Using analytical and numerical techniques we analyze the static and dynamical properties of solit... more Using analytical and numerical techniques we analyze the static and dynamical properties of solitonlike excitations in the presence of parametric disorder in the one-dimensional nonlinear Schrödinger equation with a homogeneous power nonlinearity. Both the continuum and the discrete problem are investigated. We find that otherwise unstable excitations can be stabilized by the presence of disorder in the continuum problem. For
We study the continuum limit of a nonlinear Schrödinger lattice model with both on-site and inter... more We study the continuum limit of a nonlinear Schrödinger lattice model with both on-site and inter-site nonlinearities, describing weakly coupled optical waveguides or Bose-Einstein condensates. The resulting continuum nonlinear Schrödinger-type equation includes both nonlocal and nonlinear dispersion. Looking for stationary solutions, the equation is reduced to an ordinary differential equation with a rescaled spectral parameter and a single parameter interpolating between the nonlocality and the nonlinear dispersion. It is seen that these two effects give a similar behaviour for the solutions. We find smooth solitons and, beyond a critical value of the spectral parameter, also nonanalytic solitons in the form of peakons and capons. The existence of the exotic solitons is connected to the special properties of the phase space of the equation. Stability is investigated numerically by calculating eigenvalues and eigenfunctions of the linearized problem, and we particularly find that with both nonlocal and nonlinear dispersion simultaneously present, all solutions are unstable with respect to a break-up into short-wavelength oscillations.
The paper reports on the effects of adding noise, damping and impurities to the 2-dimensional non... more The paper reports on the effects of adding noise, damping and impurities to the 2-dimensional nonlinear Schrödinger equation. The effects of including long-range dispersion in the 1-dimensional nonlinear Schrödinger equation are also examined. The framework of the reported results is the application of nonlinear Schrödinger systems in the study of molecular systems and organic thin films. The paper includes numerical as well as analytical results.
Collapse of solitary excitations in the nonlinear Schrödinger equation with nonlinear damping and... more Collapse of solitary excitations in the nonlinear Schrödinger equation with nonlinear damping and white noise. Peter L. Christiansen, Yuri B. Gaididei, Magnus Johansson, Kim Ø. Rasmussen, and Irina I. Yakimenko Institute ...
In the framework of a discrete nonlinear Schrodinger equation with long-range dispersion, we prop... more In the framework of a discrete nonlinear Schrodinger equation with long-range dispersion, we propose a general mechanism for obtaining a controlled switching between bistable localized excitations. We show that the application of a spatially symmetric kick leads to ...
We show that the NLS systems with multiplicative noise, nonlinear damping and nonlocal dispersion... more We show that the NLS systems with multiplicative noise, nonlinear damping and nonlocal dispersion exhibit a variety of interesting effects which may be useful for modelling the dynamical behavior of one-and two-dimensional systems.
An analysis of two-dimensional (2D) effects in the nonlinear Kronig-Penney model is presented. We... more An analysis of two-dimensional (2D) effects in the nonlinear Kronig-Penney model is presented. We establish an effective one-dimensional description of the 2D effects, resulting in a set of pseudodifferential equations. The stationary states of the 2D system and their ...
Vortex core reversal in magnetic particle is essentially influenced by a surface anisotropy. Unde... more Vortex core reversal in magnetic particle is essentially influenced by a surface anisotropy. Under the action of a perpendicular static magnetic field the vortex core undergoes a shape deformation of pillow-or barrel-shaped type, depending on the type of the surface anisotropy. This deformation plays a key point in the switching mechanism: We predict that the vortex polarity switching is accompanied (i) by a linear singularity in case of Heisenberg magnet with bulk anisotropy only and (ii) by a point singularities in case of surface anisotropy or exchange anisotropy. We study in details the switching process using spin-lattice simulations and propose a simple analytical description using a wired core model, which provides an adequate description of the Bloch point statics, its dynamics and the Bloch point mediated switching process. Our analytical predictions are confirmed by spin-lattice simulations for Heisenberg magnet and micromagnetic simulations for nanomagnet with account of a dipolar interaction. PACS: 75.10.Hk Classical spin models; 05.45.-a Nonlinear dynamics and chaos; 75.70.Rf Surface magnetism; 75.75.-c Magnetic properties of nanostructures; 75.78.-n Magnetization dynamics.
Under the action of an alternating perpendicular magnetic field the polarity of the vortex state ... more Under the action of an alternating perpendicular magnetic field the polarity of the vortex state nanodisk can be efficiently switched. We predict the regular and chaotic dynamics of the vortex polarity and propose a simple analytical description in terms of a reduced vortex core model. Conditions for the controllable polarity switching are analyzed.
The motion of fluxons of the same polarity in three vertically stacked Josephson junctions is stu... more The motion of fluxons of the same polarity in three vertically stacked Josephson junctions is studied. In this configuration the difference between exterior and interior junctions plays a more important role than in other configurations with several interior junctions. Below the Swihart velocity c Ϫ , the coupling between junctions leads to a repulsion of the fluxons with the same polarity. Above this critical velocity a fluxon will induce radiation in the neighboring junctions, leading to a bunching of the fluxons in the stacked junctions. Using the Sakai-Bodin-Pedersen model, three coupled perturbed sine-Gordon equations are numerically studied for different values of coupling, damping, and bias parameters. In a narrow range of velocities bunching occurs. Outside this interval the fluxons split and new fluxons may be created. I-V characteristics are presented.
The vortex core shape in the three dimensional Heisenberg magnet is essentially influenced by a s... more The vortex core shape in the three dimensional Heisenberg magnet is essentially influenced by a surface anisotropy. We predict that depending of the surface anisotropy type there appears barrel-or pillow-shaped deformation of the vortex core along the magnet thickness. Our theoretical study is well confirmed by spin-lattice simulations.
ABSTRACT A wave equation including nonlinear terms up to the second order for a thermoviscous New... more ABSTRACT A wave equation including nonlinear terms up to the second order for a thermoviscous Newtonian fluid is proposed. In the lossless case this equation results from an expansion to third order of the Lagrangian for the fundamental non-dissipative fluid dynamical equations. Thus it preserves the Hamiltonian structure, in contrast to the Kuznetsov equation, a model often used in nonlinear acoustics. An exact traveling wave front solution is derived from a generalized traveling wave assumption for the velocity potential. Numerical studies of the evolution of a number of arbitrary initial conditions as well as head-on colliding and confluent wave fronts exhibit several nonlinear interaction phenomena. These include wave fronts of changed velocity and amplitude along with the emergence of rarefaction waves. An analysis using the continuity of the solutions as well as the boundary conditions is proposed. The dynamics of the rarefaction wave is approximated by a collective coordinate approach in the energy balance equation. KeywordsThe Kuznetsov equation–Traveling wave analysis–Wave fronts–Rarefaction waves–Collective coordinate approach
A one-dimensional discrete nonlinear Schrödinger (DNLS) model with the power dependence, r −s on ... more A one-dimensional discrete nonlinear Schrödinger (DNLS) model with the power dependence, r −s on the distance r, of dispersive interactions is proposed. The stationary states of the system are studied both analytically and numerically. Two kinds of trial functions, exp-like and sech-like are exploited and the results of both approaches are compared. Both on-site and inter-site stationary states are investigated. It is shown that for s sufficiently large all features of the model are qualitatively the same as in the DNLS model with nearest-neighbor interaction. For s less than some critical value, s cr , there is an interval of bistability where two stable stationary states exist at each excitation number. The bistability of on-site solitons may occur for dipole-dipole dispersive interaction (s = 3), while s cr for inter-site solitons is close to 2.1.
We review some recent results concerning the existence and stability of spatially localized and t... more We review some recent results concerning the existence and stability of spatially localized and temporally quasiperiodic (non-stationary) excitations in discrete nonlinear Schrödinger (DNLS) models. In two dimensions, we show the existence of linearly stable, stationary and non-stationary localized vortex-like solutions. We also show that stationary on-site localized excitations can have internal 'breathing' modes which are spatially localized and symmetric. The excitation of these modes leads to slowly decaying, quasiperiodic oscillations. Finally, we show that for some generalizations of the DNLS equation where bistability occurs, a controlled switching between stable states is possible by exciting an internal breathing mode above a threshold value.
We found resonantly excited precession motions of a three-dimensional vortex core in soft magneti... more We found resonantly excited precession motions of a three-dimensional vortex core in soft magnetic nanospheres and controllable precession frequency with the sphere diameter 2R, as studied by micromagnetic numerical and analytical calculations. The precession angular frequency for an applied static field HDC is given as ωMV = γeffHDC, where γeff = γ〈mΓ〉 is the effective gyromagnetic ratio in collective vortex dynamics, with the gyromagnetic ratio γ and the average magnetization component 〈mΓ〉 of the ground-state vortex in the core direction. Fitting to the micromagnetic simulation data for 〈mΓ〉 yields a simple explicit form of 〈mΓ〉 ≈ (73.6 ± 3.4)(lex/2R)(2.20±0.14), where lex is the exchange length of a given material. This dynamic behavior might serve as a foundation for potential bio-applications of size-specific resonant excitation of magnetic vortex-state nanoparticles, for example, magnetic particle resonance imaging.
We show that the interaction of the magnetic subsystem of a curved magnet with the magnet curvatu... more We show that the interaction of the magnetic subsystem of a curved magnet with the magnet curvature results in the coupling of a topologically nontrivial magnetization pattern and topology of the object. The mechanism of this coupling is explored and illustrated by an example of a ferromagnetic Möbius ring, where a topologically induced domain wall appears as a ground state in the case of strong easy-normal anisotropy. For the Möbius geometry, the curvilinear form of the exchange interaction produces an additional effective Dzyaloshinskii-like term which leads to the coupling of the magnetochirality of the domain wall and chirality of the Möbius ring. Two types of domain walls are found, transversal and longitudinal, which are oriented across and along the Möbius ring, respectively. In both cases, the effect of magnetochirality symmetry breaking is established. The dependence of the ground state of the Möbius ring on its geometrical parameters and on the value of the easy-normal ani...
Nonlinear solitary excitations are studied in an anharmonic chain with cubic or Toda nearest-neig... more Nonlinear solitary excitations are studied in an anharmonic chain with cubic or Toda nearest-neighbor interaction and exponentially decaying harmonic long-range interactions. Analytic expressions are obtained for the cubic interatomic potential, and the results are in qualitative agreement with numerical simulations. The relation of amplitude and velocity is quite different to a lattice without long-range forces especially near the speed of sound with the possibility of multiple solutions for a given velocity.
Using analytical and numerical techniques we analyze the static and dynamical properties of solit... more Using analytical and numerical techniques we analyze the static and dynamical properties of solitonlike excitations in the presence of parametric disorder in the one-dimensional nonlinear Schrödinger equation with a homogeneous power nonlinearity. Both the continuum and the discrete problem are investigated. We find that otherwise unstable excitations can be stabilized by the presence of disorder in the continuum problem. For
We study the continuum limit of a nonlinear Schrödinger lattice model with both on-site and inter... more We study the continuum limit of a nonlinear Schrödinger lattice model with both on-site and inter-site nonlinearities, describing weakly coupled optical waveguides or Bose-Einstein condensates. The resulting continuum nonlinear Schrödinger-type equation includes both nonlocal and nonlinear dispersion. Looking for stationary solutions, the equation is reduced to an ordinary differential equation with a rescaled spectral parameter and a single parameter interpolating between the nonlocality and the nonlinear dispersion. It is seen that these two effects give a similar behaviour for the solutions. We find smooth solitons and, beyond a critical value of the spectral parameter, also nonanalytic solitons in the form of peakons and capons. The existence of the exotic solitons is connected to the special properties of the phase space of the equation. Stability is investigated numerically by calculating eigenvalues and eigenfunctions of the linearized problem, and we particularly find that with both nonlocal and nonlinear dispersion simultaneously present, all solutions are unstable with respect to a break-up into short-wavelength oscillations.
The paper reports on the effects of adding noise, damping and impurities to the 2-dimensional non... more The paper reports on the effects of adding noise, damping and impurities to the 2-dimensional nonlinear Schrödinger equation. The effects of including long-range dispersion in the 1-dimensional nonlinear Schrödinger equation are also examined. The framework of the reported results is the application of nonlinear Schrödinger systems in the study of molecular systems and organic thin films. The paper includes numerical as well as analytical results.
Collapse of solitary excitations in the nonlinear Schrödinger equation with nonlinear damping and... more Collapse of solitary excitations in the nonlinear Schrödinger equation with nonlinear damping and white noise. Peter L. Christiansen, Yuri B. Gaididei, Magnus Johansson, Kim Ø. Rasmussen, and Irina I. Yakimenko Institute ...
In the framework of a discrete nonlinear Schrodinger equation with long-range dispersion, we prop... more In the framework of a discrete nonlinear Schrodinger equation with long-range dispersion, we propose a general mechanism for obtaining a controlled switching between bistable localized excitations. We show that the application of a spatially symmetric kick leads to ...
We show that the NLS systems with multiplicative noise, nonlinear damping and nonlocal dispersion... more We show that the NLS systems with multiplicative noise, nonlinear damping and nonlocal dispersion exhibit a variety of interesting effects which may be useful for modelling the dynamical behavior of one-and two-dimensional systems.
An analysis of two-dimensional (2D) effects in the nonlinear Kronig-Penney model is presented. We... more An analysis of two-dimensional (2D) effects in the nonlinear Kronig-Penney model is presented. We establish an effective one-dimensional description of the 2D effects, resulting in a set of pseudodifferential equations. The stationary states of the 2D system and their ...
Vortex core reversal in magnetic particle is essentially influenced by a surface anisotropy. Unde... more Vortex core reversal in magnetic particle is essentially influenced by a surface anisotropy. Under the action of a perpendicular static magnetic field the vortex core undergoes a shape deformation of pillow-or barrel-shaped type, depending on the type of the surface anisotropy. This deformation plays a key point in the switching mechanism: We predict that the vortex polarity switching is accompanied (i) by a linear singularity in case of Heisenberg magnet with bulk anisotropy only and (ii) by a point singularities in case of surface anisotropy or exchange anisotropy. We study in details the switching process using spin-lattice simulations and propose a simple analytical description using a wired core model, which provides an adequate description of the Bloch point statics, its dynamics and the Bloch point mediated switching process. Our analytical predictions are confirmed by spin-lattice simulations for Heisenberg magnet and micromagnetic simulations for nanomagnet with account of a dipolar interaction. PACS: 75.10.Hk Classical spin models; 05.45.-a Nonlinear dynamics and chaos; 75.70.Rf Surface magnetism; 75.75.-c Magnetic properties of nanostructures; 75.78.-n Magnetization dynamics.
Under the action of an alternating perpendicular magnetic field the polarity of the vortex state ... more Under the action of an alternating perpendicular magnetic field the polarity of the vortex state nanodisk can be efficiently switched. We predict the regular and chaotic dynamics of the vortex polarity and propose a simple analytical description in terms of a reduced vortex core model. Conditions for the controllable polarity switching are analyzed.
The motion of fluxons of the same polarity in three vertically stacked Josephson junctions is stu... more The motion of fluxons of the same polarity in three vertically stacked Josephson junctions is studied. In this configuration the difference between exterior and interior junctions plays a more important role than in other configurations with several interior junctions. Below the Swihart velocity c Ϫ , the coupling between junctions leads to a repulsion of the fluxons with the same polarity. Above this critical velocity a fluxon will induce radiation in the neighboring junctions, leading to a bunching of the fluxons in the stacked junctions. Using the Sakai-Bodin-Pedersen model, three coupled perturbed sine-Gordon equations are numerically studied for different values of coupling, damping, and bias parameters. In a narrow range of velocities bunching occurs. Outside this interval the fluxons split and new fluxons may be created. I-V characteristics are presented.
The vortex core shape in the three dimensional Heisenberg magnet is essentially influenced by a s... more The vortex core shape in the three dimensional Heisenberg magnet is essentially influenced by a surface anisotropy. We predict that depending of the surface anisotropy type there appears barrel-or pillow-shaped deformation of the vortex core along the magnet thickness. Our theoretical study is well confirmed by spin-lattice simulations.
ABSTRACT A wave equation including nonlinear terms up to the second order for a thermoviscous New... more ABSTRACT A wave equation including nonlinear terms up to the second order for a thermoviscous Newtonian fluid is proposed. In the lossless case this equation results from an expansion to third order of the Lagrangian for the fundamental non-dissipative fluid dynamical equations. Thus it preserves the Hamiltonian structure, in contrast to the Kuznetsov equation, a model often used in nonlinear acoustics. An exact traveling wave front solution is derived from a generalized traveling wave assumption for the velocity potential. Numerical studies of the evolution of a number of arbitrary initial conditions as well as head-on colliding and confluent wave fronts exhibit several nonlinear interaction phenomena. These include wave fronts of changed velocity and amplitude along with the emergence of rarefaction waves. An analysis using the continuity of the solutions as well as the boundary conditions is proposed. The dynamics of the rarefaction wave is approximated by a collective coordinate approach in the energy balance equation. KeywordsThe Kuznetsov equation–Traveling wave analysis–Wave fronts–Rarefaction waves–Collective coordinate approach
A one-dimensional discrete nonlinear Schrödinger (DNLS) model with the power dependence, r −s on ... more A one-dimensional discrete nonlinear Schrödinger (DNLS) model with the power dependence, r −s on the distance r, of dispersive interactions is proposed. The stationary states of the system are studied both analytically and numerically. Two kinds of trial functions, exp-like and sech-like are exploited and the results of both approaches are compared. Both on-site and inter-site stationary states are investigated. It is shown that for s sufficiently large all features of the model are qualitatively the same as in the DNLS model with nearest-neighbor interaction. For s less than some critical value, s cr , there is an interval of bistability where two stable stationary states exist at each excitation number. The bistability of on-site solitons may occur for dipole-dipole dispersive interaction (s = 3), while s cr for inter-site solitons is close to 2.1.
We review some recent results concerning the existence and stability of spatially localized and t... more We review some recent results concerning the existence and stability of spatially localized and temporally quasiperiodic (non-stationary) excitations in discrete nonlinear Schrödinger (DNLS) models. In two dimensions, we show the existence of linearly stable, stationary and non-stationary localized vortex-like solutions. We also show that stationary on-site localized excitations can have internal 'breathing' modes which are spatially localized and symmetric. The excitation of these modes leads to slowly decaying, quasiperiodic oscillations. Finally, we show that for some generalizations of the DNLS equation where bistability occurs, a controlled switching between stable states is possible by exciting an internal breathing mode above a threshold value.
We found resonantly excited precession motions of a three-dimensional vortex core in soft magneti... more We found resonantly excited precession motions of a three-dimensional vortex core in soft magnetic nanospheres and controllable precession frequency with the sphere diameter 2R, as studied by micromagnetic numerical and analytical calculations. The precession angular frequency for an applied static field HDC is given as ωMV = γeffHDC, where γeff = γ〈mΓ〉 is the effective gyromagnetic ratio in collective vortex dynamics, with the gyromagnetic ratio γ and the average magnetization component 〈mΓ〉 of the ground-state vortex in the core direction. Fitting to the micromagnetic simulation data for 〈mΓ〉 yields a simple explicit form of 〈mΓ〉 ≈ (73.6 ± 3.4)(lex/2R)(2.20±0.14), where lex is the exchange length of a given material. This dynamic behavior might serve as a foundation for potential bio-applications of size-specific resonant excitation of magnetic vortex-state nanoparticles, for example, magnetic particle resonance imaging.
We show that the interaction of the magnetic subsystem of a curved magnet with the magnet curvatu... more We show that the interaction of the magnetic subsystem of a curved magnet with the magnet curvature results in the coupling of a topologically nontrivial magnetization pattern and topology of the object. The mechanism of this coupling is explored and illustrated by an example of a ferromagnetic Möbius ring, where a topologically induced domain wall appears as a ground state in the case of strong easy-normal anisotropy. For the Möbius geometry, the curvilinear form of the exchange interaction produces an additional effective Dzyaloshinskii-like term which leads to the coupling of the magnetochirality of the domain wall and chirality of the Möbius ring. Two types of domain walls are found, transversal and longitudinal, which are oriented across and along the Möbius ring, respectively. In both cases, the effect of magnetochirality symmetry breaking is established. The dependence of the ground state of the Möbius ring on its geometrical parameters and on the value of the easy-normal ani...
Nonlinear solitary excitations are studied in an anharmonic chain with cubic or Toda nearest-neig... more Nonlinear solitary excitations are studied in an anharmonic chain with cubic or Toda nearest-neighbor interaction and exponentially decaying harmonic long-range interactions. Analytic expressions are obtained for the cubic interatomic potential, and the results are in qualitative agreement with numerical simulations. The relation of amplitude and velocity is quite different to a lattice without long-range forces especially near the speed of sound with the possibility of multiple solutions for a given velocity.
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Papers by Yuri Gaididei