Nonlinear Schrodinger Type Equation Research Articles

Spectral and soliton structures of the Sasa–Satsuma higher-order nonlinear Schrödinger equation

In this paper, the Riemann–Hilbert problem of the Sasa–Satsuma higher-order nonlinear Schrodinger equation is investigated, from which spectral and soliton structures are discussed in detail. In addition, an algebra technique is developed to illustrate the soliton structures using Mathematica symbolic computations by choosing suitable parameters, including breather-bell soliton, double-hump soliton, and bell-to-breather soliton interactions. The results show that the spectral and soliton structures of the Sasa–Satsuma higher-order nonlinear Schrodinger equation are more complicated than many other nonlinear Schrodinger type equations.

Finite Dimensional Global Attractor for a Fractional Schrödinger Type Equation with Mixed Anisotropic Dispersion

We study the Cauchy problem for a class of nonlinear damped fractional Schrodinger type equation in a two dimensional unbounded domain. Then, we focus on long-time behaviour of the solutions proving that this behaviour is described by the existence of regular finite-dimensional global attractor in the energy space.

Asymptotic behaviour of the solutions for a weakly damped anisotropic sixth-order Schrödinger type equation in $ \mathbb{R}^2 $

<p style='text-indent:20px;'>We study the long-time behaviour of the solutions to a nonlinear damped anisotropic sixth-order Schrödinger type equation in <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{R}^2 $\end{document}</tex-math></inline-formula> that reads</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ u_t+i\Delta u-i \left(\partial_y^4 u-\partial_y^6 u\right)+ig(|u|^2)u+\gamma u = f\,,\;\;(t,(x,y))\in \mathbb{R}\times \mathbb{R}^2\,. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>We prove that this behaviour is described by the existence of regular global attractor in an anisotropic Sobolev space with finite fractal dimension.</p>

Local error of a splitting scheme for a nonlinear Schrödinger-type equation with random dispersion

We study a Lie splitting scheme for a nonlinear Schrodinger-type equation with random dispersion. The main result is an approximation of the local error. Then we can deduce sharp order estimates, for instance in the case of a white noise dispersion.

On the focusing generalized Hartree equation

In this paper we give a review of the recent progress on the focusing generalized Hartree equation, which is a nonlinear Schrodinger-type equation with the nonlocal nonlinearity, expressed as a convolution with the Riesz potential. We describe the local well-posedness in H1 and Hs settings, discuss the extension to the global existence and scattering, or finite time blow-up. We point out different techniques used to obtain the above results, and then show the numerical investigations of the stable blow-up in the L2 -critical setting. We finish by showing known analytical results about the stable blow-up dynamics in the L2 -critical setting.

Dust-acoustic envelope solitons in strongly coupled dusty plasmas

The propagation nature and the modulational instability (MI) criteria of low frequency amplitude modulated dust-acoustic envelope solitons are analyzed theoretically and numerically in an unmagnetized dusty plasma consisting of strongly coupled micron-sized inertial dust grains, Boltzmann distributed non-inertial electrons and ions. A hydrodynamical fluid model approach is used, based on [V. V. Yaroshenko, F. Verheest, H. M. Thomas and G.E. Morfill, New J. Phys. 11, 073013 (2009); V. V. Yaroshenko, V. Nosenko, and G. E. Morfill, Phys. Plasmas, 17, 103709 (2010)], where the strong coupling between dust grains for by an effective electrostatic pressure is accounted, which is in fact a dynamical function of the dust-density and the electrostatic potential. It is seen that the strong coupling between dust particles due to the effective electrostatic ``pressure” modifies the wavepacket stability and affects the occurrence of MI. A cubic nonlinear Schrodinger-type equation, which describes the evolution of the wavepacket’s envelope, is derived by adopting a multiscale technique for the amplitude dynamics. It is shown that long-wavelength dust-acoustic wavepackets are stable, while MI sets in beyond a carrier wavenumber threshold, which is evaluated in terms of intrinsic plasma parameters of relevance. On the other hand, the MI window is larger for higher ion density. The effect of the ion number density on MI is twofold, in that a higher ion density leads to a higher threshold (thus, the wave will be modulationally unstable for the larger values of the carrier wavenumber) but it also entails a wider instability window (hence, external perturbations must be of specific wavenumbers to trigger the instability). Numerical simulation confirms that the bright-type envelope soliton solution is stable beyond the carrier wavenumber threshold, i.e., in the modulationally unstable region, while the dark type envelope soliton is stable for wavenumbers below the threshold.

Modulation of the Nonlinear Ion Acoustic Waves in a Weakly Relativistic Warm Plasma with Nonextensively Distributed Electrons

A reductive perturbation technique (multiple scales) is applied to a weakly relativistic warm unmagnetized adiabatic plasma system consisting of inertial ions fluid and nonextensively distributed electrons. A nonlinear Schrodinger-type (NST) equation for finite wave number at the second order is derived. Using the reductive perturbation technique we derived the corresponding Korteweg-de Vries (K-dV) equation. For small wavenumber limit the K-dV equation is transformed into NST equation. It is found that the coefficient of the NST equation obtained from the K-dV equation agree with the corresponding coefficients of NST equation obtained by the multiple scales. Moreover we investigated the effect of the physical parameters of the system namely temperature ratio of the ion temperature T_i to electron temperature T_e, the relativistic factor u_0/C as well as the nonextensive parameter (q) of the distribution on the stability/instability of the system. It is found that these parameters affect strongly on the stability/instability regions. Finally, the validity of our results in astrophysical plasma is briefly discussed.

A parallelizable energy-preserving integrator MB4 and its application to quantum-mechanical wavepacket dynamics

In simulating physical systems, conservation of the total energy is often essential, especially when energy conversion between different forms of energy occurs frequently. Recently, a new fourth order energy-preserving integrator named MB4 was proposed based on the so-called continuous stage Runge–Kutta methods (Miyatake and Butcher in SIAM J Numer Anal 54(3):1993–2013, 2016). A salient feature of this method is that it is parallelizable, which makes its computational time for one time step comparable to that of second order methods. In this paper, we illustrate how to apply the MB4 method to a concrete ordinary differential equation using the nonlinear Schrodinger-type equation on a two-dimensional grid as an example. This system is a prototypical model of two-dimensional disordered organic material and is difficult to solve with standard methods like the classical Runge–Kutta methods due to the nonlinearity and the $$\delta $$ -function like potential coming from defects. Numerical tests show that the method can solve the equation stably and preserves the total energy to 16-digit accuracy throughout the simulation. It is also shown that parallelization of the method yields up to 2.8 times speedup using 3 computational nodes.

An efficient and highly accurate spectral method for modeling the propagation of solitary magnetic spin waves in thin films

In previous works, it was shown that the propagation of magnetic spin waves in thin films can be approximated by a nonlinear Schrodinger-type equation. The formulation begins with the magnetostatic equations (Gauss and Ampere’s laws of magnetism) and the Landau–Lifshitz equation. The solution of this system is a potential function whose dimensionless amplitude is the solution of a nonlinear Schrodinger. In the current work, we are demonstrating an efficient infinite series solution using the Christov functions. This is the first time the functions are used in problems involving complex arithmetics. The solutions of the time-independent and time-dependent problems are given in complex series form.

Riemann-Hilbert Approach and N-Soliton Solutions For Three-Component Coupled Hirota Equations

In this work, we consider an integrable three-component coupled Hirota (tcCH) equations in detail via the Riemann-Hilbert (RH) approach. We present some properties of the spectral problems of the tcCH equations with $4\times4$ the Lax pair. Moreover, the RH problem of the equations is established via analyzing the analyticity of the spectrum problem. By studying the symmetry of the spectral problem, we get the spatiotemporal evolution of scattering data. Finally, the $N$-soliton solution is derived by solving the RH problem with reflectionless case. According to the resulting $N$-soliton solution, the influences of each parameters on collision dynamic behaviors between solitons are discussed, and the method of how to control the interactions are suggested by some graphic analysis. In addition, some new phenomenon for soliton collision is presented including localized structures and dynamic behaviors of one- and two- soliton solutions, which can help enrich the nonlinear dynamics of the $N$-component nonlinear Schrodinger type equations.

Modulation of the Nonlinear Ion Acoustic Waves in a Weakly Relativistic Warm Plasma with Superthermally Distributed Electrons

A reductive perturbation technique (multiple scales) is applied to a weakly relativistic warm unmagnetized adiabatic plasma system consisting of inertial ions fluid and nonextensively distributed electrons. A nonlinear Schrodinger-type (NST) equation for finite wave number at the second order is derived. Using the reductive perturbation technique we derived the corresponding Korteweg-de Vries (K-dV) equation. For small wavenumber limit the K-dV equation is transformed into NST equation. It is found that the coefficient of the NST equation obtained from the K-dV equation agree with the corresponding coefficients of NST equation obtained by the multiple scales. Moreover we investigated the effect of the physical parameters of the system namely temperature ratio of the ion temperature T_i to electron temperature T_e, the relativistic factor u_0/C as well as the nonextensive parameter (q) of the distribution on the stability/instability of the system. It is found that these parameters affect strongly on the stability/instability regions. Finally, the validity of our results in astrophysical plasma is briefly discussed.

Vortex Solitons for a Class of Schrödinger Equation with Square Root Nonlinear Term

In this paper, we prove an existence theory for ring-profiled optical vortex solitons via constrained minimization, which are considered in the context of an electromagnetic light wave propagating in a nonlinear media and governed by a nonlinear Schrodinger type equation with square root nonlinear term.

Limiting cases of Sobolev inequalities on stratified groups

In this paper we present critical Gagliardo-Nirenberg, Trudinger-type and Brezis-Gallouet-Wainger inequalities concerning the limiting cases of the embedding theorems for Sobolev spaces on stratified groups. Moreover, using the critical Gagliardo-Nirenberg inequality the existence of least energy solutions of the nonlinear Schrodinger type equations can be obtained. We also express the best constant in the critical Gagliardo-Nirenberg inequality in the variational form as well as in terms of the ground state solutions of the corresponding nonlinear subelliptic equations.

Solitary wave and modulational instability in a strongly coupled semiclassical relativistic dusty pair plasma with density gradient

Using a set of fluid equations to describe the inertial dust grain component in a dense collision-less unmagnetised plasma under the influence of weakly relativistic semiclassical electrons and positrons, the propagation of dust-acoustic wave is studied in the strong coupling regime when the dust density is non-uniform. Our main aim is to analyse the role of semiclassical and relativistic environment (frequently encountered in astrophysical context) on various features of strongly coupled dusty plasma. The semiclassical environment of the electrons and positrons is assumed to be described by the Chandrasekhar equation of state. Our second aim is to see the effect of spatial variation of the dust equilibrium density, which is known to occur due to the deformation of the Debye sheath which in turn leads to polarisation force. A new type of nonlinear Schrodinger equation with spatially varying coefficient is deduced and its modulational stability is studied in detail. In the last section, we have taken recourse to Madelung picture to deduce a variable coefficient Korteweg–de Vries equation from this new nonlinear Schrodinger equation.

Regularizing Nonlinear Schrödinger Equations Through Partial Off-axis Variations

We study a class of focusing nonlinear Schrodinger-type equations derived recently by Dumas, Lannes, and Szeftel within the mathematical description of high intensity laser beams. These equations i...

Unconditional convergence of linearized implicit finite difference method for the 2D/3D Gross-Pitaevskii equation with angular momentum rotation

This paper is concerned with the time-step condition of linearized implicit finite difference method for solving the Gross-Pitaevskii equation with an angular momentum rotation term. Unlike the existing studies in the literature, where the cut-off function technique was used to establish the error estimates under some conditions of the time-step size, this paper introduces an induction argument and a ‘lifting’ technique as well as some useful inequalities to build the optimal maximum error estimate without any constraints on the time-step size. The analysis method can be directly extended to the general nonlinear Schrodinger-type equations in twoand three-dimensions and other linear implicit finite difference schemes. As a by-product, this paper defines a new type of energy functional of the grid functions by using a recursive relation to prove that the proposed scheme preserves well the total mass and energy in the discrete sense. Several numerical results are reported to verify the error estimates and conservation laws.

The Sylvester Equation and Integrable Equations: The Ablowitz—Kaup—Newell—Segur System

In this paper, we seek connections between the Sylvester equation and the Ablowitz—Kaup—Newell—Segur (AKNS) system. By the Sylvester equation KM — MK = r sT, we introduce master function S(i, j) = sT Kj (I + M)−1Kir. This function satisfies some recurrence relations. By imposing dispersion relations on r and s, we study the constructions of the AKNS system, where some AKNS type equations are investigated emphatically, including second-AKNS equation, second-modified AKNS (mAKNS) equation, third-AKNS equation, third-mAKNS equation and (—1)st-AKNS equation. The reductions of these equations to complex Korteweg—de Vries (KdV) equation, real and complex modified Korteweg—de Vries (mKdV) type equations, nonlinear Schrodinger (NLS) type equations and sine-Gordon (sG) equation are discussed.

Unconditional $$L^{\infty }$$ L ∞ -convergence of two compact conservative finite difference schemes for the nonlinear Schrödinger equation in multi-dimensions

This paper is concerned with the unconditional and optimal $$L^{\infty }$$ -error estimates of two fourth-order (in space) compact conservative finite difference time domain schemes for solving the nonlinear Schrodinger equation in two or three space dimensions. The fact of high space dimension and the approximation via compact finite difference discretization bring difficulties in the convergence analysis. The two proposed schemes preserve the total mass and energy in the discrete sense. To establish the optimal convergence results without any constraint on the time step, besides the standard energy method, the cut-off function technique as well as a ‘lifting’ technique are introduced. On the contrast, previous works in the literature often require certain restriction on the time step. The convergence rate of the proposed schemes are proved to be of $$O(h^4+\tau ^2)$$ with time step $$\tau $$ and mesh size h in the discrete $$L^{\infty }$$ -norm. The analysis method can be directly extended to other finite difference schemes for solving the nonlinear Schrodinger-type equations. Numerical results are reported to support our theoretical analysis, and investigate the effect of the nonlinear term and initial data on the blow-up solution.

Some new exact solutions of (2+1)-dimensional nonlinear Heisenberg ferromagnetic spin chain with the conformable time fractional derivative

There are various types of fractional derivatives in literature e.g. Caputo, Riemann–Liouville, conformable and so on. In the present paper, a new integrable nonlinear Schrodinger type equation in (2+1)-dimensions is investigated, which describes the spin dynamics of Heisenberg ferromagnetic spin chains. Some bright and dark solitons and other exact solutions of this equation with conformable time fractional derivative are reported. Then, fractionalize effects of this model are graphically investigated. Moreover, the simplest equation method and Nucci’s method are utilized for differential equations with conformable fractional derivatives.

Compression of Nanoslit Confined Polymer Solutions

Many systems of biophysical and technological interest consist of multiple interpenetrating chains in a confined volume, i.e., a confined polymer solution. Using nanofluidic approaches developed originally for the study of single chains in confined geometries, we develop an assay to create confined polymer solutions on-chip and then probe the solution response to applied compressive forcing. In our approach, multiple chains are introduced into a nanoslit via hydrodynamic flow and are then concentrated against a barrier that is permeable only to solvent. For sufficiently high concentration, the compressed solution profile can be described by a mean-field polymer model based on Doi’s two-fluid approach, with the chain free energy described by a Ginzburg-type free energy functional. This theory furnishes a partial differential equation based description of the concentration profile in terms of a nonlinear Schrodinger-type equation, providing a general theoretical framework for modeling confined polymer solut...