Nonlinear Schrodinger Equation Research Articles

Nonlinear Schrödinger waves in a disordered potential: Branched flow, spectrum diffusion, and rogue waves.

The problem of nonlinear Schrödinger (NLS) waves in a disordered potential arises in many physical occasions, such as hydrodynamics, optics, and cold atoms. It provides a paradigm for studying the interaction between nonlinearity and random effect, but the current results are far from perfect. In this paper, we systematically simulate the turbulent waves for the focusing NLS equation with dynamical (time-dependent) random potentials, where the enhanced branching structures evolve into branched soliton flows as the nonlinearity increases. In this process, the occurrence of rogue waves for short times results from the interplay of linear random focusing and modulation instability. While the nonlinear spectral analysis reveals that for longer times, it is due to a self-organization of larger solitons competing with breakup of intermediate solitons. On the other hand, we found that the strong nonlinearity can significantly increase the width of the linear (Fourier) spectrum for several time scales, but its spreading rate becomes suppressed, which has a dependence on the correlation length of the potential. We hope that our findings will facilitate a deeper understanding of the nonlinear waves interacting with disordered media.

Rational W-shape solitons on a nonlinear electrical transmission line with Josephson junction

This paper examines envelope solitons, rational W-shape solitons to a Nonlinear Electrical Transmission Line (NETL) with Josephson junction (JJ) that consist of the Nth cells of circuits. By employing the reductive perturbation approach in the semi-discrete approximation, we obtain the nonlinear Schrödinger equation (NLSE). From this equation, the frequency ranges of propagation of bright and dark solitons were obtained. As in most of the works, the NLSE obtained is well known as the seat of rogue waves and Peregrine solitons. The nonlinearity provided by the JJ, combined with the dispersion, made it possible to study the Modulation Instability (MI) gain spectrum. The 3D and 2D graphical representations illustrated the instability zones and the W-rational solutions have followed by using numerical simulation. The results obtained in this paper are of a very capital contribution for the study of rogue waves in nonlinear transmission lines or other physical problems.

Some solutions for a stochastic NLSE in the unstable and higher order dispersive environments

The nonlinear Schrödinger equations (NLSEs) arise from a large variety of fields, such as quantum mechanics, optical fiber communication, plasma physics, superfluid and many others. In this article, we consider the stochastic unstable nonlinear Schrödinger equation (UNLSE) and higher order dispersive nonlinear Schrödinger equation (HODNLSE). Namely, we consider these equations by a multiplicative noise in Itô sense. We applied the efficient solver in order to get the vital solutions. The forms of presented solutions are rational; trigonometric; hyperbolic functions. Some of acquired stochastic solutions are new and practical. The presented solutions perform an important factor for understanding the qualitative interpretation in various physical environments, such as optical fibers, plasma fluids and new physics. With the help of Matlab software, some graphs were given to depict the behavior of the gained solutions.

Characteristic behaviour of N-order ion acoustic rogue waves solution in electron-positron-ion plasmas

The characteristic behaviour of breathers as well as N-order Rogue wave (RW) solution in homogeneous unmagnetized electron-positron-ion plasmas (consisting of cold ions, and isothermal Maxwellian electrons and positrons) has been studied theoretically. Nonlinear Schrödinger (NLS) equation is derived employing Krylov-Bogoliubov-Mitropolosky (KBM) perturbation technique in solving the concerned problems. The analytical solution of the NLS equation in the forms of breathers such as Akhmediev breather (AB), Kuznetsov-Ma breather (KM) solitons, as well as first-, second-, third-, and fourth-order RWs solution are obtained. The characteristic behaviour of breathers’ and their dependency on the relevant physical parameters are examined. The different orders of RWs are analyzed to obtain the region of existence of ion acoustic (IA) rogue waves (IARWs) with respect to unstable region. The potential profiles concerning different orders of RWs depend on the ratio of positron to electron concentrations () and wave number (). Analysis reveals that the spatial difference between the two adjacent peaks of AB profiles are increasing with the enhancement of transformer physical parameter (), while the temporal evolution between the two neighboring peaks of KM profiles decreases with increasing and the amplitudes become large as well. The amplitudes of first-, second-, and third-order RWs profiles are decreasing with increasing while the amplitude of fourth-order RWs profile is increasing. The results obtained in this study may be useful in understanding the propagation of nonlinear waves in optical fibers, Bose-Einstein condensates, plasma physics and atmospheric physics.

Weak Langmuir turbulence in disordered multimode optical fibers

We consider the propagation of temporally incoherent waves in multimode optical fibers (MMFs) in the framework of the multimode nonlinear Schr\"odinger (NLS) equation accounting for the impact of the natural structural disorder that affects light propagation in standard MMFs (random mode coupling and polarization fluctuations). By averaging the dynamics over the fast disordered fluctuations, we derive a Manakov equation from the multimode NLS equation, which reveals that the Raman effect introduces a previously unrecognized nonlinear coupling among the modes. Applying the wave turbulence theory on the Manakov equation, we derive a very simple scalar kinetic equation describing the evolution of the multimode incoherent waves. The structure of the kinetic equation is analogous to that developed in plasma physics to describe weak Langmuir turbulence. The extreme simplicity of the derived kinetic equation provides physical insight into the multimode incoherent wave dynamics. It reveals the existence of different collective behaviors where all modes self-consistently form a multimode spectral incoherent soliton state. Such an incoherent soliton can exhibit a discrete behavior characterized by collective synchronized spectral oscillations in frequency space. The theory is validated by accurate numerical simulations: The simulations of the generalized multimode NLS equation are found in quantitative agreement with those of the derived scalar kinetic equation without using adjustable parameters.

The Focusing NLS Equation with Step-Like Oscillating Background: The Genus 3 Sector

We consider the Cauchy problem for the focusing nonlinear Schrödinger equation with initial data approaching different plane waves $$A_j\mathrm {e}^{\mathrm {i}\phi _j}\mathrm {e}^{-2\mathrm {i}B_jx}$$ , $$j=1,2$$ as $$x\rightarrow \pm \infty $$ . The goal is to determine the long-time asymptotics of the solution, according to the value of $$\xi =x/t$$ . The general situation is analyzed in a recent paper where we develop the Riemann–Hilbert approach and detect different asymptotic scenarios, depending on the relationships between the parameters $$A_1$$ , $$A_2$$ , $$B_1$$ , and $$B_2$$ . In particular, in the shock case $$B_1<B_2$$ , some scenarios include genus 3 sectors, i.e., ranges of values of $$\xi $$ where the leading term of the asymptotics is given in terms of hyperelliptic functions attached to a Riemann surface $$M(\xi )$$ of genus three. The present paper is devoted to the complete asymptotic analysis in such a sector.

Ground States for the NLS Equation with Combined Nonlinearities on Noncompact Metric Graphs

We investigate the existence of ground states with prescribed mass for the NLS energy with combined $L^2$-critical and subcritical nonlinearities, on a general non-compact metric graph $\mathcal{G}$. The interplay between the different nonlinearities creates new phenomena with respect to purely critical or subcritical problems on graphs; from a different perspective, topological and metric properties of the underlying graph drastically influence existence and non-existence of ground states with respect to the analogue problem on the real line.

Nonexistence of solutions of higher-order nonlinear non-Gauge Schrödinger equation

A nonexistence result is proved of the space higher-order nonlinear Schrodinger equation where m &gt; 1, n &gt; 1 and p &gt; n. Our method of proof rests on a judicious choice of the test function in the weak formulation of the equation. Then, we obtain an upper bound of the life span of solutions. Furthermore, the necessary conditions for the existence of local or global solutions are provided.Next, we extend our results to the 2 × 2 – system.

Lagrangian nonlocal nonlinear Schrödinger equations

In 2013 Ablowitz and Musslimani (AM) obtained a nonlocal generalization of the NLS equation which is integrable, but does not have a standard Lagrangian structure. The present paper shows that there exist two new nonlocal NLS equations, similar to the AM model, which do possess Lagrangian structures. We show that these two models (LN1 and LN2) possess solitary wave solutions which remain trapped in the neighborhood of the origin (x=0), and solitary waves which are able to escape from the origin. These two types of solutions are obtained by direct numerical solutions, and also by a variational method. In the case of LN2 model, the variational approach explains the existence of these two types of solutions. Collisions of breathers which obey the equation LN2 are numerically studied, and the results show that these breathers are robust solutions.

Role of higher order dispersion on instability criterion of saturable fiber system with Non-Kerr Nonlinearities

We analyze the modulational instability (MI) of a continuous wave (CW) in saturable fiber systems, including higher-order dispersions, self steepening, and self-frequency shift, cubic, quintic, and septic nonlinearities. We show that whenever the combined effect of higher-order dispersion and non-Kerr nonlinearities with suitable systems meets a wide variety of MI behaviors by mixing different combinations are presented. By modifying the nonlinear Schrödinger equation (NLSE) for MI analysis using Maxwell's theory, the new solution is derived for linear stability analysis to obtain an expression for the MI gain. Among different combinations of higher-order dispersion and non-Kerr nonlinearities, the critical behavior of MI gain suppressed under saturable systems, which leads to affect the nature of soliton stability. Under the activation of self steepening and self-frequency shift in the saturable system, the generated MI behavior regained its original to retain the soliton stability. This kind of new and tuning of MI behavior will be a useful instrument in nonlinear optic communications with straightforward applications.

3D-Modulational Stability of Envelope Soliton in a Quantum Electron–Ion Plasma—A Generalised Nonlinear Schrödinger Equation

In physical reality, the phenomena of plasma physics is actually a three-dimensional one. On the other hand, a vast majority of theoretical studies only analyze a one-dimensional prototype of the situation. So, in this communication, we tried to treat the quantum electron–ion plasma in a full 3D setup and the modulational stability of envelope soliton was studied in a quantum electron–ion plasma in three dimensions. The Krylov–Bogoliubov–Mitropolsky method was applied to the three-dimensional plasma governing equations. A generalized form of the nonlinear Schrödinger (NLS) equation was obtained, whose dispersive term had a tensorial character, which resulted in the anisotropic behavior of the wave propagation even in absence of a magnetic field. The stability condition was deduced ab initio and the stability zones were plotted as a function of plasma parameters. The modulational stability of such a three-dimensional NLS equation was then studied as a function of plasma parameters. It is interesting to note that the nonlinear excitation of soliton took place again here due to the balance of nonlinearity and dispersion. The zones of contour plots are given in detail.

Nonlinear physical complex hirota dynamical system: Construction of chirp free optical dromions and numerical wave solutions

The Hirota dynamical system is a modified nonlinear Schrödinger equation (NLSE). It has time-delay corrections and higher-order dispersion to the cubic nonlinearity. It describe the propagation of wave in the optical fibers and ocean; it can be viewed as an approximation which is more accurate than the NLSE. We investigate the nonlinear generalized higher-order Hirota equation, for certain ultrashort optical pulses propagating in a nonlinear inhomogeneous fiber. By implementing variational principle and computational techniques, we obtained chirp optical and numerical wave solutions. Furthermore, the existence, uniqueness and stability are studied for this model.

Logarithmic and hyperbolic spirals associated with Schrodinger's equation

We investigate the equationQ″+N−1ζQ′−Q+ia(ζQ)′=0,0<ζ<∞, where N≥2 is spatial dimension, and a>0 is a parameter. This equation plays an important role in analyzing a nonlinear Schrodinger equation self-focusing model. Setting N=3, Q=ρeiθ, we getρ″+2ζρ′=ρ((θ′)2+aζθ′+1),ρθ″+2ζρθ′+2ρ′θ′=−a(ζρ)′ We analyze three diverse solutions of this system. The first two, Q1=ρ1eiθ1 and Q2=ρ2eiθ2 exist ∀ζ>0, and become singular as ζ→0+. They rotate around (0,0) in the (Re(Q),Im(Q)) plane, with θ1′(ζ)<0 and θ2′(ζ)<0∀ζ>0, and form logarithmic and hyperbolic spirals when ζ≫1. The third solution, Q3=ρ3eiθ3 is different. First, 0<limζ→0+⁡ρ3(ζ)<∞. Second, θ3′(ζ) changes sign infinitely often on 0<ζ<∞, causing the formation of “internal oscillations” in the trajectory of the solution.

Multi-elliptic rogue wave clusters of the nonlinear Schrödinger equation on different backgrounds

In this work, we analyze the multi-elliptic rogue wave clusters of the nonlinear Schrödinger equation (NLSE) in order to understand more thoroughly the origin and appearance of optical rogue waves in this system. Such structures are obtained on uniform backgrounds by using the Darboux transformation scheme for finding analytical solutions of the NLSE under various conditions. In particular, we solve the eigenvalue problem of the Lax pair of order n in which the first m evolution shifts are equal, nonzero, and eigenvalue dependent, while the imaginary parts of all eigenvalues tend to one. We show that an Akhmediev breather of order \(n-2m\) appears at the origin of the (x, t) plane and can be considered as the central rogue wave of the so-formed cluster. We show that the high-intensity narrow peak, with the characteristic intensity distribution in its vicinity, is enclosed by m ellipses consisting of the first-order Akhmediev breathers. The number of maxima on each ellipse is determined by its index and the solution order. Since rogue waves in nature usually appear on a wavy background, we utilize the modified Darboux transformation scheme to build such solutions on a Jacobi elliptic dnoidal background. We analyze the vertical semi-axis of all ellipses in a cluster as a function of an absolute evolution shift. We show that the cluster radial symmetry in the (x, t) plane is broken when the shift value is increased above a threshold. We apply the same analysis on the Hirota equation, to examine the influence of a real parameter and Hirota’s operator on the cluster appearance. The same analysis can be applied to the infinite hierarchy of extended NLSEs. The main outcomes of this paper are the new multi-rogue wave solutions of the nonlinear Schrödinger equation and its extended family on uniform and elliptic backgrounds.

Kink soliton behavior study for systems with power-law nonlinearity

In this study, we investigate the kink soliton dynamics for power-law nonlinear systems. Based on the F-expansion method, we first derive the novel kink soliton solution of the nonlinear Schrödinger equation (NLSE) with third-order dispersion term, power-law dependent nonlinearity term, linear attenuation term, and self-steepness term under appropriate parameter settings. With pictorial demonstration, we show that the obtained kink soliton solution not only has the soliton features of the classical NLSE, but also has power-law features. The theoretical results presented in our work can be used to guide the observation of soliton behavior in power-law dependent media.

Logarithmic estimates for mean-field models in dimension two and the Schrödinger–Poisson system

In dimension two, we investigate a free energy and the ground state energy of the Schrodinger-Poisson system coupled with a logarithmic nonlinearity in terms of underlying functional inequalities which take into account the scaling invariances of the problem. Such a system can be considered as a nonlinear Schrodinger equation with a cubic but nonlocal Poisson nonlinearity, and a local logarithmic nonlinearity. Both cases of repulsive and attractive forces are considered. We also assume that there is an external potential with minimal growth at infinity, which turns out to have a logarithmic growth. Our estimates rely on new logarithmic interpolation inequalities which combine logarithmic Hardy-Littlewood-Sobolev and logarithmic Sobolev inequalities. The two-dimensional model appears as a limit case of more classical problems in higher dimensions.

Hamiltonian form of an Extended Nonlinear Schrödinger Equation for Modelling the Wave field in a System with Quadratic and Cubic Nonlinearities

We derive a Hamiltonian form of the fourth-order (extended) nonlinear Schrödinger equation (NLSE) in a nonlinear Klein–Gordon model with quadratic and cubic nonlinearities. This equation describes the propagation of the envelope of slowly modulated wave packets approximated by a superposition of the fundamental, second, and zeroth harmonics. Although extended NLSEs are not generally Hamiltonian PDEs, the equation derived here is a Hamiltonian PDE that preserves the Hamiltonian structure of the original nonlinear Klein–Gordon equation. This could be achieved by expressing the fundamental harmonic and its first derivative in symplectic form, with the second and zeroth harmonics calculated from the variational principle. We demonstrate that the non-Hamiltonian form of the extended NLSE under discussion can be retrieved by a simple transformation of variables.

Homotopy Analysis Method and Time-fractional NLSE with Double Cosine, Morse, and New Hyperbolic Potential Traps

A brief outline of the derivation of the time-fractional nonlinear Schrödinger equation (NLSE) is furnished. The homotopy analysis method (HAM) is applied to study time-fractional NLSE with three separate trapping potential models that we believe have not been investigated yet. The first potential is a double cosine potential $[V(x)=V_{1}\cos x+V_{2}\cos 2x]$, the second one is the Morse potential $[V(x)=V_{1}e^{-2\beta x}+V_{2}e^{-\beta x}]$, and a hyperbolic potential $[V(x)=V_{0}\tanh(x)\sech(x)]$ is taken as the third model. The fractional derivatives and integrals are described in the Caputo and Riemann Liouville sense, respectively. The solutions are given in the form of convergent series with easily computable components. A physical analysis with graphical representations explicitly reveals that HAM is effective and convenient for solving nonlinear differential equations of fractional order.

Integrability, Variational Principle, Bifurcation, and New Wave Solutions for the Ivancevic Option Pricing Model

The Ivancevic option pricing model comes as an alternative to the Black‐Scholes model and depicts a controlled Brownian motion associated with the nonlinear Schrodinger equation. The applicability and practicality of this model have been studied by many researchers, but the analytical approach has been virtually absent from the literature. This study intends to examine some dynamic features of this model. By using the well‐known ARS algorithm, it is demonstrated that this model is not integrable in the Painlevé sense. He’s variational method is utilized to create new abundant solutions, which contain the bright soliton, bright‐like soliton, kinky‐bright soliton, and periodic solution. The bifurcation theory is applied to investigate the phase portrait and to study some dynamical behavior of this model. Furthermore, we introduce a classification of the wave solutions into periodic, super periodic, kink, and solitary solutions according to the type of the phase plane orbits. Some 3D‐graphical representations of some of the obtained solutions are displayed. The influence of the model’s parameters on the obtained wave solutions is discussed and clarified graphically.

Grey solitons in the ultracold fermions at the full spin polarization

A minimal coupling quantum hydrodynamic model of spin-1/2 fermions at the full spin polarization corresponding to a nonlinear Schrodinger equation is considered. The nonlinearity is primarily caused by the Fermi pressure. It provides an effective repulsion between fermions. However, there is the additional contribution of the short-range interaction appearing in the third order by the interaction radius. It leads to the modification of the pressure contribution. Solitons are considered for the infinite medium with no restriction on the amplitude of the wave. The Fermi pressure leads to the soliton in form of the area of decreased concentration. However, the center of solution corresponding to the area of minimal concentration has nonzero value of concentration. Therefore, the grey soliton is found. Soliton exist if the speed of its propagation is below the Fermi velocity. Keywords: degenerate fermions, hydrodynamics, non-linear Schrodinger equation, dark soliton, Fermi pressure.