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Boris Shakarov

Publications and source records attributed to Boris Shakarov.

13 recordsLinked to original sources
arXiv2026

Transverse Instability and Bifurcation Analysis of the line soliton for the NLS equation on a Fractured Strip

We consider the nonlinear Schr\"odinger equation on a two-dimensional strip with an attractive $\delta$ interaction and power nonlinearity. We investigate the transverse stability and bifurcation of line solitons as the width of the strip varies. We first establish local well-posedness in $H^1$, conservation of mass and energy, and global existence in the $H^1$-subcritical regime. We then identify a critical width $L_*$ at which the line soliton undergoes a transverse instability. More precisely, we prove orbital stability for $L L_*$. At the critical width, a simple eigenvalue of the linearized operator crosses zero, and we construct, via the Lyapunov-Schmidt reduction, a branch of positive nontrivial stationary solutions bifurcating from the line soliton. We determine the direction of this bifurcation by computing the second-order variation of the width along the branch. Finally, we investigate the orbital stability of the bifurcating solitons and obtain a stability criterion which can be evaluated in the regime of sufficiently small interaction strength.

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arXiv2026

Scattering for Defocusing NLS with Inhomogeneous Nonlinear Damping and Nonlinear Trapping Potential

We investigate an energy-subcritical defocusing nonlinear Schr\"odinger equation in $\mathbb R^3$ subject to a lower order nonlinear trapping potential and a spatially dependent nonlinear damping: \begin{equation*} i\partial_t u + \Delta u + i a(x) |u|^{2\sigma_2} u = |u|^{2\sigma_1} u + V(x)|u|^{2\sigma_3} u. \end{equation*} We prove that if the damping acts where $V$ induces concentration effects, i.e. where $V$ is either negative or non-repulsive, solutions are global and uniformly bounded in $H^1$, and scatter in the intercritical regime. A primary challenge arises from the spatial dependence of $a(x)$, which breaks the energy's monotonicity. Consequently, a uniform in time control of the $H^1$ norm of a solution is non-trivial and represents a new result even for $V = 0$. We overcome this issue by introducing a novel energy modified by virial argument, showing simultaneously a uniform bound on the energy and local energy decay estimates, which are subsequently upgraded to scattering via interaction Morawetz estimates.

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arXiv2026

On the defocusing stationary nonlinear Schr\"odinger equation on metric graphs

We study the defocusing nonlinear Schr\"odinger equation on noncompact metric graphs under general self-adjoint vertex conditions ensuring the existence of a negative eigenvalue of the Hamiltonian operator. First, we focus on the existence of energy ground states with prescribed mass. We show that existence and stability always hold for small masses and fail for large masses in the $L^2$-subcritical regime. For $\delta$-type vertex conditions, we provide more precise results: ground states exist for all masses in the $L^2$-critical and supercritical cases, while in the subcritical case, for one vertex graphs, there exists a sharp mass threshold such that ground states exist below it and do not exist above it. Moreover, we show that the ground state bifurcates from the vanishing solution at the bottom of the Hamiltonian spectrum. Finally, we present multiplicity results for stationary solutions, both in the fixed-frequency and fixed-mass settings.

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arXiv2025

Ground States for the Nonlinear Schr{\"o}dinger Equation on Open Books and Dimensional Reduction to Metric Graphs

In this work, we study the dimensional reduction of stationary states in the shrinking limit for a broad class of two-dimensional domains, called open books, to their counterparts on metric graphs. An open book is a two-dimensional structure formed by rectangular domains sharing common boundaries. We first develop a functional-analytic framework suited to variational problems on open books and establish the existence of solutions as constrained action minimizers. For graph-based open books (i.e., those isomorphic to the product of a graph with an interval) we prove the existence of a sharp transition in the dimensionality of ground states. Specifically, there exists a critical transverse width: below this threshold, all ground states coincide with the ground states on the underlying graph trivially extended in the transverse direction; above it, ground states become genuinely two-dimensional.

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arXiv2025

Ground States for the Defocusing Nonlinear Schr\"odinger Equation on Non-Compact Metric Graphs

We investigate the existence and stability of ground states for the defocusing nonlinear Schr\"odinger equation on non-compact metric graphs. We establish a sharp criterion for the existence of action ground states in terms of the spectral properties of the underlying Hamiltonian: ground states exist if and only if the bottom of the spectrum is negative and the frequency lies within a suitable range. We further explore the relation between action and energy ground states, showing that while every action minimizer yields an energy minimizer, the converse fails in general. In particular, we prove that energy ground states may not exist for arbitrary masses. This discrepancy is illustrated through explicit examples on star graphs with $\delta$ and $\delta'$-type vertex conditions: in the mass-subcritical case, we exhibit a large interval of masses for which no energy minimizer exists, whereas in the supercritical regime, energy ground states exist for all masses.

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arXiv2025

Scattering for defocusing cubic NLS under locally damped strong trapping

We are interested in the scattering problem for the cubic 3D nonlinear defocusing Schr\"odinger equation with variable coefficients. Previous scattering results for such problems address only the cases with constant coefficients or assume strong variants of the non-trapping condition, stating that all the trajectories of the Hamiltonian flow associated with the operator are escaping to infinity. In contrast, we consider the most general setting, where strong trapping, such as stable closed geodesics, may occur, but we introduce a compactly supported damping term localized in the trapping region, to explore how damping can mitigate the effects of trapping. In addition to the challenges posed by the trapped trajectories, notably the loss of smoothing and of scale-invariant Strichartz estimates, difficulties arise from the damping itself, particularly since the energy is not, a priori, bounded. For $H^{1+\epsilon}$ initial data -- chosen because the local-in-time theory is a priori no better than for 3D unbounded manifolds, where local well-posedness of strong $H^1$ solutions is unavailable -- we establish global existence and scattering in $H^{s}$ for any $0 \leq s <1$ in positive times, the inability to reach $H^1$ being related to the loss of smoothing due to trapping.

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arXiv2024

Ground states on a fractured strip and one dimensional reduction

We consider the nonlinear Schr\''odinger equation on a strip with Neumann boundary conditions and a delta condition on the $x$-axis. First, we show the existence of ground states as minimizers of the action or of the energy under suitable constraints. Second, we prove that the energy minimizers converge to the ground state on the line with a delta condition as the amplitude of the strip shrinks to zero.

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arXiv2024

Global solutions and Asymptotic Behavior to a Norm-preserving Non-local Parabolic Flow

We consider a nonlinear parabolic model that forces solutions to stay on a $L^2$-sphere through a nonlocal term in the equation. We study the local and global well-posedness on a bounded domain and the whole Euclidean space in the energy space. Then, we consider the solutions' asymptotic behavior. We prove strong convergence to a stationary state and asymptotic convergence to the ground state in bounded domains when the initial condition is positive.

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arXiv2023

On the formation of singularities for the slightly supercritical NLS equation with nonlinear damping

We consider the focusing, mass-supercritical NLS equation augmented with a nonlinear damping term. We provide sufficient conditions on the nonlinearity exponents and damping coefficients for finite-time blow-up. In particular, singularities are formed for focusing and dissipative nonlinearities of the same power, provided that the damping coefficient is sufficiently small. Our result thus rigorously proves the non-regularizing effect of nonlinear damping in the mass-supercritical case, which was suggested by previous numerical and formal results. We show that, under our assumption, the damping term may be controlled in such a way that the self-similar blow-up structure for the focusing NLS is approximately retained even within the dissipative evolution. The nonlinear damping contributes as a forcing term in the equation for the perturbation around the self-similar profile, that may produce a growth over finite time intervals. We estimate the error terms through a modulation analysis and a careful control of the time evolution of total momentum and energy functionals.

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arXiv2023

Existence and Large Time Behavior for a Dissipative Variant of the Rotational NLS Equation

We study a dissipative variant of the Gross-Pitaevskii equation with rotation. The model contains a nonlocal, nonlinear term that forces the conservation of $L^2$-norm of solutions. We are motivated by several physical experiments and numerical simulations studying the formation of vortices in Bose-Einstein condensates. We show local and global well-posedness of this model and investigate the asymptotic behavior of its solutions. In the linear case, the solution asymptotically tends to the eigenspace associated with the smallest eigenvalue in the decomposition of the initial datum. In the nonlinear case, we obtain weak convergence to a stationary state. Moreover, for initial energies in a specific range, we prove strong asymptotic stability of ground state solutions.

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arXiv2022

Stability of Cnoidal Waves for the Damped Nonlinear Schr\"odinger Equation

We consider the cubic nonlinear Schr\"odinger (NLS) equation with a linear damping on the one dimensional torus and we investigate the stability of some solitary wave profiles within the dissipative dynamics. The undamped cubic NLS equation is well known to admit a family of periodic waves given by Jacobi elliptic functions of cnoidal type. We show that the family of cnoidal waves is orbitally stable. More precisely, by considering a sufficiently small perturbation of a given cnoidal wave at initial time, the evolution will always remain close (up to symmetries of the equation) to the cnoidal wave whose mass is modulated according to the dissipative dynamics. This result extends the concept of orbital stability to this non-Hamiltonian evolution. Since cnoidal waves are not exact solutions to the damped NLS, the perturbation is forced away from the family of solitary wave profiles. In order to control this secular growth of the error, we find a first order approximation of the solitary wave that takes into account the dissipative term. Then we use a suitable, exponentially decreasing Lyapunov functional that controls the $H^1$-norm of the perturbation around the approximated solitons.

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arXiv2022

Existence and asymptotic behavior for $L^2$-norm preserving nonlinear heat equations

We consider a nonlinear parabolic equation with a nonlocal term, which preserves the $L^2$-norm of the solution. We study the local and global well posedness on a bounded domain, as well as the whole Euclidean space, in $H^1$. Then we study the asymptotic behavior of solutions. In general, we obtain weak convergence in H^1 to a stationary state. For a ball, we prove strong asymptotic convergence to the ground state when the initial condition is positive.

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