Discrete Nonlinear Schrodinger Equation Research Articles

Discrete and Semi-Discrete Multidimensional Solitons and Vortices: Established Results and Novel Findings.

This article presents a concise survey of basic discrete and semi-discrete nonlinear models, which produce two- and three-dimensional (2D and 3D) solitons, and a summary of the main theoretical and experimental results obtained for such solitons. The models are based on the discrete nonlinear Schrödinger (DNLS) equations and their generalizations, such as a system of discrete Gross-Pitaevskii (GP) equations with the Lee-Huang-Yang corrections, the 2D Salerno model (SM), DNLS equations with long-range dipole-dipole and quadrupole-quadrupole interactions, a system of coupled discrete equations for the second-harmonic generation with the quadratic (χ(2)) nonlinearity, a 2D DNLS equation with a superlattice modulation opening mini-gaps, a discretized NLS equation with rotation, a DNLS coupler and its PT-symmetric version, a system of DNLS equations for the spin-orbit-coupled (SOC) binary Bose-Einstein condensate, and others. The article presents a review of the basic species of multidimensional discrete modes, including fundamental (zero-vorticity) and vortex solitons, their bound states, gap solitons populating mini-gaps, symmetric and asymmetric solitons in the conservative and PT-symmetric couplers, cuspons in the 2D SM, discrete SOC solitons of the semi-vortex and mixed-mode types, 3D discrete skyrmions, and some others.

Global attractor and l^p solutions to initial value problems of discrete nonlinear Schrodinger equations complex potential

In this article, we investigate the global well-posedness of initial value problems of the time-dependent discrete nonlinear Schrodinger equation with a complex potential and sufficiently general nonlinearity on a multidimensional lattice in weighted \( l^p\) spaces for \( 1< p <\infty\). Thanks to our improved estimates we are able to prove the existence of global attractor for \( l^p\) solutions to the initial value problem.
 For more information see https://ejde.math.txstate.edu/Volumes/2024/12/abstr.html

Propagation of spin waves in a lattice of laterally and vertically coupled YIG microwaveguides by changing the magnetization angle in linear and nonlinear modes

Purpose. Investigation of the joint manifestation of the effects of anisotropic signal propagation, coupling, and nonlinear power dependence of the medium parameters in a lattice of laterally and vertically coupled spin-wave (SW) microwaveguides. Consideration of the case of the influence of the rotation of the magnetization angle and the change of the lateral gap between microwaveguides located on the same substrate on the transverse profile of the spin-wave beam and the spatial localization of the SW amplitude. Methods. The method of micromagnetic modeling based on the numerical solution of the Landau–Lifshitz–Hilbert equation shows the possibility of controlling the direction of propagation of SW in an ensemble of laterally and vertically coupled iron yttrium garnet (YIG) microwaveguides by changing the magnetization angle. By the method of numerical integration of the system of coupled discrete nonlinear Schrodinger equations, the possibility of changing the transverse profile of the spin-wave beam by changing the level of the initial signal amplitude is shown. Results. The spatial distributions of the components of the dynamic magnetization of the SW excited in two microwaveguides located on the same substrate obtained in micromagnetic simulations indicate a change in the character of localization of the SW power in the output sections of the microwaveguides. At variation of the lattice magnetization angle, a shift of the threshold power value is observed, at which a characteristic curbing of the transverse width of the spin-wave beam in the nonlinear mode appears. Conclusion. When excitation of surface magnetostatic SW in a lattice of laterally and vertically coupled microwaveguides, a transformation of the transverse profile of the wave is observed at a deviation of the magnetization angle of the structure by 15º , which is manifested in the change of the SW length and its localization in each of the microwaveguides. The combined effects of dipole coupling, gyrotropy, and nonlinearity of the medium make it possible to control the value of the threshold power of the SW, at which the mode of diffractionless propagation of the spin-wave beam is realized in a single layer of the structure.

2-D exotic soliton in 2-D Heisenberg ferromagnetic chains with Dzyaloshinsky-Moriya interaction

The two-dimensional (2D) version of ferromagnet XXZ spin chain with DzyaloshinskyMoriya (DM) interaction, recently introduced by F. Kenmogne and coworkers is reexplored. Firstly by using the Dyson-Maleev transformation, the 2-D discrete nonlinear Schrödinger (DNLS) equation, governing the quantum states behaviors is found. Next using the semidiscrete multiple-scale method, the 2-D DNLS equation is reduced to the 2D extended nonlinear Schrödinger (ENLS) equation which consists of the basic 2-D NLS equation with additional nonlinear dispersive terms. This equation admits the classical 2D pulse quantum states, when additional terms vanish. In addition, this equation admits the 2D compacton and 2D peakon-like boson quantum states. Furthermore, we notice that on the contrary to the classical outcomes where amplitudes of both solutions are free parameters, the amplitudes for two dimensional quantum states are not free parameters since the obtained solutions need to be normalized.

Numerical generation and investigation of rogue waves for discrete nonlinear Schrödinger equations

We have numerically generated the rogue waves supported by discrete nonlinear Schrödinger equation (DNLSE). The investigation has been made on pure cubic DNLSE as well as on cubic-quintic discrete nonlinear Schrödinger equation (CQDNLSE). Runge–Kutta fourth (RK4) order method has been employed to obtain these results. For the cubic DNLSE breathers are found to be localized in one of the dimensions and periodic in the other dimensions, whereas rogue waves are found to be appearing from nowhere and disappearing without a trace. For CQDNLSE it is reported that the sign of quintic term possesses a significant effect on the intensity of the rogue waves. Further these waves are appearing from nowhere but instead of disappearing these show periodic reappearance. For larger value of quintic coefficient this phenomenon is visible at shorter length scale. Breathers are reported for the first time for the cubic DNLSE. The obtained results for CQDNLSE are unobserved previously in any constant coefficient discrete nonlinear equation.

General rogue wave solution to the discrete nonlinear Schrödinger equation

In the present paper, we are concerned with the rogue wave solution to an integrable discrete nonlinear Schrödinger (NLS) equation. First, through the KP–Toda reduction method, the general breather solution of the discrete NLS equation is deduced from two bilinear equations in discrete two-dimensional Toda-lattice hierarchy. Then, by taking an ingenious and successive limit to the breather solution, we derive the higher order rogue wave solution to the discrete NLS equation.

Analisis Kestabilan Solusi Soliton pada Persamaan Schrodinger Nonlinier Diskrit Nonlokal

In this paper, the Nonlocal Discrete Nonlinear Schrodinger (DNLS) equation that interpolates the Nonlocal Ablowitz-Ladik DNLS and the Nonlocal Cubic DNLS equations and its stability are studied in detail. The solution of the Nonlocal SNLD equation is a soliton wave in the form of a Gaussian ansatz obtained using the method of Variational Approximation (VA). The stability of the solution is also analyzed using the VA. These semi-analytical results are then compared to numerical results. The soliton and its stability obtained via VA is concluded to be having a fairly good conformity with numerical results.

Frozen dynamics of a breather induced by an adiabatic invariant

The discrete nonlinear Schrödinger (DNLS) equation is a Hamiltonian model displaying an extremely slow relaxation process when discrete breathers appear in the system. In (Iubini et al 2019 Phys. Rev. Lett. 122 084102), it was conjectured that the frozen dynamics of tall breathers is due to the existence of an adiabatic invariant (AI). Here, we prove the conjecture in the simplified context of a unidirectional DNLS equation, where the breather is ‘forced’ by a background unaffected by the breather itself. We first clarify that the nonlinearity of the breather dynamics and the deterministic nature of the forcing term are both necessary ingredients for the existence of a frozen dynamics. We then derive perturbative expressions of the AI by implementing a canonical perturbation theory and via a more phenomenological approach based on the estimate of the energy flux. The resulting accurate identification of the AI allows revealing the presence and role of sudden jumps as the main breather destabilization mechanism, with an unexpected similarity with Lévy processes.

Localization in the Discrete Non-linear Schrödinger Equation and Geometric Properties of the Microcanonical Surface

It is well known that, if the initial conditions have sufficiently high energy density, the dynamics of the classical Discrete Non-Linear Schr\"odinger Equation (DNLSE) on a lattice shows a form of breaking of ergodicity, with a finite fraction of the total charge accumulating on a few sites and residing there for times that diverge quickly in the thermodynamic limit. In this paper we show that this kind of localization can be attributed to some geometric properties of the microcanonical potential energy surface, and that it can be associated to a phase transition in the lowest eigenvalue of the Laplacian on said surface. We also show that the approximation of considering the phase space motion on the potential energy surface only, with effective decoupling of the potential and kinetic partition functions, is justified in the large connectivity limit, or fully connected model. In this model we further observe a synchronization transition, with a synchronized phase at low temperatures.

Homoclinic Orbits and Localized Solutions in Discrete Nonlinear Schrodinger Equation with Long-Range Interaction

In this paper, we use the homoclinic orbit approach without using small perturbations to prove the existence of soliton solutions of the discrete nonlinear Schrodinger equations with long-range interaction by employing the properties of the symmetries of reversible planar maps. Moreover, the long-range interaction by a potential proportional to $1/l^{1+alpha} $ with fractional $alpha < 1 $ and $l $ as natural number.

From a generalized discrete NLS equation in discrete alpha helical proteins to the fourth-order NLS equation

In this paper, we consider a generalized integrable discrete nonlinear Schrödinger (NLS) equation, which can describe the dynamics of discrete alpha helical proteins with higher-order excitations and lead to the higher-order NLS equation in the continuum limit. The Darboux transformation (DT) and the soliton solutions of this generalized discrete NLS equation are implemented. It is shown that the integrable properties of the generalized discrete NLS equation, including the discrete Lax pair, the DT and the discrete soliton solutions, give rise to their continuous counterparts as the discrete space step tends to zero.

Multi-rogue wave solutions for a generalized integrable discrete nonlinear Schrödinger equation with higher-order excitations

In this paper, we construct the discrete higher-order rogue wave (RW) solutions for a generalized integrable discrete nonlinear Schrödinger (NLS) equation. First, based on the modified Lax pair, the discrete version of generalized Darboux transformation is constructed. Second, the dynamical behaviors of first-, second- and third-order RW solutions are investigated in corresponding to the unique spectral parameter, higher-order term coefficient, and free constants. The differences between the RW solution of the higher-order discrete NLS equation and that of the Ablowitz–Ladik (AL) equation are illustrated in figures. Moreover, we explore the numerical experiments, which demonstrates that strong-interaction RWs are stabler than the weak-interaction RWs. Finally, the modulation instability of continuous waves is studied.

Nonlinear non-Hermitian skin effect

Distant boundaries in linear non-Hermitian lattices can dramatically change energy eigenvalues and corresponding eigenstates in a nonlocal way. This effect is known as non-Hermitian skin effect (NHSE). Combining non-Hermitian skin effect with nonlinear effects can give rise to a host of novel phenomenas, which may be used for nonlinear structure designs. Here we study nonlinear non-Hermitian skin effect and explore nonlocal and substantial effects of edges on stationary nonlinear solutions. We show that fractal and continuum bands arise in a long lattice governed by a nonreciprocal discrete nonlinear Schrodinger equation. We show that stationary solutions are localized at the edge in the continuum band. We consider a non-Hermitian Ablowitz-Ladik model and show that nonlinear exceptional point disappears if the lattice is infinitely long.

Multiple homoclinic solutions for discrete Schrödinger equations with perturbed and sublinear terms

A class of perturbed discrete nonlinear Schrodinger equations with sublinear terms is studied. The new features are: (1) negative and positive energy homoclinic solutions are obtained; (2) the continuity of solutions in the perturbed terms at zero is studied; (3) the positive-definite condition can be removed.

Modulational instabilities and chaotic-like behaviors in repulsive lattices

In this paper, we study the dynamics of granular mediums. By comparing each grain of the medium with a ball, and the interaction between various balls like linear elastic interactions, we have established that the dynamics of the system of pendulum used by Fermi–Pasta–Ulam can be governed by a discrete nonlinear Schrodinger equation. Like second advanced, we also established under which conditions this system could be prone to a modulational instability. The long-time dynamics of modulated waves is examined using numerical simulations which are corroborated with analytical results leading to the generation of nonlinear modulated waves which have the shape of a soliton. At the end, we compare our results with what have been carried out previously.

Kink and Anti-Kink Soliton for Stationary Cubic-Quintic Discrete Nonlinear Schrodinger Equation

Kink and Anti-Kink Soliton for Stationary Cubic-Quintic Discrete Nonlinear Schrodinger Equation

Modulational instability induced generation of solitary wave profile of an anisotropic-ferromagnetic nanowire with asymmetric Dzyaloshinskii-Moriya interaction

We study the nanoscale magnetization evolution of a one dimensional ferromagnetic nanowire with the inclusion of next-nearest-neighbor interaction and Dzyaloshinkii-Moriya (DM) interaction. The system is modelled using Heisenberg Hamiltonian in the quasi classical approximation of spin operators. The dynamics of the system is found to be governed by generalized discrete nonlinear Schrodinger equation. Further we investigate analytically the modulational instability in ferromagnetic nanowire system, where the effect of next-nearest neighbour (NNN) interaction and DM interaction are taken into an account to show graphically the effect of interactions on Modulational Instability (MI) gain spectra.

Soliton Excitation in ferromagnetic Nanotubes with Higher Order Single Ion Anisotropy

Dynamics of soliton in classical discrete Heisenberg ferromagnetic nanotubes (FN) with the effect of higher order single ion anisotropy was studied. The Hirota Bilinearization (HB) method was applied to the discrete nonlinear Schrodinger (DNLS) equation and obtains the soliton solution for FN system. The effects of exchange interaction and single ion anisotropy on soliton evolution along FN system were discussed through graphically.

Plane wave solution of extended discrete nonlinear Schrödinger equation

In this paper, we considered the extended discrete nonlinear Schrodinger equation (EDNLSE) which includes the nearest neighbour nonlinear interaction in addition to the on-site cubic and quintic nonlinearities. The objective of this study is to investigate the modulational instability of plane matter-wave solution in dipolar Bose-Einstein Condensates (BEC) in a periodic optical lattice and to compare the analytical results with numerical. Analytically, the problem is solved by using perturbed solution of the plane wave where the instability of the gain can be obtained. The conditions of the stability of the plane wave had been analysed and confirmed numerically, by applications of Runge-Kutta method. Three specific cases were studied where only cubic-quintic nonlinearity is considered, only quintic-dipolar ( ) is considered and lastly non-zero for all terms. The numerical results are aligned with the analytical results.

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.