications.

Quantum versus classical effects in the chirped-drive discrete nonlinear Schrödinger equation

A chirped parametrically driven discrete nonlinear Schrodinger equation is discussed. It is shown that the system allows two resonant excitation mechanisms, i.e., successive two-level transitions (ladder climbing) or a continuous classical-like nonlinear phase-locking (autoresonance). Two-level arguments are used to study the ladder-climbing process, and semiclassical theory describes the autoresonance effect. The regimes of efficient excitation in the problem are identified and characterized in terms of three dimensionless parameters describing the driving strength, the dispersion nonlinearity, and the Kerr-type nonlinearity, respectively. The nonlinearity alters the borderlines between the regimes, and their characteristics.

On the nonexistence of degenerate phase-shift multibreathers in Klein–Gordon models with interactions beyond nearest neighbors

Abstract In this work, we study the existence of, low amplitude, phase-shift multibreathers for small values of the linear coupling in Klein–Gordon chains with interactions beyond the classical nearest-neighbor (NN) ones. In the proper parameter regimes, the considered lattices bear connections to models beyond one spatial dimension, namely the so-called zigzag lattice, as well as the two-dimensional square lattice or coupled chains. We examine initially the necessary persistence conditions of the system derived by the so-called Effective Hamiltonian Method, in order to seek for unperturbed solutions whose continuation is feasible. Although this approach provides useful insights, in the presence of degeneracy, it does not allow us to determine if they constitute true solutions of our system. In order to overcome this obstacle, we follow a different route. By means of a Lyapunov–Schmidt decomposition, we are able to establish that the bifurcation equation for our models can be considered, in the small energy and small coupling regime, as a perturbation of a corresponding, beyond nearest-neighbor, discrete nonlinear Schrodinger equation. There, nonexistence results of degenerate phase-shift discrete solitons can be demonstrated by an additional Lyapunov–Schmidt decomposition, and translated to our original problem on the Klein–Gordon system. In this way, among other results, we can prove nonexistence of four-site vortex-like waveforms in the zigzag Klein–Gordon model. Finally, briefly considering a one-dimensional model bearing similarities to the square lattice, we conclude that the above strategy is not efficient for the proof of the existence or nonexistence of vortices due to the higher degeneracy of this configuration.

Dynamics of soliton collision phenomena on classical discrete Heisenberg weak ferromagnetic spin chain

We study the collision dynamics of soliton in discrete classical ferromagnetic spin chain with Dzyaloshinskii-Moriya (DM) interaction in the classical limit. The discrete spin dynamics are found to be governed by discrete nonlinear Schrödinger (DNLS) equation. One soliton and two soliton solutions of DNLS equation are obtained by employing the discrete Hirota bilinearization method. Through the graphical illustrations for the one soliton and two soliton solution, we discuss the effect of DM interaction in ferromagnetic spin chain. Inelastic collision phenomenon occurring in this system and both amplitudes of each soliton get suppressed or enhanced after collision. We also find the discreteness effect on the soliton excitation in ferromagnetic spin chain. The modulation in the soliton excitation induced by DM interaction and discreteness is also discussed briefly.

Variational Approximations for Intersite Soliton in a Ablowitz-Ladik-Cubic Discrete Nonlinear Schrödinger Equation

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Standing Waves for Discrete Nonlinear Schrödinger Equations with Nonperiodic Bounded Potentials

In this paper, we investigate standing waves in discrete nonlinear Schrodinger equations with nonperiodic bounded potentials. By using the critical point theory and the spectral theory of self-adjoint operators, we prove the existence and infinitely many sign-changing solutions of the equation. The results on the exponential decay of standing waves are also provided.

Ground State Solutions of Discrete Asymptotically Linear Schrödinger Equations with Bounded and Non-periodic Potentials

We study the existence of ground state solutions for a class of discrete nonlinear Schrodinger equations with a sign-changing potential V that converges at infinity and a nonlinear term being asymptotically linear at infinity. The resulting problem engages two major difficulties: one is that the associated functional is strongly indefinite and the other is that, due to the convergency of V at infinity, the classical methods such as periodic translation technique and compact inclusion method cannot be employed directly to deal with the lack of compactness of the Cerami sequence. New techniques are developed in this work to overcome these two major difficulties. This enables us to establish the existence of a ground state solution and derive a necessary and sufficient condition for a special case. To the best of our knowledge, this is the first attempt in the literature on the existence of a ground state solution for the strongly indefinite problem under no periodicity condition on the bounded potential and the nonlinear term being asymptotically linear at infinity. Moreover, our conditions can also be used to significantly improve the well-known results of the corresponding continuous nonlinear Schrodinger equation.

Riemann-Hilbert factorization of matrices invariant under inversion in a circle

We consider matrix functions with certain invariance under inversion in the unit circle. If such a function satisfies a positivity assumption on the unit circle, then only zero partial indices appear in its Riemann-Hilbert (Wiener-Hopf) factorization. It implies the unique solvability of a certain class of Riemann-Hilbert boundary value problems. It includes the ones associated with the inverse scattering transform of the focusing/defocusing integrable discrete nonlinear Schrodinger equations.

Strong Convergence for Discrete Nonlinear Schrödinger equations in the Continuum Limit

We consider discrete nonlinear Schrodinger equations (DNLS) on the lattice $h\Bbb{Z}^d$ whose linear part is determined by the discrete Laplacian which accounts only for nearest neighbor interactio...

Computational efficiency of numerical integration methods for the tangent dynamics of many-body Hamiltonian systems in one and two spatial dimensions

We investigate the computational performance of various numerical methods for the integration of the equations of motion and the variational equations for some typical classical many-body models of condensed matter physics: the Fermi-Pasta-Ulam-Tsingou (FPUT) chain and the one- and two-dimensional disordered, discrete nonlinear Schrodinger equations (DDNLS). In our analysis we consider methods based on Taylor series expansion, Runge-Kutta discretization and symplectic transformations. The latter have the ability to exactly preserve the symplectic structure of Hamiltonian systems, which results in keeping bounded the error of the system's computed total energy. We perform extensive numerical simulations for several initial conditions of the studied models and compare the numerical efficiency of the used integrators by testing their ability to accurately reproduce characteristics of the systems' dynamics and quantify their chaoticity through the computation of the maximum Lyapunov exponent. We also report the expressions of the implemented symplectic schemes and provide the explicit forms of the used differential operators. Among the tested numerical schemes the symplectic integrators $ABA864$ and $SRKN^a_{14}$ exhibit the best performance, respectively for moderate and high accuracy levels in the case of the FPUT chain, while for the DDNLS models $s9\mathcal{ABC}6$ and $s11\mathcal{ABC}6$ (moderate accuracy), along with $s17\mathcal{ABC}8$ and $s19\mathcal{ABC}8$ (high accuracy) proved to be the most efficient schemes.

Wannier functions and discrete NLS equations for nematicons$^\dagger$

We derive nonlocal discrete nonlinear Schrodinger (DNLS) equations for laser beam propagation in optical waveguide arrays that use a nematic liquid crystal substrate. We start with an NLS-elliptic model for the problem and propose a simplified version that incorporates periodicity in one of the directions transverse to the propagation of the beam. We use Wannier basis functions for an associated Schrodinger operator with periodic potential to derive discrete equations for Wannier modes and propose some possible simplified systems for interactions of modes within the first energy band of the periodic Schrodinger operator. In particular, we present the simplest generalization of a model proposed by Fratalocchi and Assanto by including a linear nonlocal term, and see evidence for parameter regimes where nonlinearity is more pronounced.

Existence and Stability of Traveling Waves for Discrete Nonlinear Schrödinger Equations Over Long Times

We consider the problem of existence and stability of solitary traveling waves for the one-dimensional discrete nonlinear Schrodinger equation with cubic nonlinearity, near the continuum limit. We ...

Supratransmission in discrete one-dimensional lattices with the cubic–quintic nonlinearity

We numerically analyzed the supratransmission phenomenon in the discrete nonlinear Schrodinger equation with the cubic–quintic nonlinearity. It has been reported that the homoclinic nonlinear band-gap threshold matches very well with the model. In the case of the cooperation between the nonlinearities (self-focusing cubic and quintic terms), the train of discrete band-gap waves overcomes the potential barrier of the first sites before merging or rebounding. In the case of competing self-focusing cubic and defocusing quintic nonlinearities, it is found that the lattice induces the generation of the train of dark solitons carried by a traveling kink and the traveling kink for chosen driving amplitude.