arXiv subjects

U. Al Khawaja

Publications and source records attributed to U. Al Khawaja.

At least 19 recordsLinked to original sources

arXiv2026

Dynamics of Flat-Top--Bubble Vector Solitons

We investigate the dynamics of two-component flat-top--bubble vector solitons using a variational approximation and numerical simulations. The system is described by coupled cubic-quintic nonlinear Schrödinger equations with repulsive intercomponent coupling. We introduce a self-induced mechanism in which a localized flat-top soliton (FTS) generates a density bubble in a second component: the localized density acts as a repulsive potential for the background field, creating a composite bound state. A background-subtracted formulation and normalized super-Gaussian profiles yield collective-coordinate equations for the component centers and an effective interaction potential for their relative separation. The potential forms a binding well that is nearly triangular over its global displacement range but smooth and locally parabolic near its minimum. The variational approximation consequently predicts harmonic internal oscillations whose frequency is independent of amplitude in the small-oscillation regime. Real-time simulations verify this weak-kick behavior with good agreement between the variational and numerical frequencies. For strong phase imprints, equating the exact kick energy to the variational binding depth predicts a kick scale $K_{\mathrm{escape}}$ for the onset of partial FTS escape. Real-time simulations support this prediction: below this scale, the composite remains nearly intact, whereas above it, a substantial fraction of the localized component leaves the bubble.

nlin.PS↗

arXiv2026

Phase-controlled elastic, inelastic, and coalescent collisions of two-dimensional flat-top solitons

We investigate elastic, inelastic, and coalescent collisions between two-dimensional flat-top solitons supported by the cubic-quintic nonlinear Schrödinger equation. Numerical simulations reveal distinct collision regimes ranging from nearly elastic scattering to strongly inelastic interactions leading to long-lived merged states. We demonstrate that the transition between these regimes is primarily controlled by the relative phase of the solitons at the collision point, with out-of-phase collisions suppressing overlap and in-phase collisions promoting strong interaction. Kinetic-energy diagnostics are introduced to quantitatively characterize collision outcomes and to identify phase- and separation-dependent windows of elasticity. To interpret the observed dynamics, we extract effective phase-dependent interaction potentials from collision trajectories, providing a mechanical picture of attraction and repulsion between flat-top solitons. The stability of merged states formed after strongly inelastic collisions is explained by their lower energetic cost, arising from interfacial energetics, where a balance between internal pressure and edge tension plays a central role. A variational analysis based on direct energy minimization supports this picture by revealing robust energetic minima associated with stationary two-dimensional flat-top solitons.

nlin.PS↗

arXiv2026

Single impurity atom embedded in a dipolar two-soliton molecule as a qubit

We consider a single impurity atom trapped in a double well (DW) potential created by a dipolar two-soliton molecule in a quasi-one-dimensional geometry. By solving the eigenvalue problem for the impurity atom in the DW potential, we find that its ground and first excited states are well separated from higher excited states. This allows it to be approximated by a desirable two-level quantum system. Numerical simulations of the Schrödinger equation, governing impurity atom, demonstrate periodic oscillations in the probability of finding the impurity confined either to the ``left" or to the ``right" side of the DW potential. An analytic expression for the coherent oscillations of the population imbalance between the two wells of the DW potential has been derived using the two-mode approximation. Theoretical predictions of the mathematical model are in good agreement with the results of numerical simulations. Potential usage of the developed setup as a physical realization of ``qubit" has been discussed.

cond-mat.quant-gas↗

arXiv2025

Perturbative nonlinear J-matrix method of scattering in two dimensions

We introduce a perturbative formulation for a nonlinear extension of the J-matrix method of scattering in two dimensions. That is, we obtain the scattering matrix for the time-independent nonlinear Schrödinger equation in two dimensions with circular symmetry. The formulation relies on the linearization of products of orthogonal polynomials and on the utilization of the tools of the J-matrix method. Gauss quadrature integral approximation is instrumental in the numerical implementation of the approach. We present the theory for a general ψ^{2n + 1} nonlinearity, where n is a natural number, and obtain results for the cubic and quintic nonlinearities, ψ^3 and ψ^5. At certain value(s) of the energy, we observe the occurrence of bifurcation with two stable solutions. This curious and interesting phenomenon is a clear signature and manifestation of the underlying nonlinearity.

quant-ph↗

arXiv2025

A Search Method for Hirota Bilinear Systems of Nonlinear Evolution Equations

We present a systematic search method for finding Hirota bilinear systems of nonlinear evolution equations, with emphasis on the nonlinear Schrödinger equation (NLSE). Using a known exact solution, couplings between the different terms of the differential equation are identified, which are then used to derive the bilinear system. We show that a nonlinear evolution equation may have many solution-dependent Hirota bilinear systems. Nonetheless, all solution members of a certain solution class are associated with a single bilinear system. This has been demonstrated for the known solutions of the NLSE. For instance, all solution members of the N-bright soliton solutions class lead to the same bilinear system. Similarly, the class of N-dark soliton solutions and the class of breathers solutions are associated with their own distinctive bilinear systems. Identifying the bilinear system of a known seed solution will thus help in finding the rest of the solution members of the same class using the Hirota method. We apply our method to the nonintegrable NLSE with dual nonlinearity and external potential.

nlin.SI↗

arXiv2024

Elastic and Inelastic Scattering of Flat-Top Solitons

We investigate the interaction between two flat-top solitons within the cubic-quintic nonlinear Schrödinger equation framework. Our study results point towards a significant departure of flat-top solitons collisional characteristics from the conventional behaviors exhibited in the scattering dynamics of two bright solitons. Our investigation outlines specific regimes corresponding to the dual flat-top solitons' elastic and inelastic collisions. We determine regimes in the parameter space where exchange in the widths of the interacting flat-top solitons is possible, even within the elastic collision domain. We find a periodic occurrence of completely elastic scattering of flat-top solitons in terms of the solitons' parameters. Investigating the internal energy transfers between the solitons and the emitted radiation reveals the origin of inelasticity in flat-top solitons collisions. We perform a variational calculation that accounts for the amount of radiation produced by the collision and hence provides further insight on the physics underlying the loss of elasticity of collisions.

nlin.PS↗

arXiv2024

Manifestation of superfluidity in atom-number-imbalanced two-component Bose-Einstein condensates

Superfluid and dissipative regimes in the dynamics of a two-component quasi-one-dimensional Bose-Einstein condensate (BEC) with unequal atom numbers in the components have been explored. The system supports localized waves of the symbiotic type owing to the same-species repulsion and cross-species attraction. The minority BEC component moves through the majority component and creates excitations. To quantify the emerging excitations we introduce a time-dependent function called disturbance. Through numerical simulations of the coupled Gross-Pitaevskii equations with periodic boundary conditions, we have revealed a critical velocity of the localized wave, above which a transition from superfluid to dissipative regime occurs, evidenced by a sharp increase in the disturbance function. The factors responsible for the discrepancy between the actual critical velocity and the speed of sound, expected from theoretical arguments, have been discussed.

cond-mat.quant-gas↗

arXiv2023

Unidirectional flow of flat-top solitons

We numerically demonstrate the unidirectional flow of flat-top solitons when interacting with two reflectionless potential wells with slightly different depths. The system is described by a nonlinear Schrödinger equation with dual nonlinearity. The results show that for shallow potential wells, the velocity window for unidirectional flow is larger than for deeper potential wells. A wider flat-top solitons also have a narrow velocity window for unidirectional flow than those for thinner flat-top solitons.

nlin.PS↗

arXiv2022

Reflectionless potentials and resonant scattering of flat-top and thin-top solitons

We identify a class of potentials for which the scattering of flat-top solitons and thin-top solitons of the nonlinear Schrödinger equation with dual nonlinearity can be reflectionless. The scattering is characterized by sharp resonances between regimes of full transmission and full quantum reflection. Perturbative expansion in terms of the magnitude of radiation losses leads to the general form of reflectionless potentials. Simulating the scattering of flat-top solitons and thin-top solitons confirms the reflectionless feature of these potentials.

nlin.PS↗

arXiv2022

Quantum reflection of dark solitons scattered by reflectionless potential barrier and position-dependent dispersion

We investigate theoretically and numerically quantum reflection of dark solitons propagating through an external reflectionless potential barrier or in the presence of a position-dependent dispersion. We confirm that quantum reflection occurs in both cases with sharp transition between complete reflection and complete transmission at a critical initial soliton speed. The critical speed is calculated numerically and analytically in terms of the soliton and potential parameters. Analytical expressions for the critical speed were derived using the exact trapped mode, a time-independent, and a time-dependent variational calculations. It is then shown that resonant scattering occurs at the critical speed, where the energy of the incoming soliton is resonant with that of a trapped mode. Reasonable agreement between analytical and numerical values for the critical speed is obtained as long as a periodic multi-soliton ejection regime is avoided.

nlin.PS↗

arXiv2022

Bound states spectrum of the nonlinear Schrödinger equation with Pöschl-Teller and square potential wells

We obtain the spectrum of bound states for a modified Pöschl-Teller and square potential wells in the nonlinear Schrödinger equation. For a fixed norm of bound states, the spectrum for both potentials turns out to consist of a finite number of multi-node localized states. We use modulational instability analysis to derive the relation that gives the number of possible localized states and the maximum number of nodes in terms of the width of the potential. Soliton scattering by these two potentials confirmed the existence of the localized states which form as trapped modes. Critical speed for quantum reflection was calculated using the energies of the trapped modes.

nlin.PS↗

arXiv2021

High accuracy power series method for solving scalar, vector, and inhomogeneous nonlinear Schrödinger equations

We develop a high accuracy power series method for solving partial differential equations with emphasis on the nonlinear Schrödinger equations. The accuracy and computing speed can be systematically and arbitrarily increased to orders of magnitude larger than those of other methods. Machine precision accuracy can be easily reached and sustained for long evolution times within rather short computing time. In-depth analysis and characterisation for all sources of error are performed by comparing the numerical solutions with the exact analytical ones. Exact and approximate boundary conditions are considered and shown to minimise errors for solutions with finite background. The method is extended to cases with external potentials and coupled nonlinear Schrödinger equations.

math.NA↗

arXiv2021

Amplifying Optical Signals with Discrete Solitons in Waveguide Arrays

We present a design and protocol to achieve an essential feature of an optical transistor, namely the amplification of input signal with the use of discrete solitons in waveguide arrays. We consider the scattering of a discrete soliton by a reflectionless potential in the presence of a control soliton. We show that at the sharp transition region between full reflectance and full transmittance, the intensity of the reflected or transmitted soliton is highly sensitive to the intensity of the control soliton. This suggests a setup of signal amplifier. For realistic purposes, we modulate the parameters of the reflectionless potential well to achieve a performance of amplifier with a controllable amplification. To facilitate the experimental realization, we calculate the amplification factor in terms of the parameters of the potential well and the input power of the control soliton. The suggested signal amplifier device will be an important component in the all-optical data processing.

physics.optics↗

arXiv2021

Adding Binary Numbers with Discrete Solitons in Waveguide Arrays

We present a design and protocol to add binary numbers using discrete solitons in waveguide arrays. We show that the nonlinear interaction between discrete solitons in waveguide arrays can be exploited to design half and full adders. By modulating the separation between waveguides and introducing control solitons, we achieve the performance of an XOR gate. We construct the half and full adders using the XOR gate together with the previously-designed OR and AND gates. To facilitate the experimental realization, we calculate the profile of separations between the waveguides that will lead to the performance of the XOR gate.

physics.optics↗

arXiv2021

Families of Skyrmions in Two-Dimensional Spin-1/2 Systems

We find Skyrmion-like topological excitations for a two-dimensional spin-1/2 system. Expressing the spinor wavefunction in terms of a rotation operator maps the spin-1/2 system to a Manakov system. We employ both analytical and numerical methods to solve the resulting Manakov system. Using a generalized similarity transformation, we reduce the two-dimensional Manakov system to the integrable one-dimensional Manakov system. Solutions obtained in this manner diverge at the origin. We employ a power series method to obtain an infinite family of localized and nondiverging solutions characterized by a finite number of nodes. A numerical method is then used to obtain a family of localized oscillatory solutions with an infinite number of nodes corresponding to a skyrmion composed of concentric rings with intensities alternating between the two components of the spinor. We investigate the stability of the skyrmion solutions found here by calculating their energy functional in terms of their effective size. It turns out that indeed the skyrmion is most stable when the phase difference between the concentric rings is $π$, i.e., alternating between spin up and spin down. Our results are also applicable to doubly polarized optical pulses.

nlin.PS↗

arXiv2021

Superposition principle and composite solutions to coupled nonlinear Schrödinger equations

We show that the superposition principle applies to coupled nonlinear Schrödinger equations with cubic nonlinearity where exact solutions may be obtained as a linear combination of other exact solutions. This is possible due to the cancellation of cross terms in the nonlinear coupling. First, we show that a {\it composite} solution which is a linear combination of the two components of a {\it seed} solution is another solution to the same coupled nonlinear Schrödinger equation. Then, we show that a linear combination of two composite solutions is also a solution to the same equation. With emphasis on the case of Manakov system of two-coupled nonlinear Schrödinger equations, the superposition is shown to be equivalent to a rotation operator in a two-dimensional function space with components of the seed solution being its coordinates. Repeated application of the rotation operator, starting with a specific seed solution, generates a series of composite solutions which may be represented by a generalized solution that defines a family of composite solutions. Applying the rotation operator to almost all known exact seed solutions of the Manakov system, we obtain for each seed solution the corresponding family of composite solutions. Composite solutions turn out, in general, to possess interesting features that do not exist in the seed solution. Using symmetry reductions, we show that the method applies also to systems of $N$-coupled nonlinear Schrödinger equations. Specific examples for the three coupled-nonlinear Schrödinger equation are given.

nlin.SI↗

arXiv2021

Unidirectional flow of composite bright-bright solitons through asymmetric double potential barriers and wells

We investigate the dynamics of two component bright-bright (BB) solitons through reflectionless double barrier and double well potentials in the framework of a Manakov system governed by the coupled nonlinear Schrödinger equations. The objective is to achieve unidirectional flow and unidirectional segregation/splitting, which may be used in the design of optical data processing devices. We observe how the propagation of composite BB soliton is affected by the presence of interaction coupling between the two components passing through the asymmetric potentials. We consider Gaussian and Rosen-Morse double potential barriers in order to achieve the unidirectional flow. Moreover, we observe a novel phenomenon which we name "\textit{Polarity Reversal}" in the unidirectional flow. In this situation, the polarity of the diode is reversed. To understand the physics underlying these phenomena, we perform a variational calculation where we also achieve unidirectional segregation/splitting using an asymmetric double square potential well. Our comparative study between analytical and numerical analysis lead to an excellent agreement between the two methods.

nlin.PS↗

arXiv2021

High-speed soliton ejection generated from the scattering of bright solitons by modulated reflectionless potential wells

We investigate numerically and theoretically the conditions leading to soliton ejection stimulated through the scattering of bright solitons by modulated reflectionless potential wells. Such potential wells allow for the possibility of controlled ejection of solitons with significantly high speeds. At the outset, we describe the scattering setup and characterise the soliton ejection in terms of the different parameters of the system. Then, we formulate a theoretical model revealing the underlying physics of soliton ejection. The model is based on energy and norm exchange between the incident soliton and a stable trapped mode corresponding to an exact solution of the governing nonlinear Schrodinger equation. Remarkably, stationary solitons can lead to high-speed soliton ejection where part of the nonlinear interaction energy transforms to translational kinetic energy of the ejected soliton. Our investigation shows that soliton ejection always occurs whenever the incident soliton norm is greater than that of the trapped mode while their energy is almost the same. Once the incident soliton is trapped, the excess in norm turns to an ejected soliton in addition to a small amount of radiation that share translational kinetic energy. We found that higher ejection speeds are obtained with multi-node trapped modes that have larger binding energy. Simultaneous two-soliton ejection has been also induced by two solitons scattering with the potential from both of its sides. An ejection speed almost twice as that of single soliton ejection was obtained. Ejection outcome and ejection speed turn out to be sensitive to the relative phase between the two incoming solitons, which suggests a tool for soliton phase interferometry.

nlin.PS↗