Soliton Splitting by External Delta Potentials

We show that in the dynamics of the nonlinear Schrodinger equation a soliton scattered by an external delta potential splits into two solitons and a radiation term. Theoretical analysis gives the amplitudes and phases of the reflected and transmitted solitons with errors going to zero as the velocity of the incoming soliton tends to infinity. Numerical analysis shows that this asymptotic relation is valid for all but very slow solitons. We also show that the total transmitted mass, that is, the square of the L² norm of the solution restricted on the transmitted side of the delta potential, is in good agreement with the quantum transmission rate of the delta potential.     

Expanding solitons with non-negative curvature operator coming out of cones

We consider Ricci flow of complete Riemannian manifolds which have bounded non-negative curvature operator, non-zero asymptotic volume ratio and no boundary. We prove scale invariant estimates for these solutions. Using these estimates, we show that there is a limit solution, obtained by scaling down this solution at a fixed point in space. This limit solution is an expanding soliton coming out of the asymptotic cone at infinity.     

The conjugate heat equation and Ancient solutions of the Ricci flow

We prove Gaussian type bounds for the fundamental solution of the conjugate heat equation evolving under the Ricci flow. As a consequence, for dimension 4 and higher, we show that the backward limit of Type I κ-solutions of the Ricci flow must be a non-flat gradient shrinking Ricci soliton. This extends Perelman?s previous result on backward limits of κ-solutions in dimension 3, in which case the curvature operator is nonnegative (it follows from Hamilton–Ivey curvature pinching estimate). As an application, this also addresses an issue left in Naber (2010) [23], where Naber proves the interesting result that there exists a Type I dilation limit that converges to a gradient shrinking Ricci soliton, but that soliton might be flat. The Gaussian bounds that we obtain on the fundamental solution of the conjugate heat equation under evolving metric might be of independent interest.     

The KdV soliton crosses a dissipative and dispersive border

We demonstrate the behavior of the soliton which, while moving in non-dissipative and dispersion-constant medium encounters a finite-width barrier with varying dissipation and/or dispersion; beyond the layer dispersion is constant (but not necessarily of the same value) and dissipation is null. The transmitted wave either retains the form of a soliton (though of different parameters) or scatters a into a number of them. And a reflection wave may be negligible or absent. This models a situation similar to a light passing from a humid air to a dry one through the vapor saturation/condensation area. Some rough estimations for a prediction of an output are given using the relative decay (or accumulation) of the KdV conserved quantities in a dissipative area; in particular for a restriction for a number of solitons in the transmitted signal.     

On the Potential Function of Gradient Steady Ricci Solitons

In this paper, we study the potential function of gradient steady Ricci solitons. We prove that the infimum of the potential function decays linearly. As a consequence, we show that a gradient steady Ricci soliton with bounded potential function must be trivial, and that no gradient steady Ricci soliton admits uniformly positive scalar curvature.     

Dark solitons in external potentials

We consider the persistence and stability of dark solitons in the Gross–Pitaevskii (GP) equation with a small decaying potential. We show that families of black solitons with zero speed originate from extremal points of an appropriately defined effective potential and persist for sufficiently small strength of the potential. We prove that families at the maximum points are generally unstable with exactly one real positive eigenvalue, while families at the minimum points are generally unstable with exactly two complex-conjugated eigenvalues with positive real part. This mechanism of destabilization of the black soliton is confirmed in numerical approximations of eigenvalues of the linearized GP equation and full numerical simulations of the nonlinear GP equation. We illustrate the monotonic instability associated with the real eigenvalues and the oscillatory instability associated with the complex eigenvalues and compare the numerical results of evolution of a dark soliton with the predictions of Newton’s particle law for its position.      

Spatiotemporal Chaotic Dynamics of Solitons with Internal Structure in the Presence of Finite-Width Inhomogeneities

We present an analytical and numerical study of the Klein–Gordon kink-soliton dynamics in inhomogeneous media. In particular, we study an external field that is almost constant for the whole system but that changes its sign at the center of coordinates and a localized impurity with finite-width. The soliton solution of the Klein–Gordon-like equations is usually treated as a structureless point-like particle. A richer dynamics is unveiled when the extended character of the soliton is taken into account. We show that interesting spatiotemporal phenomena appear when the structure of the soliton interacts with finite-width inhomogeneities. We solve an inverse problem in order to have external perturbations which are generic and topologically equivalent to well-known bifurcation models and such that the stability problem can be solved exactly. We also show the different quasiperiodic and chaotic motions the soliton undergoes as a time-dependent force pumps energy into the traslational mode of the kink and relate these dynamics with the excitation of the shape modes of the soliton.     

On the Chern‐Ricci flow and its solitons for Lie groups

This paper is concerned with Chern‐Ricci flow evolution of left‐invariant hermitian structures on Lie groups. We study the behavior of a solution, as t is approaching the first time singularity, by rescaling in order to prevent collapsing and obtain convergence in the pointed (or Cheeger‐Gromov) sense to a Chern‐Ricci soliton. We give some results on the Chern‐Ricci form and the Lie group structure of the pointed limit in terms of the starting hermitian metric and, as an application, we obtain a complete picture for the class of solvable Lie groups having a codimension one normal abelian subgroup. We have also found a Chern‐Ricci soliton hermitian metric on most of the complex surfaces which are solvmanifolds, including an unexpected shrinking soliton example.     

Dark solitons in external potentials

We consider the persistence and stability of dark solitons in the Gross–Pitaevskii (GP) equation with a small decaying potential. We show that families of black solitons with zero speed originate from extremal points of an appropriately defined effective potential and persist for sufficiently small strength of the potential. We prove that families at the maximum points are generally unstable with exactly one real positive eigenvalue, while families at the minimum points are generally unstable with exactly two complex-conjugated eigenvalues with positive real part. This mechanism of destabilization of the black soliton is confirmed in numerical approximations of eigenvalues of the linearized GP equation and full numerical simulations of the nonlinear GP equation. We illustrate the monotonic instability associated with the real eigenvalues and the oscillatory instability associated with the complex eigenvalues and compare the numerical results of evolution of a dark soliton with the predictions of Newton’s particle law for its position.     

Invariant Measures and the Soliton Resolution Conjecture

The soliton resolution conjecture for the focusing nonlinear Schrödinger equation (NLS) is the vaguely worded claim that a global solution of the NLS, for generic initial data, will eventually resolve into a radiation component that disperses like a linear solution, plus a localized component that behaves like a soliton or multisoliton solution. Considered to be one of the fundamental open problems in the area of nonlinear dispersive equations, this conjecture has eluded a proof or even a precise formulation to date. This paper proves a “statistical version” of this conjecture at mass‐subcritical nonlinearity, in the following sense: The uniform probability distribution on the set of all functions with a given mass and energy, if such a thing existed, would be a natural invariant measure for the NLS flow and would reflect the long‐term behavior for “generic initial data” with that mass and energy. Unfortunately, such a probability measure does not exist. We circumvent this problem by constructing a sequence of discrete measures that, in principle, approximate this fictitious probability distribution as the grid size goes to 0. We then show that a continuum limit of this sequence of probability measures does exist in a certain sense, and in agreement with the soliton resolution conjecture, the limit measure concentrates on the unique ground state soliton. Combining this with results from ergodic theory, we present a tentative formulation and proof of the soliton resolution conjecture in the discrete setting. The above results, following in the footsteps of a program of studying the long‐term behavior of nonlinear dispersive equations through their natural invariant measures initiated by Lebowitz, Rose, and Speer and carried forward by Bourgain, McKean, Tzvetkov, Oh, and others, are proved using a combination of techniques from large deviations, PDEs, harmonic analysis, and bare‐hands probability theory. It is valid in any dimension. © 2014 Wiley Periodicals, Inc.     

Soliton versus the gas: Fredholm determinants,analysis, and the rapid oscillations behind the kinetic equation

We analyse the case of a dense modified Korteweg–de Vries (mKdV) soliton gas and its large time behaviour in the presence of a single trial soliton. We show that the solution can be expressed in terms of Fredholm determinants as well as in terms of a Riemann–Hilbert problem. We then show that the solution can be decomposed as the sum of the background gas solution (a modulated elliptic wave), plus a soliton solution: the individual expressions are however quite convoluted due to the interaction dynamics. Additionally, we are able to derive the local phase shift of the gas after the passage of the soliton, and we can trace the location of the soliton peak as the dynamics evolves. Finally, we show that the soliton peak, while interacting with the soliton gas, has an oscillatory velocity whose leading order average value satisfies the kinetic velocity equation analogous to the one posited by V. Zakharov and G. El.     

Dynamics of Embedded Solitons in the Extended Korteweg–de Vries Equations

Embedded solitons are solitary waves residing inside the continuous spectrum of a wave system. They have been discovered in a wide array of physical situations recently. In this article, we present the first comprehensive theory on the dynamics of embedded solitons and nonlocal solitary waves in the framework of the perturbed fifth-order Korteweg–de Vries (KdV) hierarchy equation. Our method is based on the development of a soliton perturbation theory. By obtaining the analytical formula for the tail amplitudes of nonlocal solitary waves, we demonstrate the existence of single-hump embedded solitons for both Hamiltonian and non-Hamiltonian perturbations. These embedded solitons can be isolated (existing at a unique wave speed) or continuous (existing at all wave speeds). Under small wave speed limit, our results show that the tail amplitudes of nonlocal waves are exponentially small, and the product of the amplitude and cosine of the phase is a constant to leading order. This qualitatively reproduces the previous results on the fifth-order KdV equation obtained by exponential asymptotics techniques. We further study the dynamics of embedded solitons and prove that, under Hamiltonian perturbations, a localized wave initially moving faster than the embedded soliton will asymptotically approach this embedded soliton, whereas a localized wave moving slower than the embedded soliton will decay into radiation. Thus, the embedded soliton is semistable. Under non-Hamiltonian perturbations, stable embedded solitons are found for the first time.     

Some Applications of the Hodge-de Rham Decomposition to Ricci Solitons

The aim of this paper is to present a link between the Perelman potential for a compact Ricci soliton M ⁿ and the Hodge-de Rham decomposition theorem, we shall use this result to present an integral formula which enables us to establish conditions under which the Ricci soliton is trivial. Moreover, given a Ricci soliton such that its associated vector field X is a conformal vector field we show that in the compact case X is a Killing vector field, while for the non-compact case, either the soliton is Gaussian or X is a Killing vector field.     

A numerical method for calculating solitons of the nonlinear Schödinger equation in the axially symmetric case

A numerical method is proposed for determination of the eigenfunctions and eigenvalues of the nonlinear Schrödinger equation in the axially symmetric case. Optical solitons interpreted in the physical sense are found for various values of the nonlinearity coefficient by means of the developed method. As has previously been shown by other authors, such solitons are unstable under small perturbations of their shape. Since the considered problem finds numerous applications, methods providing for soliton stabilization are widely discussed in the literature. One of these methods involves strong modulation of the medium nonlinearity or even the reversal of the nonlinearity sign, which necessitates taking into account the wave reflected from irregularities and analyzing additionally the applicability of the mathematical model. We show that, theoretically, it is possible to stabilize a soliton via weak modulation of the cubic-nonlinearity coefficient. Such modulation ensures alternation of the length of nonlinear layers and enables one to increase the path length by a factor of 70 without a beam collapse.     

How to excite the internal modes of sine-Gordon solitons

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 behaves as a deformable (non-rigid) object. A soliton moving in an array of inhomogeneities can also present sustained oscillations of its width. There are very important phenomena (like the soliton–antisoliton collisions) where the existence of internal modes plays a crucial role. We show that, under some conditions, the dynamics of the soliton shape modes can be chaotic. A short report of some of our results has been published in [Phys. Rev. E 65 (2002) 065601(R)].     

Chiral solitons in a quantum potential

We study chiral solitons in a quantum potential using a dimensional reduction of the problem for (2+1)-dimensional anyons. We show that the integrable version of the model is described by a family of the resonant derivative nonlinear Schrödinger equations. For a quantum potential strength s > 1, we show that the chiral soliton interaction has a resonance. We investigate the semiclassical quantization procedure for solitons.     

Soliton chaos and lengthscale competition in nonlinear dynamics

Perturbing soliton-bearing completely integrable dynamics can give rise to rich and fascinating behaviour. If the perturbation introduces a lengthscale which is large compared to the spatial extent of the solitons present in the system, the solitons move like particles in an effective potential. Taking into account two-soliton interaction can result in chaotic behaviour called ‘soliton chaos’. In the opposite limit of a small-lengthscale perturbation the solitons acquire a dressing which effectively shields them from the perturbation. If the resulting ‘dressed solitons’ are subject to an additional long-wavelength perturbation they move like renormalised particles. Furthermore they can scatter nearly elastically. If the perturbation contains lengthscales which are comparable to one of the soliton's typical lengthscales then lengthscale competition can occur. Neither the particle approximation nor the dressed-particle approximation for the soliton is valid and complicated spatio-temporal behaviour is observed. We illustrate this scenario by means of the perturbed nonlinear Schrödinger equation. The perturbed sine-Gordon equation and the Ablowitz-Ladik equation are also discussed.     

Bright soliton solution of a Gross–Pitaevskii equation

We theoretically and numerically study the bright soliton solutions of a Gross–Pitaevskii equation governing one-dimensional (1D)(cigar-shaped) Bose–Einstein condensates (BEC) trapped in an optical lattice of 1D structure. The analytical expression of bright soliton is derived by using the variational approximation, which completely matches the numerical results with a range of potential’s parameters. Moreover, we determined the parameter domains for the persistence and non-persistence of bright soliton solutions.