arXiv subjects

U. Al Khawaja

Publications and source records attributed to U. Al Khawaja.

At least 37 records · Page 2Linked to original sources

arXiv2020

Singular Soliton Molecules of the Nonlinear Schrodinger Equation

We derive an exact solution to the local nonlinear Schrödinger equation (NLSE) using the Darboux transformation method. The new solution describes the profile and dynamics of a two-soliton molecule. Using an algebraically-decaying seed solution, we obtain a two-soliton solution with diverging peaks, which we denote as singular soliton molecule. We find that the new solution has a finite binding energy. We calculate the force and potential of interaction between the two solitons, which turn out to be of molecular-type. The robustness of the bond between the two solitons is also verified. Furthermore, we obtain a new solution to the nonlocal NLSE using the same method and seed solution. The new solution in this case corresponds to an elastic collision of a soliton, a breather soliton on flat background, and a breather soliton on a background with linear ramp. Finally, we consider an NLSE which is nonlocal in time rather than space. Although we did not find a Lax pair to this equation, we derive three exact solutions.

nlin.PS↗

arXiv2019

Interwires polar soliton molecules in a biwire system

We investigate stability the dynamical properties of one-dimensional interwires polar soliton molecules in a biwire setup with dipole moments aligned perpendicularly to the line of motion and in opposite directions in different wires. Numercial results based on a nonlocal nonlinear Schrödinger equation reveal the existence of a bound state leading to the formation of a molecule. We show that the degree of the nonlocality and the interwire separation play an important role in stabilizing the molecules. Dynamics of the width and center-of-mass of a solitons is also analyzed in terms of the system parameters.

nlin.PS↗

arXiv2018

Weakly bound solitons and two-soliton molecules in dipolar Bose-Einstein condensates

Analytic expressions have been derived for the interaction potential between dipolar bright solitons and the binding energy of a two-soliton molecule. The properties of these localized structures are explored with a focus on their behavior in the weakly bound regime, with a small binding energy. Using the variational approach a coupled system of ordinary differential equations for the parameters of a soliton molecule is obtained for the description of their evolution. Predictions of the model are compared with numerical simulations of the governing nonlocal Gross-Pitaevskii equation and good qualitative agreement between them is demonstrated.

cond-mat.quant-gas↗

arXiv2018

Nonlinear conformal-degree-preserving Dirac equations in 2+1 space-time

We propose nonlinear Dirac equations where the conformal degree of the self-interaction terms are equal to that of the Dirac operator and the coupling parameters are dimensionless. As such, the massless equation is conformally invariant and preserves the conformal degree for both the linear and nonlinear components. In 2+1 space-time, we use these features to propose three- and four-parameter nonlinear Dirac equation models that might be useful for applications in 2D systems such as graphene sheets, ribbons and nanotubes.

quant-ph↗

arXiv2018

New classes of solutions in the Coupled PT Symmetric Nonlocal Nonlinear Schrodinger Equations with Four Wave Mixing

We investigate generalized nonlocal coupled nonlinear Schroedinger equation containing Self-Phase Modulation, Cross-Phase Modulation and Four-Wave Mixing involving nonlocal interaction. By means of Darboux transformation, we obtained a family of exact breathers and solitons including the Peregrine soliton, Kuznetsov-Ma breather, Akhmediev breather along with all kinds of soliton-soliton and breather-soliton interactions. We analyze and emphasize the impact of the four-wave mixing on the nature and interaction of the solutions. We found that the presence of Four-Wave Mixing converts a two-soliton solution into an Akhmediev breather. In particular, the inclusion of Four-Wave Mixing results in the generation of a new solution which is spatially and temporally periodic called "Soliton (Breather) lattice".

nlin.PS↗

arXiv2018

Enhanced mobility of discrete solitons in anisotropic two-dimensional waveguide arrays with modulated separations

We consider two-dimensional waveguide arrays with anisotropic coupling coefficients. We show using numerical and variational calculations that four stationary soliton types exist: Site-Centered, Bond-Centered, Hybrid-X and Hybrid-Y. For the isotropic case the last two modes become identical and equivalent to the known hybrid soliton. With a variational calculation using a gaussian trial function and six variational parameters corresponding to the soliton's position, width, and velocity components, the four stationary soliton types are reproduced and their equilibrium widths are accounted for accurately for a wide range of anisotropy ratios. We obtained using the variational calculation the Peierls-Nabarro potential and barrier heights for the four soliton types and different anisotropy ratios. We have also obtained a phase diagram showing regions of soliton stability against collapse and subregions of mobility in terms of the initial kick-in speed and anisotropy ratio. The phase diagram shows that 2D solitons become highly mobile for anisotropy ratios larger than some critical values that depend on the initial kick-in speed. This fact was then exploited to design tracks within the 2D waveguide array along which the soliton can be accelerated and routed. We have calculated the actual waveguide separations needed to realist the proposed guided trajectories of 2D solitons.

nlin.PS↗

arXiv2018

Expansion of fractional derivatives in terms of an integer derivative series: physical and numerical applications

We use the displacement operator to derive an infinite series of integer order derivatives for the Grünwald-Letnikov fractional derivative and show its correspondence to the Riemann-Liouville and Caputo fractional derivatives. We demonstrate that all three definitions of a fractional derivative lead to the same infinite series of integer order derivatives. We find that functions normally represented by Taylor series with a finite radius of convergence have a corresponding integer derivative expansion with an infinite radius of convergence. Specifically, we demonstrate robust convergence of the integer derivative series for the hyperbolic secant (tangent) function, characterized by a finite radius of convergence of the Taylor series $R=π/2$, which describes bright (dark) soliton propagation in non-linear media. We also show that for a plane wave, which has a Taylor series with an infinite radius of convergence, as the number of terms in the integer derivative expansion increases, the truncation error decreases. Finally, we illustrate the utility of the truncated integer derivative series by solving two linear fractional differential equations, where the fractional derivative is replaced by an integer derivative series up to the second order derivative. We find that our numerical results closely approximate the exact solutions given by the Mittag-Leffler and Fox-Wright functions. Thus, we demonstrate that the truncated expansion is a powerful method for solving linear fractional differential equations, such as the fractional Schrödinger equation.

math.NA↗

arXiv2017

Collisional Dynamics of Solitons in the Coupled PT symmetric Nonlocal nonlinear Schrodinger equations

We investigate the focusing coupled PT-symmetric nonlocal nonlinear Schrodinger equation employing Darboux transformation approach. We find a family of exact solutions including pairs of Bright-Bright, Dark-Dark and Bright-Dark solitons in addition to solitary waves. We show that one can convert bright bound state onto a dark bound state in a two-soliton solution by selectively fine tuning the amplitude dependent parameter. We also show that the energy in each mode remains conserved unlike the celebrated Manakov model. We also characterize the behaviour of the soliton solutions in detail. We emphasize that the above phenomenon occurs due to the nonlocality of the model.

nlin.SI↗

arXiv2016

High-accuracy power series solutions with arbitrarily large radius of convergence for fractional nonlinear differential equations

Fractional nonlinear differential equations present an interplay between two common and important effective descriptions used to simplify high dimensional or more complicated theories: nonlinearity and fractional derivatives. These effective descriptions thus appear commonly in physical and mathematical modeling. We present a new series method providing systematic controlled accuracy for solutions of fractional nonlinear differential equations. The method relies on spatially iterative use of power series expansions. Our approach permits an arbitrarily large radius of convergence and thus solves the typical divergence problem endemic to power series approaches. We apply our method to the fractional nonlinear Schrödinger equation and its imaginary time rotation, the fractional nonlinear diffusion equation. For the fractional nonlinear Schrödinger equation we find fractional generalizations of cnoidal waves of Jacobi elliptic functions as well as a fractional bright soliton. For the fractional nonlinear diffusion equation we find the combination of fractional and nonlinear effects results in a more strongly localized solution which nevertheless still exhibits power law tails, albeit at a much lower density.

cond-mat.stat-mech↗

arXiv2013

Averaged dynamics of soliton molecules in dispersion-managed optical fibers

The existence regimes and dynamics of soliton molecules in dispersion-managed (DM) optical fibers have been studied. Initially we develop a variational approximation (VA) for description of periodic dynamics of a soliton molecule within each unit cell of the dispersion map. The obtained system of coupled equations for the pulse width and chirp allows to find the parameters of DM soliton molecules for the given dispersion map and pulse energy. Then by means of a scaling transformation and averaging procedure we reduce the original nonlinear Schrödinger equation (NLSE) with piecewise-constant periodic dispersion to its counterpart with constant coefficients and additional parabolic potential. The obtained averaged NLSE with expulsive potential can explain the essential features of solitons and soliton molecules in DM fibers related to their energy loss during propagation. Also, the model of averaged NLSE predicts the instability of the temporal position of the soliton, which may lead to difficulty in holding the pulse in the middle of its time slot. All numerical simulations are performed using the parameters of the existing DM fiber setup, and illustrated via pertinent examples.

nlin.PS↗

arXiv2013

Stability of $N$-soliton molecules in dispersion-managed optical fibers

We investigate the stability of $N$-soliton molecules in dispersion-managed optical fibers with focus on the recently realized 2- and 3-soliton molecules. We calculate their binding energy using an averaged nonlinear Schr$\rm{\ddot o}$dinger equation. A combination of variational and numerical solutions to this equation shows that it describes well the intensity profiles and relative separations of the experimental molecules. Extending the calculation to larger values of $N$, the binding energy per soliton is found to saturate at N $N\ge7$.

nlin.PS↗

arXiv2013

Directional flow of solitons with asymmetric potential wells: Soliton diode

We study the flow of bright solitons through two asymmetric potential wells. The scattering of a soliton by certain type of single potential wells, e.g., Gaussian or Rosen-Morse, is distinguished by a critical velocity above which solitons can transmit almost completely and below which solitons can reflect nearly perfectly. For two such wells in series with certain parameter combinations, we find that there is an appreciable velocity range for which solitons can propagate in one direction only. Our study shows that this directional propagation or diode behavior is due to a combined effect of the sharp transition in the transport coefficients at the critical velocity and a slight reduction in the center-of-mass speed of the soliton while it travels across a potential well.

nlin.PS↗

arXiv2010

Formation of matter-wave soliton molecules

We propose a method of forming matter-wave soliton molecules that is inspired by the recent experiment of Dris {\it et al.}. In the proposed set-up we show that if two solitons are initially prepared in phase and with a sufficiently small separation and relative velocity, a bound pair will always form. This is verified by direct numerical simulation of the Gross-Pitaevskii equation and by the derivation of the exact interaction energy of two solitons, which takes the form of a Morse potential. This interaction potential depends not only on the separation but also on the relative phase of the solitons and is essential for an analytical treatment of a host of other problems, such as the soliton gas and the Toda lattice of solitons.

cond-mat.quant-gas↗

arXiv2010

Stability and dynamics of two-soliton molecules

The problem of soliton-soliton force is revisited. From the exact two solitons solution of a nonautonomous Gross-Pitaevskii equation, we derive a generalized formula for the mutual force between two solitons. The force is given for arbitrary solitons amplitude difference, relative speed, phase, and separation. The latter allows for the investigation of soliton molecules formation, dynamics, and stability. We reveal the role of the time-dependent relative phase between the solitons in binding them in a soliton molecule. We derive its equilibrium bond length, spring constant, frequency, effective mass, and binding energy of the molecule. We investigate the molecule's stability against perturbations such as reflection from surfaces, scattering by barriers, and collisions with other solitons.

cond-mat.quant-gas↗

arXiv2010

A comparative analysis of Painlevé, Lax Pair, and Similarity Transformation methods in obtaining the integrability conditions of nonlinear Schrödinger equations

We derive the integrability conditions of nonautonomous nonlinear Schr$\rm\ddot o$dinger equations using the Lax Pair and Similarity Transformation methods. We present a comparative analysis of these integrability conditions with those of the Painlev$\rm\acute{e}$ method. We show that while the Painlev$\rm\acute{e}$ integrability conditions restrict the dispersion, nonlinearity, and dissipation/gain coefficients to be space-independent and the external potential to be only a quadratic function of position, the Lax Pair and the Similarity Transformation methods allow for space-dependent coefficients and an external potential that is not restricted to the quadratic form. The integrability conditions of the Painlev$\rm\acute{e}$ method are retrieved as a special case of our general integrability conditions. We also derive the integrability conditions of nonautonomous nonlinear Schr$\rm\ddot o$dinger equations for two- and three-spacial dimensions.

math-ph↗

arXiv2009

Soliton localization in Bose-Einstein condensates with time-dependent harmonic potential and scattering length

We derive exact solitonic solutions of a class of Gross-Pitaevskii equations with time-dependent harmonic trapping potential and interatomic interaction. We find families of exact single-solitonic, multi-solitonic, and solitary wave solutions. We show that, with the special case of an oscillating trapping potential and interatomic interaction, a soliton can be localized indefinitely at an arbitrary position. The localization is shown to be experimentally possible for sufficiently long time even with only an oscillating trapping potential and a constant interatomic interaction.

cond-mat.quant-gas↗

arXiv2007

Exact Solitonic Solutions of the Gross-Pitaevskii Equation with a Linear Potential

We derive classes of exact solitonic solutions of the time-dependent Gross-Pitaevskii equation with repulsive and attractive interatomic interactions. The solutions correspond to a string of bright solitons with phase difference between adjacent solitons equal to $π$. While the relative phase, width, and distance between adjacent solitons turn out to be a constant of the motion, the center of mass of the string moves with a constant acceleration arising from the inhomogeneity of the background.

cond-mat.other↗

arXiv2007

Exact Solitonic Solutions of the One-Dimensional Gross-Pitaevskii Equation with a Time-Dependent Harmonic Potential and Interatomic Interaction

We derive exact solitonic solutions of the one-dimensional time-dependent Gross-Pitaevskii equation with time-dependent strengths of the harmonic external potential and the interatomic interaction. The time-dependence of the external potential and interatomic interaction are given in terms of a general function of time. For an oscillating strength of the external potential, the solutions correspond to breathing single and multiple solitons. The amplitude and frequency of the oscillating potential can be used to control the dynamics of the center of mass of the solitons. For certain values of these parameters, the solitons can be {\it trapped} at the center of the harmonic potential.

cond-mat.other↗