We find exact monotonic solitons in the nonlocal Gordon equation u_{tt}=J*u-u-f(u), in the case J(x)=1/2 e^{-|x|}. To this end we come up with an inverse method, which gives a representation of the set of nonlinearities admitting such solutions. We also study u''''+łu''-sin u=0, which arises from the above when we write it in traveling wave coordinates and pass to a certain limit. For this equation we find an exact 4π-kink and show the nonexistence of 2π-kinks, using the analytic continuation method of Amick and McLeod.
Inspired by the well known sine-Gordon equation, we present a symmetric coupled system of two real scalar fields in $1+1$ dimensions. There are three different topological soliton solutions which be labelled according to their topological charges. These solitons can absorb some localized non-dispersive wave packets in collision processes. It will be shown numerically, during collisions between solitons, there will be an uncertainty which originates from the amount of the maximum amplitude and arbitrary initial phases of the trapped wave packets.
We investigate the dynamics of the sine-Gordon solitons perturbed by spatiotemporal external forces. We prove the existence of internal (shape) modes of sine-Gordon solitons when they are in the presence of inhomogeneous space-dependent external forces, provided some conditions (for these forces) hold. Additional periodic time-dependent forces can sustain oscillations of the soliton width. We show that, in some cases, the internal mode even can become unstable, causing the soliton to decay in an antisoliton and two solitons. In general, in the presence of spatiotemporal forces the soliton beha…
The paper presents soliton-breather models of particles tunneling on the example of Klein-Gordon and Schrodinger equation nonlinear breathers. It is shown that in this case the non-linearity registration should lead to spatial restrictions in breathers wave properties observation and to appearance of radiation during tunneling. The paper also presents a new type of Schrodinger equation, which leads to the solutions localization in a space three-dimensional spherically symmetric case.
We investigate the dynamics of the sine-Gordon solitons perturbed by spatiotemporal external forces. We prove the existence of internal (shape) modes of sine-Gordon solitons when they are in the presence of inhomogeneous space-dependent external forces, provided some conditions (for these forces) hold. Additional periodic time-dependent forces can sustain oscillations of the soliton width. We show that, in some cases, the internal mode even can become unstable, causing the soliton to decay in an antisoliton and two solitons. In general, in the presence of spatiotemporal forces the soliton beha…
Topological solitons can propagate without radiation in discrete media. These solutions are known as embedded solitons (ES). They come as isolated solutions and exist despite their resonance with the linear spectrum of the respective lattices. In this paper the properties of embedded solitons in the discrete double sine-Gordon equation with the next-neighbor and second-neighbor interactions are investigated. Depending on the sign of these interactions they can be either destructive or favorable for the ES creation. The ES existence area depends on the width of the linear spectrum: narrowing of…
Nonlinear wave propagation plays a crucial role in the functioning of many physical and biophysical systems. In the propagation regime, disturbances due to the presence of local external perturbations, such as localised defects or boundary interphase walls have gained great attention. In this article, the complex phenomena that occur when sine-Gordon line solitons collide with localised inhomogeneities are investigated. By a one-dimensional theory, it is shown that internal modes of two-dimensional sine-Gordon solitons can be activated depending on the topological properties of the inhomogenei…
We show that quadratic solitons are equivalent to solitons of a nonlocal Kerr medium. This provides new physical insight into the properties of quadratic solitons, often believed to be equivalent to solitons of an effective saturable Kerr medium. The nonlocal analogy also allows for novel analytical solutions and the prediction of novel bound states of quadratic solitons.
We investigate theoretically the interaction of dark solitons in materials with a spatially nonlocal nonlinearity. In particular we do this analytically and for arbitrary degree of nonlocality. We employ the variational technique to show that nonlocality induces an attractive force in the otherwise repulsive soliton interaction.
We investigate properties of dark solitons in nonlocal materials with an arbitrary degree of nonlocality. We employ the variational technique and describe the dark solitons, for the first time, in the whole range of degree of nonlocality.
We have introduced a class of spiraling elliptic solitons in generic nonlocal nonlinear media. The spiraling elliptic solitons carry the orbital angular momentum. This class solitons are stable for any degree of nonlocality except for the local case when the response function of the material is Gaussian function.
Random jitter in the soliton arrival time (the Gordon-Haus effect) is analyzed for solitons being solutions of the integrable modified nonlinear Schroedinger equation. It is shown that the mean square fluctuation of the soliton position depends on the soliton parameters which can be properly adjusted to suppress the Gordon-Haus jitter.
We investigate the propagation of one-dimensional bright and dark spatial solitons in a nonlocal Kerr-like media, in which the nonlocality is of general form. We find an exact analytical solution to the nonlinear propagation equation in the case of weak nonlocality. We study the properties of these solitons and show their stability.
We investigate the propagation of partially coherent beams in spatially nonlocal nonlinear media with a logarithmic type of nonlinearity. We derive analytical formulas for the evolution of the beam parameters and conditions for the formation of nonlocal incoherent solitons.
Nonlinear waves that collide with localized defects exhibit complex behavior. Apart from reflection, transmission, and annihilation of an incident wave, a local inhomogeneity can activate internal modes of solitons, producing many impressive phenomena. In this work, we investigate a two-dimensional sine-Gordon model perturbed by a family of localized forces. We observed the formation of bubble-like and drop-like structures due to local internal shape modes instabilities. We describe the formation of such structures on the basis of a one-dimensional theory of activation of internal modes of sG …
We predict random-phase spatial solitons in instantaneous nonlocal nonlinear media. The key mechanism responsible for self-trapping of such incoherent wave-packets is played by the non-local (rather than the traditional non-instantaneous) nature of the nonlinearity. This new kind of incoherent solitons has profoundly different features than other incoherent solitons.
A collective coordinate theory is develop for soliton ratchets in the damped discrete sine-Gordon model driven by a biharmonic force. An ansatz with two collective coordinates, namely the center and the width of the soliton, is assumed as an approximated solution of the discrete non-linear equation. The evolution of these two collective coordinates, obtained by means of the Generalized Travelling Wave Method, explains the mechanism underlying the soliton ratchet and captures qualitatively all the main features of this phenomenon. The theory accounts for the existence of a non-zero depinning th…
We reveal that nonlocality can provide a simple physical mechanism for stabilization of multi-hump optical solitons, and present the first example of stable rotating dipole solitons and soliton spiraling, known to be unstable in all types of realistic nonlinear media with local response.
We address the impact of nonlocality in the physical features exhibited by solitons supported by Kerr-type nonlinear media with an imprinted optical lattice. We discover that nonlocality of nonlinear response can profoundly affect the soliton mobility, hence all the related phenomena. Such behavior manifests itself in significant reductions of the Peierls-Nabarro potential with increase of the degree of nonlocality, a result that opens the rare possibility in nature of almost radiationless propagation of highly localized solitons across the lattice.
We address the stability of multipole-mode solitons in nonlocal Kerr-type nonlinear media. Such solitons comprise several out-of-phase peaks packed together by the forces acting between them. We discover that dipole-, triple-, and quadrupole-mode solitons can be made stable, whereas all higher-order soliton bound states are unstable.
We reveal that spatially localized vortex solitons become stable in self-focusing nonlinear media when the vortex symmetry-breaking azimuthal instability is eliminated by a nonlocal nonlinear response. We study the main properties of different types of vortex beams and discuss the physical mechanism of the vortex stabilization in spatially nonlocal nonlinear media.
The sine-Gordon equation exhibits a gap in its linear spectrum. That gives rise to memory, or non-Markovian, effects in the soliton formation processes. The generalized variational approach is suggested to derive the model equation that governs the solitary wave evolution. The detailed analytical and numerical studies show that the soliton relaxation dynamics exhibits the main specific features of quantum emitters decay processes in photonic band gap materials. In particular, the non-Markovian effects lead to the extremely long-lived oscillations of the sine-Gordon solitons. That results in th…
We discuss existence and stability of two-dimensional solitons in media with spatially nonlocal nonlinear response. We show that such systems, which include thermal nonlinearity and dipolar Bose Einstein condensates, may support a variety of stationary localized structures - including rotating spatial solitons. We also demonstrate that the stability of these structures critically depends on the spatial profile of the nonlocal response function.
The optical spatial solitons with ellipse-shaped spots have generally been considered to be a result of either linear or nonlinear anisotropy. In this paper, we introduce a class of spiraling elliptic solitons in the nonlocal nonlinear media without both linear and nonlinear anisotropy. The spiraling elliptic solitons carry the orbital angular momentum, which plays a key role in the formation of such solitons, and are stable for any degree of nonlocality except the local case when the response function of the material is Gaussian function. The formation of such solitons can be attributable to …
We investigate the properties of localized waves in systems governed by nonlocal nonlinear Schrodinger type equations. We prove rigorously by bounding the Hamiltonian that nonlocality of the nonlinearity prevents collapse in, e.g., Bose-Einstein condensates and optical Kerr media in all physical dimensions. The nonlocal nonlinear response must be symmetric, but can be of completely arbitrary shape. We use variational techniques to find the soliton solutions and illustrate the stabilizing effect of nonlocality.
Nonlocality is a key feature of many physical systems since it prevents a catastrophic collapse and a symmetry-breaking azimuthal instability of intense wave beams in a bulk self-focusing nonlinear media. This opens up an intriguing perspective for stabilization of complex topological structures such as higher-order solitons, vortex rings and vortex ring-on-line complexes. Using direct numerical simulations, we find a class of cylindrically-symmetric $n$-th order spatial solitons having the intensity distribution with a central bright spot surrounded by $n$ bright rings of varying size. We inv…
We have performed numerical analysis of the two-dimensional (2D) soliton solutions in Bose-Einstein condensates with nonlocal dipole-dipole interactions. For the modified 2D Gross-Pitaevski equation with nonlocal and attractive local terms, we have found numerically different types of nonlinear localized structures such as fundamental solitons, radially symmetric vortices, nonrotating multisolitons (dipoles and quadrupoles), and rotating multisolitons (azimuthons). By direct numerical simulations we show that these structures can be made stable.
Existence and stability of PT-symmetric gap solitons in a periodic structure with defocusing nonlocal nonlinearity are studied both theoretically and numerically. We find that, for any degree of nonlocality, gap solitons are always unstable in the presence of an imaginary potential. The instability manifests itself as a lateral drift of solitons due to an unbalanced particle flux. We also demonstrate that the perturbation growth rate is proportional to the amount of gain (loss), thus predicting the observability of stable gap solitons for small imaginary potentials.
We address (2+1)D surface solitons occurring at the interface between a linear medium and a nonlocal nonlinear medium whose nonlinear contribution to the refractive index has a initial value at the interface. We find that there exist stable single and dipole surface solitons which do not exhibit a power threshold. The properties of the surface solitons can be affected by the initial value and the degree of nonlocality. When a laser beam is launched away from the interface, the beam will be periodic oscillations.
The interaction of solitons with external potentials in nonlinear Klein-Gordon field theory is investigated using an improved model. The presented model has been constructed with a better approximation for adding the potential to the Lagrangian through the metric of background space-time. The results of the model are compared with another model and the differences are discussed.
Interaction of Double sine-Gordon solitons with a space dependent potential wall and also a potential well has been investigated by employing an analytical model based on the collective coordinate approach. The potential has been added to the model through a suitable nontrivial metric for the background space-time. The model is able to predict most of the features of the soliton-potential interaction. It is shown that a soliton can pass through a potential barrier if its velocity is greater than a critical velocity which is a function of soliton initial conditions and also characters of the po…
Nonlinear space-time dynamics, defined in terms of celebrated 'solitonic' equations, brings indispensable tools for understanding, prediction and control of complex behaviors in both physical and life sciences. In this paper, we review sine-Gordon solitons, kinks and breathers as models of nonlinear excitations in complex systems in physics and in living cellular structures, both intra-cellular (DNA, protein folding and microtubules) and inter-cellular (neural impulses and muscular contractions). Key words: Sine-Gordon solitons, kinks and breathers, DNA, Protein folding, Microtubules, Neural c…
We consider the reflectionless transport of sine-Gordon solitons on a line. Transparent boundary conditions for the sine-Gordon equation on a line are derived using the so-called potential approach. Our numerical implementation of these novel boundary conditions proves the absence of the backscattering in transmission of sine-Gordon solitons through the boundary of the considered finite domains.
Observation of sine-Gordon-like solitons in a spinor Bose-Einstein condensate
Kirchhoff-Institut für Physik, Universität Heidelberg（海德堡大学基尔霍夫物理研究所）；Department of Mathematics and Statistics, and Department of Physics, University of Massachusetts Amherst（马萨诸塞大学阿默斯特分校数学与统计系及物理系）；Theoretical Sciences Visiting Program, Okinawa Institute of Science and Technology Graduate University（冲绳科学技术大学院大学理论科学访问项目）
We investigate the possibility of dc soliton motion sustained by pure ac driving in the sine-Gordon model. We show by means of the collective coordinate formalism that ac driving induces a net dc velocity whose modulus and direction depend on the driving phase. Numerical simulations of the full sine-Gordon equation confirm the correctness and accuracy of this prediction. Non trivial cases when dc soliton motion is transformed into oscillatory as well as the effects of damping are analyzed. Our results settle a long standing issue about the existence and characteristics of this phenomenon, whos…
The particular type of four-kink multi-solitons (or quadrons) adiabatic dynamics of the sine-Gordon equation in a model with two identical point attracting impurities has been studied. This model can be used for describing magnetization localized waves in multilayer ferromagnet. The quadrons structure and properties has been numerically investigated. The cases of both large and small distances between impurities has been viewed. The dependence of the localized in impurity region nonlinear high-amplitude waves frequencies on the distance between the impurities has been found. For an analytical …
We investigate one-dimensional Helmholtz solitons with nonlocal nonlinearity. We show an exact analytical solution to the nonlocal nonlinear nonparaxial propagation equation in the cases of high and weak nonlocality. We also numerically find that the degree of nonlocality can affect the width of nonlocal Helmholtz soliton beams, but have no effect on their stability. Contrarily, nonparaxiality can affect their stability, but have no effect on their width.
We study gray solitons in nonlocal nonlinear media and show that they are stable and can form bound states. We reveal that gray soliton velocity depends on the nonlocality degree, and that it can be drastically reduced in highly nonlocal media. This is in contrast to the case of local media, where the maximal velocity is dictated solely by the asymptotic soliton amplitude.
We predict the existence of surface dark solitons at the interface between a self-defocusing nonlocal nonlinear medium and a linear medium. The fundamental and higher-order surface dark solitons can exist when the linear refractive index of the self-defocusing media is much larger than that of the linear media. The fundamental solitons are stable and the stabilities of higher-order solitons depend on both nonlocality degree and propagation constant.
We reveal that lattice interfaces imprinted in nonlocal nonlinear media support surface solitons that do not exist in other similar settings, including interfaces of local and nonlocal uniform materials. We show the impact of nonlocality on the domains of existence and stability of the surface solitons, focusing on new types of dipole solitons residing partially inside the optical lattice. We find that such solitons feature strongly asymmetric shapes and that they are stable in large parts of their existence domain.
The large phase shift of strongly nonlocal spatial optical soliton(SNSOS) in the (1+1)-dimensional [(1+1)D] lead glass is investigated using the perturbation method. The fundamental soliton solution of the nonlocal nonlinear Schodinger equation(NNLSE) under the second approximation in strongly nonlocal case is obtained. It is found that the phase shift rate along the propagation direction of such soliton is proportional to the degree of nonlocality, which indicates that one can realize Pi-phase-shift within one Rayleigh distance in (1+1)D lead glass. A full comprehension of the nonlocality-enh…
General $N$-solitons in three recently-proposed nonlocal nonlinear Schrödinger equations are presented. These nonlocal equations include the reverse-space, reverse-time, and reverse-space-time nonlinear Schrödinger equations, which are nonlocal reductions of the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy. It is shown that general $N$-solitons in these different equations can be derived from the same Riemann-Hilbert solutions of the AKNS hierarchy, except that symmetry relations on the scattering data are different for these equations. This Riemann-Hilbert framework allows us to identify new t…
A new integrable (2+1)-dimensional nonlocal nonlinear Schrödinger equation is proposed. The $N$-soliton solution is given by Gram type determinant. It is found that the localized N-soliton solution has interesting interaction behavior which shows change of amplitude of localized pulses after collisions.
We investigate the validity of collective coordinate approaximations to the scattering of solitons in several classes of models in (1+1) dimensional field theory models. We look at models which are deformations of the sine-Gordon (SG) or the nonlinear Schrödinger (NLS) model as they posses solitons which are topological (SG) or non-topological (NLS). Our deformations preserve their topology (SG), but changes their integrability properties, either completely or partially (models become `quasi-integrable'). As our collective coordinate approximation does not allow for the radiation of energy out…
We present the experimental observation of scalar multi-pole solitons in highly nonlocal nonlinear media, including dipole-, tri-pole, quadru-pole, and necklace-type solitons, organized as arrays of out-of-phase bright spots. These complex solitons are meta-stable, but with a large parameters range where the instability is weak, enabling their experimental observation.
We address the evolution of higher-order excited states, such as vortex and multi-hump solitons, in nonlocal media with synthetic, competing focusing and defocusing nonlinearities with different nonlocal transverse scales. We reveal that introduction of suitable competing effects makes possible the stabilization of vortex solitons with topological charge m>2, as well as one-dimensional multi-hump solitons with number of humps p>4, all of which are highly unstable in natural nonlocal materials with focusing nonlinearities.
We show that multipole solitons can be made stable via vectorial coupling in bulk nonlocal nonlinear media. Such vector solitons are composed of mutually incoherent nodeless and multipole components jointly inducing a nonlinear refractive index profile. We found that stabilization of the otherwise highly unstable multipoles occurs below a maximum energy flow. Such threshold is determined by the nonlocality degree.
We analyze the existence, stability, and mobility of gap solitons (GSs) in a periodic photonic structure built into a nonlocal self-defocusing medium. Counter-intuitively, the GSs are supported even by a highly nonlocal nonlinearity, which makes the system quasi-linear. Unlike local models, the variational approximation (VA) predicts the GSs in a good agreement with numerical findings, due to the suppression of undulating tails of the solitons.
We analyze the diffusive motion of kink solitons governed by the thermal sine-Gordon equation. We analytically calculate the correlation function of the position of the kink center as well as the diffusion coefficient, both up to second-order in temperature. We find that the kink behavior is very similar to that obtained in the overdamped limit: There is a quadratic dependence on temperature in the diffusion coefficient that comes from the interaction among the kink and phonons, and the average value of the wave function increases with $\sqrt{t}$ due to the variance of the centers of individua…
We consider the sine-Gordon equation on metric graphs with simple topologies and derive vertex boundary conditions from the fundamental conservation laws, such as energy and current conservation. Traveling wave solutions for star and tree graphs are obtained analytically in the form of kink, antikink and breather solitons for a special case. It is shown that these solutions provide reflectionless soliton transmission at the graph vertex. We find the sum rule for bond-dependent coefficients making the sine-Gordon equation embedded on the graph completely integrable. For the general case the pro…
Profiles of static solitons in one-dimensional scalar field theory satisfy the same equations as trajectories of a fictitious particle in multidimensional mechanics. We argue that the structure and properties of the solitons are essentially different if the respective mechanical motions are chaotic. This happens in multifield models and models with spatially dependent potential. We illustrate our findings using one-field sine-Gordon model in external Dirac comb potential. First, we show that the number of different "chaotic" solitons grows exponentially with their length, and the growth rate i…
We address the impact of asymmetric nonlocal diffusion nonlinearity on the properties of gap solitons supported by photorefractive crystal with an imprinted optical lattice. We reveal how the asymmetric nonlocal response alters the domains of existence and the stability of solitons originating from different gaps. We find that in such media gap solitons cease to exist above a threshold of the nonlocality degree. We discuss how the interplay between nonlocality and lattice strength modifies the gap soliton mobility.
We consider the continuum version of an elastic chain supporting topological and non-topological degrees of freedom; this generalizes a model for the dynamics of DNA recently proposed and investigated by ourselves. In a certain limit, the non-topological degrees of freedom are frozen, and the model reduces to the sine-Gordon equations and thus supports well-known topological soliton solutions. We consider a (singular) perturbative expansion around this limit and study in particular how the non-topological field assume the role of an auxiliary field. This provides a more general framework for t…
We introduce vector soliton complexes in nonlocal Kerr-type nonlinear media. We discover that under proper conditions the combination of nonlocality and vectorial coupling has a remarkable stabilizing action on multi-humped solitons. In particular, we find that stable bound states featuring several field oscillations in each soliton component do exist. This affords stabilization of vector soliton trains incorporating large number of humps, a class of structures known to self-destroy via strong instabilities in scalar settings.
We study the transverse instability and dynamics of bright soliton stripes in two-dimensional nonlocal nonlinear media. Using a multiscale perturbation method, we derive analytically the first-order correction to the soliton shape, which features an exponential growth in time -- a signature of the transverse instability. The soliton's characteristic timescale associated with its exponential growth,is found to depend on the square root of the nonlocality parameter. This, in turn, highlights the nonlocality-induced suppression of the transverse instability. Our analytical predictions are corrobo…
In this report, the various 1D single soliton and multi-soliton solutions of the Sine-Gordon equation are explored. First the topological kink solitons and their properties for the Sine-Gordon, as well as the $ϕ^{4}$ model are investigated. This includes energy and momentum conservation laws for both models, as well as static kinks and antikinks. The Bogomol'nyi equation and the importance of Lorentz invariance are also discussed. Then in the second half, the focus is on Sine-Gordon multi-kink solutions, which are found by a separation of variables method rather than using inverse scattering o…
The ac driven sine-Gordon equation is studied analytically and numerically, with the aim of providing a full description of how soliton solutions behave. To date, there is much controversy about when ac driven dc motion is possible. Our work shows that kink solitons exhibit dc or oscillatory motion depending on the relation between their initial velocity and the force parameters. Such motion is proven to be impossible in the presence of damping terms. For breathers, the force amplitude range for which they exist when dissipation is absent is found. All the analytical results are compared with …
We study in detail the ratchet-like dynamics of topological solitons in homogeneous nonlinear Klein-Gordon systems driven by a bi-harmonic force. By using a collective coordinate approach with two degrees of freedom, namely the center of the soliton, $X(t)$, and its width, $l(t)$, we show, first, that energy is inhomogeneously pumped into the system, generating as result a directed motion; and, second, that the breaking of the time shift symmetry gives rise to a resonance mechanism that takes place whenever the width $l(t)$ oscillates with at least one frequency of the external ac force. In ad…
We report localized nonlinear modes of the self-focusing and defocusing nonlocal nonlinear Schroedinger equation with the generalized PT-symmetric Scarf-II, Rosen-Morse, and periodic potentials. Parameter regions are presented for broken and unbroken PT-symmetric phases of linear bounded states and the linear stability of the obtained solitons. Moreover, we numerically explore the dynamical behaviors of solitons and find stable solitons for some given parameters.
