Reflection Of Solitons Research Articles

Evolution of solitons and their reflection and transmission in a plasma having negatively charged dust grains

Theoretical calculations are carried out for studying soliton's reflection and transmission in an inho- mogeneous plasma comprising ions, two temperature electrons and negatively charged dust grains. Using reductive perturbation technique, relevant modified KdV equations are derived for the incident, reflected and trans- mitted solitons. Then a coupled equation is obtained based on these mKdV equations, which is solved for the reflected soliton under the use of solutions of mKdV equations corresponding to the incident and transmitted solitons. Based on the ratio of amplitudes of reflected and incident solitons, reflection coefficient is examined under the effect of dust grain density; the same is done for the transmission coefficient which is the ratio of amplitudes of transmitted and incident solitons. The transmission of the solitons becomes weaker under the effect of stronger magnetic field and higher dust density. However, this leads to the stronger reflection of the soliton.

Conditions for reflection and transmission of an ion acoustic soliton in a dusty plasma with variable charge dust

Modified Korteweg-de Vries (mKdV) equations are derived for the incident, reflected, and transmitted waves in order to examine the soliton reflection and its transmission through an inhomogeneous plasma comprising ions, dust grains with fluctuating charge and two types of electrons, namely nonisothermal electrons and isothermal electrons. All the mKdV equations are coupled at the point of reflection and solved for the reflected soliton. Unlike others, a relation is established between the velocity shifts of the incident, reflected and transmitted solitons, and based on a critical value of the shift of incident soliton the strengths of the soliton reflection and transmission are talked about. Conditions are obtained for the soliton reflection and its transmission, and a comparative study is made for the two cases of fixed and fluctuating charges on the dust grains.

Modified Korteweg-de Vries soliton reflection in a magnetized plasma with dust grains and trapped electrons

This article aims at studying the reflection of solitons in an inhomogeneous magnetized warm plasma having dust grains with positive or negative charge and trapped electrons (low temperature nonisothermal electrons). In order to study the soliton reflection, a coupled modified Korteweg-de Vries equation is derived and solved along with the use of incident soliton solution. The expressions for the reflected soliton amplitude, width, and reflection coefficient are obtained, and examined under different parameter regimes. The combined effect of the dust grain density with their charge polarity and trapping of the electrons is largely studied on the soliton reflection characteristics under the influence of magnetic field.

Trapping and reflection of solitons with different frequencies upon concurrent collisions in an anisotropic medium

The variational method and numerical simulation have been used to investigate the modes of velocity and frequency transformation upon concurrent collisions of soliton laser pulses in an anisotropic medium. Excitation of bound states of optical and THz solitons affects decisively collisions at a small group velocity mismatch.

Discrete optical soliton scattering by local inhomogeneities

The nonlinear wave scattering by local inhomogeneities in discrete optical systems is studied both analytically and numerically. The presented theory describes the reflection and transmission of discrete optical solitons at a point defect. In particular, the derived expressions determine the reflected and transmitted pulses from the incident one. In the range of validity, the analytical results are in excellent agreement with the numerical simulations. It is demonstrated that the point defects in structured optical materials represent effective tool for controlling and manipulation of the nonlinear light pulses.

Tunneling of optical lattice solitons at interfaces

The nonlinear wave propagation and scattering processes in discrete optical heterostructures is studied both analytically and numerically. The presented theory describes the reflection and tunneling of lattice solitons at interfaces. In particular, the derived expressions allow us to construct the reflected and transmitted pulses from the incident one. In the range of validity, the analytical results are in very good agreement with the numerical simulations. It is demonstrated that optical heterostructures represent an effective tool for controlling and manipulating nonlinear light pulses and beams.

On the Nonlinear Schrödinger Equation on the Half Line with Homogeneous Robin Boundary Conditions

Boundary value problems for the nonlinear Schrödinger equations on the half line with homogeneous Robin boundary conditions are revisited using Bäcklund transformations. In particular: relations are obtained among the norming constants associated with symmetric eigenvalues; a linearizing transformation is derived for the Bäcklund transformation; the reflection‐induced soliton position shift is evaluated and the solution behavior is discussed. The results are illustrated by discussing several exact soliton solutions, which describe the soliton reflection at the boundary with or without the presence of self‐symmetric eigenvalues.

Reflection of Channel-Guided Solitons at Corners and Junctions in Two-Dimensional Nonlinear Schrödinger Equation

Solitons confined in channels are studied in the two-dimensional nonlinear Schr\"odinger equation. We study the dynamics of two channel-guided solitons near the junction where two channels are merged. The two solitons merge into one soliton, when there is no phase shift. If a phase difference is given to the two solitons, the Josephson oscillation is induced. The Josephson oscillation is amplified near the junction. The two solitons are reflected when the initial velocity is below a critical value.

Solitons in dipolar Bose–Einstein condensates with a trap and barrier potential

The propagation of solitons in the dipolar Bose–Einstein condensate in a trap potential with a barrier potential is investigated. The regimes of soliton transmission, reflection and splitting as a function of the ratio between the local and dipolar nonlocal interactions are analysed analytically and numerically. Coherent splitting and fusion of the soliton by the defect are observed. The conditions for the fusion of split solitons are found. In addition, the delocalization transition governed by the strength of the nonlocal dipolar interaction is presented. Predicted phenomena can be useful for the design of a matter-wave splitter and interferometers using matter-wave solitons.

Phase interaction of short vector solitons

An interaction of vector solitons in the frame of coupled third-order nonlinear Schrödinger equations taking into account third-order linear dispersion, nonlinear dispersion, and cross-phase modulation terms is considered. Phase nature of the solitonsʼ interaction is shown. In particular, dependence of solitonsʼ trajectories on initial distance between solitons is shown. Conditions of reflection and propagation of solitons through each other are obtained.

Generalized q-Onsager algebras and dynamical K-matrices

A procedure to construct K-matrices from the generalized q-Onsager algebra is proposed. This procedure extends the intertwiner techniques used to obtain scalar (c-number) solutions of the reflection equation to dynamical (non-c-number) solutions. It shows the relation between soliton non-preserving reflection equations or twisted reflection equations and the generalized q-Onsager algebras. These dynamical K-matrices are important to quantum integrable models with extra degrees of freedom located at the boundaries; for instance, in the quantum affine Toda field theories on the half-line, they yield the boundary amplitudes. As examples, the cases of and are treated in detail.

General N‐Dark–Dark Solitons in the Coupled Nonlinear Schrödinger Equations

N‐dark–dark solitons in the integrable coupled NLS equations are derived by the KP‐hierarchy reduction method. These solitons exist when nonlinearities are all defocusing, or both focusing and defocusing nonlinearities are mixed. When these solitons collide with each other, energies in both components of the solitons completely transmit through. This behavior contrasts collisions of bright–bright solitons in similar systems, where polarization rotation and soliton reflection can take place. It is also shown that in the mixed‐nonlinearity case, two dark–dark solitons can form a stationary bound state.

Soliton propagation, reflection, and transmission in an inhomogeneous plasma with trapped electrons

Transmission and reflection of solitons from a semitransparent grid in an inhomogeneous plasma in the presence of trapped electrons is studied analytically. Conditions are obtained for the obliqueness of the propagation and the drift velocity of ions for the soliton transmission and reflection. Also, a transmission–reflection conservation law is derived. The contribution of trapped electrons to the solitons’ propagation and their reflection and transmission is examined through energy, amplitude, and width of the solitons, in addition to the effect of temperature and drift of the ions.

Scattering of a matter-wave single soliton and a two-soliton molecule by an attractive potential

Scattering of a matter-wave single soliton and two-soliton molecule incident on the modified Pöschl-Teller potential well has been studied by means of a collective coordinate approach and numerical simulations of the Gross-Pitaevskii equation. Despite the attractive nature of the potential we observe total reflection of solitons in particular ranges of parameters, which is the signature of quantum behavior displayed by the matter-wave soliton. For other particular sets of parameters unscathed transmission of solitons and molecules through the potential well has been identified. A specific feature of this process is that the soliton passing through the potential well overtakes the freely propagating counterpart; i.e., its mean position appears to have been advanced in time. An array of such potentials makes the "time advance" effect even more pronounced, so that scattered solitons move well ahead of nonscattered ones, fully preserving their initial shape and velocity. A possible application of the obtained results is pointed out.

Oblique Reflection of Solitons in an Inhomogeneous Plasma: Effect of Trapped Electrons

This paper aims at studying the oblique reflection of solitons in an inhomogeneous plasma having finite-temperature ions and trapped electrons (two-temperature nonisothermal electrons). In order to study the soliton reflection, a coupled equation is derived based on modified Korteweg-deVries equations for the incident and reflected solitons, and then, it is solved along with the use of incident soliton solution. The expressions for the reflected soliton amplitude, width, and reflection coefficient are obtained and examined for different parameter regimes. The reflection coefficient, which is the ratio of reflected and incident soliton amplitudes, is found to be independent of the ion and electron temperatures. It infers that both the amplitudes change in the same proportion with the ion and electron temperatures. It is observed that the soliton reflects with higher amplitude when the obliqueness is smaller. It is also seen that the reflected soliton undergoes a downshift in its original line of propagation. This downshift is further investigated in detail. The effect of the trapped electrons on the soliton reflection characteristics and on the downshift is also analyzed.

Soliton reflection in a plasma with trapped electrons: The effect of dust concentration

Considering an inhomogeneous plasma that has trapped electrons and dust grains, we investigate the possible modes, their evolution as solitons and the reflection of solitons from the density gradient. The usual form of the Korteweg–de Vries (KdV) equation, which governs the behaviour of solitons, is modified for the present plasma, and is solved by finding appropriate transformations. The role of the charge Z d (positive: α = + 1 or negative: α = − 1 ) of the dust and dust concentration n d 0 in the presence of trapped electrons (density n e l 0 and temperature T e l 0 ) is investigated for the soliton reflection. The mode that evolves as a soliton can propagate in the plasma only when ion temperature T i ≥ α n d 0 Z d / ( n e h 0 T e h + n e l 0 T e l ) , where n e h 0 and T e h 0 are the density and temperature of nonisothermal electrons. Only the compressive solitons are found to propagate and reflect in the plasma (along with a shift) when the dust charge density remains smaller than the total charge density of the electrons of both types.

Enigmas of optical and matter-wave soliton nonlinear tunneling

The method of mathematical analogies opens the possibility to study optical and matter-wave solitons in parallel and, due to the evident complexity of experiments with matter wave solitons, offers remarkable possibilities in studies of the BEC systems by performing experiments in the nonlinear optical systems and vice versa. We discuss previous misinterpretations of the binding soliton energy and investigate all possible scenarios of nonlinear soliton tunneling of Schrödinger solitons including soliton splitting on a potential barrier, leading to partial reflection and partial transmission, sub-barrier soliton tunneling and over-barrier soliton reflection. We reveal the most significant modifications of soliton tunneling scenarios due to increasing the binding energy, and show that the self-compressing soliton resembles more the classical particle case than a quantum mechanical behavior. The enhancement of the soliton binding energy is of decisive importance in the testing of a long-standing theoretical result which predicts that an optical soliton can tunnel between two regions of anomalous dispersion across a forbidden region of normal dispersion (enhanced soliton spectral tunneling effect).

Effect of Ion Temperature on Soliton Reflection in a Magnetized Positive-Ion–Negative-Ion Plasma With Two Types of Electrons

Analytical investigations are carried out for the soliton propagation and reflection in a magnetized positive-ion-negative-ion plasma having two types of electrons based on the reductive perturbation technique. It is found that this type of plasma does not support rarefactive solitons, which are otherwise found to propagate in negative-ion-containing plasmas with isothermal electrons. This is the consequence of the low-temperature electron component present in the plasma. Two types of compressive solitons, namely, the fast wave soliton and the slow wave soliton, are found to exist, but only the fast compressive solitons will reflect. The effect of ion temperature is to enhance the soliton reflection; the same is the case for an increased number of negative ions. However, the effect of low-temperature electrons is to weaken the soliton reflection. Interestingly, the solitons are found to downshift after their reflection, and this shift is less under the effects of stronger magnetic field, smaller wave propagation angle, and a higher concentration of low-temperature electrons. However, larger amplitude solitons are more downshifted, and the shift is more for higher ion temperature and when the wave propagates with larger phase velocity.

Painlevé integrability, similarity reductions, new soliton and soliton-like similarity solutions for the (2+1)-dimensional BKP equation

The (2+1)-dimensional Kadomtsev–Petviashvili (KP) equation of B-type (BKP) is hereby investigated. New soliton solutions and soliton-like similarity solutions are constructed for the (2+1)-dimensional BKP equation. The similarity solutions are not travelling wave solutions when the arbitrary functions involved are chosen appropriately. Painleve test shows that there are two solution branches, one of which has the resonance −2. And four similarity reductions for the BKP equation are given out through nontrivial variable transformations. Moreover, abundant soliton behaviour modes of the solutions, such as soliton fusion and soliton reflection, are discussed in detail.

The Ablowitz-Ladik system on the natural numbers with certain linearizable boundary conditions

We solve the initial-boundary value problem (IBVP) for the Ablowitz–Ladik system on the natural numbers with certain linearizable boundary conditions. We do so by employing a nonlinear method of images, namely, by extending the scattering potential to all integers in such a way that the extended potential satisfies certain symmetry relations. Using these extensions and the solution of the initial value problem (IVP), we then characterize the symmetries of the discrete spectrum of the scattering problem, and we show that discrete eigenvalues in the IBVP appear in octets, as opposed to quartets in the IVP. Furthermore, we derive explicit relations between the norming constants associated with symmetric eigenvalues, and we identify a new kind of linearizable IBVP. Finally, we characterize the soliton solutions of these IBVPs, which describe the soliton reflection at the boundary of the lattice.