arXiv subjects

Dmitry Pelinovsky

Publications and source records attributed to Dmitry Pelinovsky.

51 records · Page 3Linked to original sources

arXiv2008

Eigenvalues of a nonlinear ground state in the Thomas-Fermi approximation

We study a nonlinear ground state of the Gross-Pitaevskii equation with a parabolic potential in the hydrodynamics limit often referred to as the Thomas--Fermi approximation. Existence of the energy minimizer has been known in literature for some time but it was only recently when the Thomas-Fermi approximation was rigorously justified. The spectrum of linearization of the Gross-Pitaevskii equation at the ground state consists of an unbounded sequence of positive eigenvalues. We analyze convergence of eigenvalues in the hydrodynamics limit. Convergence in norm of the resolvent operator is proved and the convergence rate is estimated. We also study asymptotic and numerical approximations of eigenfunctions and eigenvalues using Airy functions.

math-ph↗

arXiv2008

Coupled-mode equations and gap solitons in a two-dimensional nonlinear elliptic problem with a separable periodic potential

We address a two-dimensional nonlinear elliptic problem with a finite-amplitude periodic potential. For a class of separable symmetric potentials, we study the bifurcation of the first band gap in the spectrum of the linear Schrödinger operator and the relevant coupled-mode equations to describe this bifurcation. The coupled-mode equations are derived by the rigorous analysis based on the Fourier--Bloch decomposition and the Implicit Function Theorem in the space of bounded continuous functions vanishing at infinity. Persistence of reversible localized solutions, called gap solitons, beyond the coupled-mode equations is proved under a non-degeneracy assumption on the kernel of the linearization operator. Various branches of reversible localized solutions are classified numerically in the framework of the coupled-mode equations and convergence of the approximation error is verified. Error estimates on the time-dependent solutions of the Gross--Pitaevskii equation and the coupled-mode equations are obtained for a finite-time interval.

math.AP↗

arXiv2007

Bounds on the tight-binding approximation for the Gross-Pitaevskii equation with a periodic potential

We justify the validity of the discrete nonlinear Schrodinger equation for the tight-binding approximation in the context of the Gross-Pitaevskii equation with a periodic potential. Our construction of the periodic potential and the associated Wannier functions is based on the previous work, while our analysis involving energy estimates and Gronwall's inequality addresses time-dependent localized solutions on large but finite time intervals.

math-ph↗

arXiv2007

Moving gap solitons in periodic potentials

We address existence of moving gap solitons (traveling localized solutions) in the Gross-Pitaevskii equation with a small periodic potential. Moving gap solitons are approximated by the explicit localized solutions of the coupled-mode system. We show however that exponentially decaying traveling solutions of the Gross-Pitaevskii equation do not generally exist in the presence of a periodic potential due to bounded oscillatory tails ahead and behind the moving solitary waves. The oscillatory tails are not accounted in the coupled-mode formalism and are estimated by using techniques of spatial dynamics and local center-stable manifold reductions. Existence of bounded traveling solutions of the Gross--Pitaevskii equation with a single bump surrounded by oscillatory tails on a finite large interval of the spatial scale is proven by using these technique. We also show generality of oscillatory tails in other nonlinear equations with a periodic potential.

math.AP↗

arXiv2007

Justification of the coupled-mode approximation for a nonlinear elliptic problem with a periodic potential

Coupled-mode systems are used in physical literature to simplify the nonlinear Maxwell and Gross-Pitaevskii equations with a small periodic potential and to approximate localized solutions called gap solitons by analytical expressions involving hyperbolic functions. We justify the use of the one-dimensional stationary coupled-mode system for a relevant elliptic problem by employing the method of Lyapunov--Schmidt reductions in Fourier space. In particular, existence of periodic/anti-periodic and decaying solutions is proved and the error terms are controlled in suitable norms. The use of multi-dimensional stationary coupled-mode systems is justified for analysis of bifurcations of periodic/anti-periodic solutions in a small multi-dimensional periodic potential.

math.AP↗

arXiv2007

Spectrum of a non-self-adjoint operator associated with the periodic heat equation

We study the spectrum of the linear operator $L = - \partial_θ - ε\partial_θ (\sin θ\partial_θ)$ subject to the periodic boundary conditions on $θ\in [-π,π]$. We prove that the operator is closed in $L^2([-π,π])$ with the domain in $H^1_{\rm per}([-π,π])$ for $|ε| < 2$, its spectrum consists of an infinite sequence of isolated eigenvalues and the set of corresponding eigenfunctions is complete. By using numerical approximations of eigenvalues and eigenfunctions, we show that all eigenvalues are simple, located on the imaginary axis and the angle between two subsequent eigenfunctions tends to zero for larger eigenvalues. As a result, the complete set of linearly independent eigenfunctions does not form a basis in $H^1_{\rm per}([-π,π])$.

math-ph↗

arXiv2006

Lyapunov--Schmidt reduction algorithm for three-dimensional discrete vortices

We address persistence and stability of three-dimensional vortex configurations in the discrete nonlinear Schrödinger (NLS) equation and develop a symbolic package based on Wolfram's MATHEMATICA for computations of the Lyapunov--Schmidt reduction method. The Lyapunov--Schmidt reduction method is a theoretical tool which enables us to study continuations and terminations of the discrete vortices for small coupling between lattice nodes as well as the spectral stability of the persistent configurations. The method was developed earlier in the context of the two-dimensional NLS lattice and applied to the on-site and off-site configurations (called the vortex cross and the vortex cell) by using semi-analytical computations. The present treatment develops a full symbolic computational package which takes a desired waveform at the anti-continuum limit of uncoupled sites, performs a required number of Lyapunov--Schmidt reductions and outputs the predictions on whether the configuration persists, for finite coupling, in the three-dimensional lattice and whether it is stable or unstable. It also provides approximations for the eigenvalues of the linearized stability problem. We report a number of applications of the algorithm to important multi-site configurations, such as the simple cube, the double cross, and the diamond. For each three-dimensional configuration, we identify exactly one solution, which is stable for small coupling between lattice nodes.

nlin.PS↗

arXiv2006

Nonlinear instability of a critical traveling wave in the generalized Korteweg -- de Vries equation

We prove the instability of a ``critical'' solitary wave of the generalized Korteweg -- de Vries equation, the one with the speed at the border between the stability and instability regions. The instability mechanism involved is ``purely nonlinear'', in the sense that the linearization at a critical soliton does not have eigenvalues with positive real part. We prove that critical solitons correspond generally to the saddle-node bifurcation of two branches of solitons.

math.AP↗

arXiv2006

Two-pulse solutions in the fifth-order KdV equation : rigorous theory and numerical approximations

We revisit existence and stability of two-pulse solutions in the fifth-order Korteweg--de Vries (KdV) equation with two new results. First, we modify the Petviashvili method of successive iterations for numerical (spectral) approximations of pulses and prove convergence of iterations in a neighborhood of two-pulse solutions. Second, we prove structural stability of embedded eigenvalues of negative Krein signature in a linearized KdV equation. Combined with stability analysis in Pontryagin spaces, this result completes the proof of spectral stability of the corresponding two-pulse solutions. Eigenvalues of the linearized problem are approximated numerically in exponentially weighted spaces where embedded eigenvalues are isolated from the continuous spectrum. Approximations of eigenvalues and full numerical simulations of the fifth-order KdV equation confirm stability of two-pulse solutions related to the minima of the effective interaction potential and instability of two-pulse solutions related to the maxima points.

nlin.PS↗

arXiv2006

Translationally invariant nonlinear Schrodinger lattices

Persistence of stationary and traveling single-humped localized solutions in the spatial discretizations of the nonlinear Schrodinger (NLS) equation is addressed. The discrete NLS equation with the most general cubic polynomial function is considered. Constraints on the nonlinear function are found from the condition that the second-order difference equation for stationary solutions can be reduced to the first-order difference map. The discrete NLS equation with such an exceptional nonlinear function is shown to have a conserved momentum but admits no standard Hamiltonian structure. It is proved that the reduction to the first-order difference map gives a sufficient condition for existence of translationally invariant single-humped stationary solutions and a necessary condition for existence of single-humped traveling solutions. Other constraints on the nonlinear function are found from the condition that the differential advance-delay equation for traveling solutions admits a reduction to an integrable normal form given by a third-order differential equation. This reduction also gives a necessary condition for existence of single-humped traveling solutions. The nonlinear function which admits both reductions defines a two-parameter family of discrete NLS equations which generalizes the integrable Ablowitz--Ladik lattice.

nlin.PS↗

arXiv2005

Normal form for travelling kinks in discrete Klein-Gordon lattices

We study travelling kinks in the spatial discretizations of the nonlinear Klein--Gordon equation, which include the discrete $ϕ^4$ lattice and the discrete sine--Gordon lattice. The differential advance-delay equation for travelling kinks is reduced to the normal form, a scalar fourth-order differential equation, near the quadruple zero eigenvalue. We show numerically non-existence of monotonic kinks (heteroclinic orbits between adjacent equilibrium points) in the fourth-order equation. Making generic assumptions on the reduced fourth-order equation, we prove the persistence of bounded solutions (heteroclinic connections between periodic solutions near adjacent equilibrium points) in the full differential advanced-delay equation with the technique of center manifold reduction. Existence and persistence of multiple kinks in the discrete sine--Gordon equation are discussed in connection to recent numerical results of \cite{ACR03} and results of our normal form analysis.

math.DS↗

arXiv2005

Instabilities of multi-hump vector solitons in coupled nonlinear Schroedinger equations

Spectral stability of multi-hump vector solitons in the Hamiltonian system of coupled nonlinear Schrödinger (NLS) equations is investigated both analytically and numerically. Using the closure theorem for the negative index of the linearized Hamiltonian, we classify all possible bifurcations of unstable eigenvalues in the systems of coupled NLS equations with cubic and saturable nonlinearities. We also determine the eigenvalue spectrum numerically by the shooting method. In case of cubic nonlinearities, all multi-hump vector solitons in the non-integrable model are found to be linearly unstable. In case of saturable nonlinearities, stable multi-hump vector solitons are found in certain parameter regions, and some errors in the literature are corrected.

nlin.PS↗

arXiv2005

Spectral stability and time evolution of N-solitons in KdV hierarchy

This paper concerns spectral stability and time evolution of $N$-solitons in the KdV hierarchy with mixed commuting time flows. Spectral stability problem is analyzed by using a pair of self-adjoint operators with finite numbers of negative eigenvalues. We show that the absence of unstable eigenvalues in the stability problem is related to the absence of negative eigenvalues of these operators in the constrained function spaces. Time evolution of $N$-solitons is uniquely characterized from the inverse scattering transform technique.

nlin.SI↗

arXiv2002

Semi-stability of embedded solitons in the general fifth-order KdV equation

Evolution of perturbed embedded solitons in the general Hamiltonian fifth-order Korteweg--de Vries (KdV) equation is studied. When an embedded soliton is perturbed, it sheds a one-directional continuous-wave radiation. It is shown that the radiation amplitude is not minimal in general. A dynamical equation for velocity of the perturbed embedded soliton is derived. This equation shows that a neutrally stable embedded soliton is in fact semi-stable. When the perturbation increases the momentum of the embedded soliton, the perturbed state approaches asymptotically the embedded soliton, while when the perturbation reduces the momentum of the embedded soliton, the perturbed state decays into radiation. Classes of initial conditions to induce soliton decay or persistence are also determined. Our analytical results are confirmed by direct numerical simulations of the fifth-order KdV equation.

nlin.PS↗

arXiv2000

Instabilities of dispersion-managed solitons in the normal dispersion regime

Dispersion-managed solitons are reviewed within a Gaussian variational approximation and an integral evolution model. In the normal regime of the dispersion map (when the averaged path dispersion is negative), there are two solitons of different pulse duration and energy at a fixed propagation constant. We show that the short soliton with a larger energy is linearly (exponentially) unstable. The other (long) soliton with a smaller energy is linearly stable but hits a resonance with excitations of the dispersion map. The results are compared with the results from the recent publications

nlin.PS↗