Soliton dynamics and elastic collisions in a spin chain with an external time-dependent magnetic field
01 Feb 2010-Physica A-statistical Mechanics and Its Applications (North-Holland)-Vol. 389, Iss: 3, pp 367-374
TL;DR: In this article, the nonlinear dynamics of a Heisenberg spin chain with an external time-oscillating magnetic field were investigated by applying the Darboux transformation method to the linear eigenvalue problem associated with this equation.
Abstract: In this paper, we investigate the nonlinear dynamics of a Heisenberg spin chain with an external time-oscillating magnetic field. Such dynamics can be described by a Landau–Lifshitz-type equation. We apply the Darboux transformation method to the linear eigenvalue problem associated with this equation, and obtain the multi-soliton solution with a purely algebraic iterative procedure. Through the analytical analysis and graphical illustrations for the solutions obtained, we discuss in detail the effects of an external magnetic field on the nonlinear wave. Under the action of an external field, although the amplitude, width and depth of soliton vary periodically with time and its symmetry property is changeable, the soliton can also propagate stably and it possesses particle-like behavior.
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TL;DR: In this paper, the authors apply the multiple exp-function method to construct the exact multiple-wave solutions of a (3 + 1)-dimensional soliton equation, which has potential application in optical communication systems and switching devices.
Abstract: The multiple exp-function method is a new approach to obtain multiple-wave solutions of nonlinear partial differential equations (NLPDEs). By this method, one can obtain multi-soliton solutions of NLPDEs. Hence, in this paper, using symbolic computation, we apply the multiple exp-function method to construct the exact multiple-wave solutions of a (3 + 1)-dimensional soliton equation. Based on this application, we obtain mobile single-wave, double-wave and multi-wave solutions for this equation. In addition, we employ the straightforward and algebraic Hirota bilinearization method to construct the multi-soliton solutions of NLPDEs, and we reveal the remarkable property of soliton–soliton collision through this approach. Further, we investigate the one- and two-soliton solutions of a (3 + 1)-dimensional soliton equation using the Hirota’s method. We explore the particle-like behavior or elastic interaction of solitons, which has potential application in optical communication systems and switching devices.
26 citations
TL;DR: In this article, a derivative nonlinear Schrodinger equation with variable coefficients is investigated, which governs the propagation of the sub-picosecond soliton pulses in inhomogeneous optical fibers.
Abstract: Under investigation in this paper is a derivative nonlinear Schrodinger equation with variable coefficients, which governs the propagation of the subpicosecond soliton pulses in inhomogeneous optical fibers. Through the nonisospectral Kaup–Newell scheme, the Lax pair is constructed with some constraints on the variable coefficients. Under the integrable conditions, bright one- and multi-soliton-like solutions are derived via the Hirota method. By suitably choosing the dispersion coefficient function, several types of inhomogeneous solitons are obtained in, respectively: (1) exponentially decreasing dispersion profile, (2) linearly decreasing dispersion profile, (3) exponentially increasing dispersion profile, and (4) periodically fluctuating dispersion profile. The intensity of the inhomogeneous soliton can be controlled by means of modifying the loss/gain term. Asymptotic analysis of the two-soliton-like solution is performed, which shows that the changes of the widths, amplitudes, and energies before and after the collision are completely caused by the variable coefficients, but have nothing to do with the collision between two soliton-like envelopes. Through suitable choices of variable coefficients, figures are plotted to illustrate the collision behavior between two inhomogeneous solitons, which has some potential applications in the real optical communication systems.
22 citations
TL;DR: Three transformations have been obtained from such a equation to the known standard and cylindrical nonlinear Schrodinger equations with the relevant constraints on the variable coefficients presented, which turn out to be more general than those previously published in the literature.
Abstract: Describing the dispersion decreasing fiber, a variable-coe fficient nonlinear Schrodinger equation is hereby under investigation. Three transformations have been obtained from such a equation to the known standard and cylindrical nonlinear Schrodinger equations with the relevant constraints on the variable coefficients presented, which t urn out to be more general than those previously published in the literature. Meanwhile, several families of exact dark-soliton-like and bright-soliton-like solutions are constructed. Also, we o btain some similarity solutions, which can be illustrated in terms of the elliptic and the second Painle ve transcendent equations.
8 citations
Cites background from "Soliton dynamics and elastic collis..."
...Optical solitons, which form with the balance between group velocity dispersion and self-phase modulation, have attracted much attention in the optical fiber communication systems, soliton laser, and switching devices, with the nonlinear Schrödinger-typed models as the focal point[12-18]....
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TL;DR: A (2+1)-dimensional modified Heisenberg ferromagnetic system, which arises in the motion of magnetization vector of the isotropic ferromagnet and biological pattern formation, is investigated and it is proved that the system possesses the N-soliton solutions expressed in terms of the double Wronskian determinant.
Abstract: A (2+1)-dimensional modified Heisenberg ferromagnetic system, which arises in the motion of magnetization vector of the isotropic ferromagnet and biological pattern formation, is investigated. Via the Hirota bilinear method, multi-soliton solutions of such a system are derived. It is proved that the system possesses the N-soliton solutions expressed in terms of the double Wronskian determinant. Head-on and overtaking elastic interactions are exhibited. Elastic interaction behavior between the two solitons has been interpreted through the asymptotic analysis, namely, amplitude and velocity of each soliton remain unchanged except for the phase shift after the interaction. Inelastic interactions including the soliton fusion and fission between two solitons are shown. During the soliton propagation, for the product of two fields, the soliton with the smaller amplitude can travel faster than with the larger, while for the third field, the soliton with the larger amplitude can travel faster than with the smaller. On the other hand, the soliton for the third field may exhibit the solitoff-like property. With respect to the three solitons, head-on elastic interaction can be found.
6 citations
TL;DR: In this paper, a modified sine-Gordon equation is solved for a soliton diode, which is a long Josephson junction in which the magnetic field of the control current reverses its direction at the center.
Abstract: Modified sine-Gordon equation is solved for a soliton diode, which is a long Josephson junction in which the magnetic field of the control current reverses its direction at the center. Applying a properly directed bias current can start the spontaneous generation of soliton and antisolitons at the transition region of the diode and leads to a flux-flow voltage. The threshold bias current at which such a voltage generation starts is a function of the control current and the size of the transition region. Such phenomena can find applications in high-speed decision making electronic comparators.
1 citations
References
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Book•
30 Sep 1992
TL;DR: In this paper, the authors developed a systematic algebraic approach to solve linear and non-linear partial differential equations arising in soliton theory, such as the non-stationary linear Schrodinger equation, Korteweg-de Vries and Kadomtsev-Petviashvili equations, the Davey Stewartson system, Sine-Gordon and nonlinearSchrodinger equations 1+1 and 2+1 Toda lattice equations, and many others.
Abstract: In 1882 Darboux proposed a systematic algebraic approach to the solution of the linear Sturm-Liouville problem. In this book, the authors develop Darboux's idea to solve linear and nonlinear partial differential equations arising in soliton theory: the non-stationary linear Schrodinger equation, Korteweg-de Vries and Kadomtsev-Petviashvili equations, the Davey Stewartson system, Sine-Gordon and nonlinear Schrodinger equations 1+1 and 2+1 Toda lattice equations, and many others. By using the Darboux transformation, the authors construct and examine the asymptotic behaviour of multisoliton solutions interacting with an arbitrary background. In particular, the approach is useful in systems where an analysis based on the inverse scattering transform is more difficult. The approach involves rather elementary tools of analysis and linear algebra so that it will be useful not only for experimentalists and specialists in soliton theory, but also for beginners with a grasp of these subjects.
2,999 citations
TL;DR: The results show that, under a safe range of parameters, the bright soliton can be compressed into very high local matter densities by increasing the absolute value of the atomic scattering length, which can provide an experimental tool for investigating the range of validity of the one-dimensional Gross-Pitaevskii equation.
Abstract: We present a family of exact solutions of the one-dimensional nonlinear Schrodinger equation which describes the dynamics of a bright soliton in Bose-Einstein condensates with the time-dependent interatomic interaction in an expulsive parabolic potential. Our results show that, under a safe range of parameters, the bright soliton can be compressed into very high local matter densities by increasing the absolute value of the atomic scattering length, which can provide an experimental tool for investigating the range of validity of the one-dimensional Gross-Pitaevskii equation. We also find that the number of atoms in the bright soliton keeps dynamic stability: a time-periodic atomic exchange is formed between the bright soliton and the background.
362 citations
TL;DR: In this article, the dynamical behavior of an easy plane magnetic chain in a magnetic field is discussed in terms of magnons and solitons, which are responsible for a zero frequency peak in the longitudinal part of the dynamic structure factor.
Abstract: An easy plane magnetic chain in a magnetic field is equivalent to a Sine-Gordon system. At low temperatures and moderate fields the dynamical behaviour of this system can be discussed in terms of magnons and solitons. Solitons are responsible for a zero frequency peak in the longitudinal part of the dynamic structure factor. They are thus observable in neutron scattering experiments.
269 citations
TL;DR: In this article, the functional form of the solitons in the continuous Heisenberg spin system is studied in one dimension. And the results for soliton-soliton scattering in the isotropic case are presented.
Abstract: Solitons in the continuous Heisenberg spin system are studied in one dimension. We present results for soliton-soliton scattering in the isotropic case. For the anisotropic results we derive the functional form of the solitons. In both cases we investigated the linearized stability equations and found no evidence of instability.
243 citations
TL;DR: In this paper, the activation energy of the quasiparticle soliton was determined via the temperature and field dependence of the intensities of the soliton's intensities, and it was shown that at the cost of a higher activation energy, the solitons can move along the ferromagnetic chains in CsNi${\mathrm{F}}_{3}$.
Abstract: Evidence for solitons moving along the ferromagnetic chains in CsNi${\mathrm{F}}_{3}$ has been obtained by inelastic neutron scattering. As predicted by Mikeska the scattering is found at low $q$, around zero energy. The soliton activation energy, $8m$, is determined via the temperature and field dependence of the intensities ($m$ is the effective mass of the quasiparticle soliton). At $H=5$ kG we find $\frac{8m}{{k}_{\mathrm{B}}}=27$ K in reasonable agreement with the predicted value, as is the energy width at $q=0.1$.
219 citations