arXiv subjects

Dmitry Pelinovsky

Publications and source records attributed to Dmitry Pelinovsky.

At least 19 recordsLinked to original sources

arXiv2025

On the ground state of the nonlinear Schr{\"o}dinger equation: asymptotic behavior at the endpoint powers

We consider the ground states of the nonlinear Schr{\"o}dinger equation, which stand for radially symmetric and exponentially decaying solutions on the full space. We investigate their behaviors at both endpoint powers of the nonlinearity, up to some rescaling to infer non-trivial limits. One case corresponds to the limit towards a Gaussian function called Gausson, which is the ground state of the stationary logarithmic Schr{\"o}dinger equation. The other case, for dimension at least three, corresponds to the limit towards the Aubin-Talenti algebraic soliton. We prove strong convergence with explicit bounds for both cases, and provide detailed asymptotics. These theoretical results are illustrated with numerical approximations.

math.AP↗

arXiv2025

Stability of dark solitons in a bubble Bose-Einstein condensate

The stability of nonlinear waves on curved surfaces is a problem of widespread interest across physics. Here, we establish the stability criteria for dark solitons on a spherical Bose-Einstein condensate. We demonstrate a sharp instability threshold in the nonlinear parameter, beyond which solitons decay into vortex dipoles via snake instabilities. Analytically and numerically, we prove this decay is dictated by a single unstable mode for each angular momentum $m \geq 2$, which is a universal mechanism that controls the resulting vortex state. Unlike in the full three-dimensional case, where snake instabilities lead to vortex rings, a dark soliton confined to the surface of a bubble can only decay into vortex pairs.

cond-mat.quant-gas↗

arXiv2023

Pinning in the extended Lugiato-Lefever equation

We consider a variant of the Lugiato-Lefever equation (LLE), which is a nonlinear Schrödinger equation on a one-dimensional torus with forcing and damping, to which we add a first-order derivative term with a potential $εV(x)$. The potential breaks the translation invariance of LLE. Depending on the existence of zeroes of the effective potential $V_\text{eff}$, which is a suitably weighted and integrated version of $V$, we show that stationary solutions from $ε=0$ can be continued locally into the range $ε\not =0$. Moreover, the extremal points of the $ε$-continued solutions are located near zeros of $V_\text{eff}$. We therefore call this phenomenon \emph{pinning} of stationary solutions. If we assume additionally that the starting stationary solution at $ε=0$ is spectrally stable with the simple zero eigenvalue due to translation invariance being the only eigenvalue on the imaginary axis, we can prove asymptotic stability or instability of its $ε$-continuation depending on the sign of $V_\text{eff}'$ at the zero of $V_\text{eff}$ and the sign of $ε$. The variant of the LLE arises in the description of optical frequency combs in a Kerr nonlinear ring-shaped microresonator which is pumped by two different continuous monochromatic light sources of different frequencies and different powers. Our analytical findings are illustrated by numerical simulations.

math.AP↗

arXiv2021

Linear instability of breathers for the focusing nonlinear Schr{ö}dinger equation

Relying upon tools from the theory of integrable systems, we discuss the linear instability of the Kuznetsov-Ma breathers and the Akhmediev breathers of the focusing nonlinear Schr{ö}dinger equation. We use the Darboux transformation to construct simultaneously the breathers and the exact solutions of the Lax system associated with the breathers. We obtain a full description of the Lax spectra for the two breathers, including multiplicities of eigenvalues. Solutions of the linearized NLS equations are then obtained from the eigenfunctions and generalized eigenfunctions of the Lax system. While we do not attempt to prove completeness of eigenfunctions, we aim to determine the entire set of solutions of the linearized NLS equations generated by the Lax system in appropriate function spaces.

math.AP↗

arXiv2017

Ground state of the conformal flow on $\mathbb{S}^3$

We consider the conformal flow model derived by Bizoń, Craps, Evnin, Hunik, Luyten, and Maliborski [Commun. Math. Phys. 353 (2017) 1179-1199] as a normal form for the conformally invariant cubic wave equation on $\mathbb{S}^3$. We prove that the energy attains a global constrained maximum at a family of particular stationary solutions which we call the ground state family. Using this fact and spectral properties of the linearized flow (which are interesting on their own due to a supersymmetric structure) we prove nonlinear orbital stability of the ground state family. The main difficulty in the proof is due to the degeneracy of the ground state family as a constrained maximizer of the energy.

math.AP↗

arXiv2016

Energy criterion for the spectral stability of discrete breathers

Discrete breathers are ubiquitous structures in nonlinear anharmonic models ranging from the prototypical example of the Fermi-Pasta-Ulam model to Klein-Gordon nonlinear lattices, among many others. We propose a general criterion for the emergence of instabilities of discrete breathers analogous to the well-established Vakhitov-Kolokolov criterion for solitary waves. The criterion involves the change of monotonicity of the discrete breather's energy as a function of the breather frequency. Our analysis suggests and numerical results corroborate that breathers with increasing (decreasing) energy-frequency dependence are generically unstable in soft (hard) nonlinear potentials.

nlin.PS↗

arXiv2016

Long-time stability of small FPU solitary waves

Small-amplitude waves in the Fermi-Pasta-Ulam (FPU) lattice with weakly anharmonic interaction potentials are described by the generalized Korteweg-de Vries (KdV) equation. Justification of the small-amplitude approximation is usually performed on the time scale, for which dynamics of the KdV equation is defined. We show how to extend justification analysis on longer time intervals provided dynamics of the generalized KdV equation is globally well-posed in Sobolev spaces and either the Sobolev norms are globally bounded or they grow at most polynomially. The time intervals are extended respectively by the logarithmic or double logarithmic factors in terms of the small amplitude parameter. Controlling the approximation error on longer time intervals allows us to deduce nonlinear metastability of small FPU solitary waves from orbital stability of the KdV solitary waves.

math.DS↗

arXiv2014

Bifurcations and stability of standing waves in the nonlinear Schrödinger equation on the tadpole graph

We develop a detailed rigorous analysis of edge bifurcations of standing waves in the nonlinear Schrödinger (NLS) equation on a tadpole graph (a ring attached to a semi-infinite line subject to the Kirchhoff boundary conditions at the junction). It is shown in the recent work [7] by using explicit Jacobi elliptic functions that the cubic NLS equation on a tadpole graph admits a rich structure of standing waves. Among these, there are different branches of localized waves bifurcating from the edge of the essential spectrum of an associated Schrödinger operator. We show by using a modified Lyapunov-Schmidt reduction method that the bifur- cation of localized standing waves occurs for every positive power nonlinearity. We distinguish a primary branch of never vanishing standing waves bifurcating from the trivial solution and an infinite sequence of higher branches with oscillating behavior in the ring. The higher branches bifurcate from the branches of degenerate standing waves with vanishing tail outside the ring. Moreover, we analyze stability of bifurcating standing waves. Namely, we show that the primary branch is composed by orbitally stable standing waves for subcritical power nonlinearities, while all nontrivial higher branches are linearly unstable near the bifurcation point. The stability character of the degenerate branches remains inconclusive at the analytical level, whereas heuristic arguments based on analysis of embedded eigenvalues of negative Krein signatures support the conjecture of their linear instability at least near the bifurcation point. Numerical results for the cubic NLS equation show that this conjecture is valid and that the degenerate branches become spectrally stable far away from the bifurcation point.

math-ph↗

arXiv2014

Orbital stability in the cubic defocusing NLS equation: I. Cnoidal periodic waves

Periodic waves of the one-dimensional cubic defocusing NLS equation are considered. Using tools from integrability theory, these waves have been shown in [Bottman, Deconinck, and Nivala, 2011] to be linearly stable and the Floquet-Bloch spectrum of the linearized operator has been explicitly computed. We combine here the first four conserved quantities of the NLS equation to give a direct proof that cnoidal periodic waves are orbitally stable with respect to subharmonic perturbations, with period equal to an integer multiple of the period of the wave. Our result is not restricted to the periodic waves of small amplitudes.

math.AP↗

arXiv2014

Orbital stability in the cubic defocusing NLS equation: II. The black soliton

Combining the usual energy functional with a higher-order conserved quantity originating from integrability theory, we show that the black soliton is a local minimizer of a quantity that is conserved along the flow of the cubic defocusing NLS equation in one space dimension. This unconstrained variational characterization gives an elementary proof of the orbital stability of the black soliton with respect to perturbations in $H^2(\mathbb{R})$.

math.AP↗

arXiv2014

Transverse instability of line solitons in massive Dirac equations

Working in the context of localized modes in periodic potentials, we consider two systems of the massive Dirac equations in two spatial dimensions. The first system, a generalized massive Thirring model, is derived for the periodic stripe potentials. The second one, a generalized massive Gross--Neveu equation, is derived for the hexagonal potentials. In both cases, we prove analytically that the line solitons suffer from instability with respect to periodic transverse perturbations of large periods. The instability is induced by the spatial translation for the massive Thirring model and by the gauge rotation for the massive Gross--Neveu model. We also observe numerically that the instability holds for the transverse perturbations of any period in the massive Thirring model and exhibits a finite threshold on the period of the transverse perturbations in the massive Gross--Neveu model.

math.AP↗

arXiv2014

On the orbital stability of Gaussian solitary waves in the log-KdV equation

We consider the logarithmic Korteweg-de Vries (log-KdV) equation, which models solitary waves in anharmonic chains with Hertzian interaction forces. By using an approximating sequence of global solutions of the regularized generalized KdV equation in $H^1(\mathbb{R})$ with conserved $L^2$ norm and energy, we construct a weak global solution of the log-KdV equation in a subset of $H^1(\mathbb{R})$. This construction yields conditional orbital stability of Gaussian solitary waves of the log-KdV equation, provided uniqueness and continuous dependence of the constructed solution holds. Furthermore, we study the linearized log-KdV equation at the Gaussian solitary wave and prove that the associated linearized operator has a purely discrete spectrum consisting of simple purely imaginary eigenvalues in addition to the double zero eigenvalue. The eigenfunctions, however, do not decay like Gaussian functions but have algebraic decay. Nevertheless, using numerical approximations, we show that the Gaussian initial data do not spread out and preserve their spatial Gaussian decay in the time evolution of the linearized log-KdV equation.

math.AP↗

arXiv2014

On the Thomas-Fermi approximation of the ground state in a PT-symmetric confining potential

For the stationary Gross-Pitaevskii equation with harmonic real and linear imaginary potentials in the space of one dimension, we study the ground state in the limit of large densities (large chemical potentials), where the solution degenerates into a compact Thomas-Fermi approximation. We prove that the Thomas-Fermi approximation can be constructed with an invertible coordinate transformation and an unstable manifold theorem for a planar dynamical system. The Thomas-Fermi approximation can be justified by reducing the existence problem to the Painlevé-II equation, which admits a unique global Hastings-McLeod solution. We illustrate numerically that an iterative approach to solving the existence problem converges but give no analytical proof of this result. Generalizations are discussed for the stationary Gross-Pitaevskii equation with harmonic real and localized imaginary potentials.

math.AP↗

arXiv2014

Justification of the log-KdV equation in granular chains: the case of precompression

For travelling waves with nonzero boundary conditions, we justify the logarithmic Korteweg-de Vries equation as the leading approximation of the Fermi-Pasta-Ulam lattice with Hertzian nonlinear potential in the limit of small anharmonicity. We prove control of the approximation error for the travelling solutions satisfying differential advance-delay equations, as well as control of the approximation error for time-dependent solutions to the lattice equations on long but finite time intervals. We also show nonlinear stability of the travelling waves on long but finite time intervals.

math.AP↗

arXiv2013

Domain walls in the coupled Gross-Pitaevskii equations

A thorough study of domain wall solutions in coupled Gross-Pitaevskii equations on the real line is carried out including existence of these solutions; their spectral and nonlinear stability; their persistence and stability under a small localized potential. The proof of existence is variational and is presented in a general framework: we show that the domain wall solutions are energy minimizing within a class of vector-valued functions with nontrivial conditions at infinity. The admissible energy functionals include those corresponding to coupled Gross--Pitaevskii equations, arising in modeling of Bose-Einstein condensates. The results on spectral and nonlinear stability follow from properties of the linearized operator about the domain wall. The methods apply to many systems of interest and integrability is not germane to our analysis. Finally, sufficient conditions for persistence and stability of domain wall solutions are obtained to show that stable pinning occurs near maxima of the potential, thus giving rigorous justification to earlier results in the physics literature.

math.AP↗

arXiv2013

Universal power law for the energy spectrum of breaking Riemann waves

The universal power law for the spectrum of one-dimensional breaking Riemann waves is justified for the simple wave equation. The spectrum of spatial amplitudes at the breaking time $t = t_b$ has an asymptotic decay of $k^{-4/3}$, with corresponding energy spectrum decaying as $k^{-8/3}$. This spectrum is formed by the singularity of the form $(x-x_b)^{1/3}$ in the wave shape at the breaking time. This result remains valid for arbitrary nonlinear wave speed. In addition, we demonstrate numerically that the universal power law is observed for long time in the range of small wave numbers if small dissipation or dispersion is accounted in the viscous Burgers or Korteweg-de Vries equations.

math-ph↗

arXiv2013

Rigorous justification of the short-pulse equation

We prove that the short-pulse equation, which is derived from Maxwell equations with formal asymptotic methods, can be rigorously justified. The justification procedure applies to small-norm solutions of the short-pulse equation. Although the small-norm solutions exist for infinite times and include modulated pulses and their elastic interactions, the error bound for arbitrary initial data can only be controlled over finite time intervals.

math-ph↗

arXiv2013

Global existence of small-norm solutions in the reduced Ostrovsky equation

We use a novel transformation of the reduced Ostrovsky equation to the integrable Tzitzéica equation and prove global existence of small-norm solutions in Sobolev space $H^3(R)$. This scenario is an alternative to finite-time wave breaking of large-norm solutions of the reduced Ostrovsky equation. We also discuss a sharp sufficient condition for the finite-time wave breaking.

nlin.SI↗