arXiv subjects

Nikola Stoilov

Publications and source records attributed to Nikola Stoilov.

9 recordsLinked to original sources

arXiv2026

Nonlinear Schrödinger equation on a unit ball in one and two dimensions

We investigate the nonlinear Schrödinger equation on a unit ball in one and two dimensions with Dirichlet boundary condition, focusing on ground states, their limiting behavior, stability and perturbation dynamics. Our computations illustrate the stabilizing effect of the Dirichlet boundary compared with the whole-space setting. In the subcritical and critical cases, the mass curves have positive slope and the considered perturbations show recurrent dynamics near the ground state family. In the critical case, the mass of the ground states increases towards the mass of the whole-space ground state. Staying below the whole-space ground state mass makes perturbations of the bounded-domain ground states stable. In the supercritical case, the computations reveal a {\it new} branching phenomenon: the mass curve develops a turning point, separating stable and unstable branches. In 1D we prove existence of a global maximum in the mass curve and derive an integral equation for the turning points. Perturbations of the stable branch remain close to the corresponding ground state, while perturbations of the unstable branch lead to finite-time blow-up or recurrent oscillations between neighborhoods of at least two coherent profiles. We prove sufficient blow-up conditions in the critical and supercritical cases, in particular, for every fixed $A>1$, the data $A Q_b$ blows up for sufficiently large $b$ in these cases. Furthermore, the Dirichlet boundary reflects the solution, preventing mass from escaping via radiation, while coherent oscillations persist even for small initial data, thus, supporting a bounded-domain soliton-resolution conjecture.

math.AP↗

arXiv2026

The 3D critical Zakharov--Kuznetsov equation: blow-up and soliton dynamics

We study the full three-dimensional dynamics of the $L^2$-critical Zakharov-Kuznetsov equation with the fractional nonlinearity $|u|^{4/3}u$, equivalently $u^{7/3}$ for real-valued functions. This equation is a higher-dimensional extension of the generalized Korteweg-de Vries equation. In the critical setting solutions to this 3D ZK equation may blow up in finite time or exhibit global time dynamics. The novelties of this work is to treat a non-integer power and to study the dynamics of solutions in a higher dimension. We first review the finite time blow-up in 2D critical ZK, then do a formal analysis of the slightly mass-supercritical blow-up dynamics, deriving the corrections to the blow-up rate and profile for the critical ZK equation in any dimension. We then perform a computational study of solutions, utilizing full 3D numerical simulations. In particular, we use a Fourier pseudospectral discretization and an integrating factor fourth-order Runge-Kutta method on a full three-dimensional grid. A multi-GPU implementation makes it possible to follow blow-up solutions in a full 3D setting. We examine perturbations of the ground state, Gaussian data, and nonsymmetric two-bump configurations. The computations show dispersive and concentrating regimes, in both cases with radiation emitted in a conic-type region opposite to the direction of propagation and convergence of the concentrating core toward a rescaled ground-state profile. The two-bump experiments also demonstrate that total mass alone does not determine the blow-up dynamics. We discuss the numerical evidence for the predicted blow-up rate and identify the pre-asymptotic and resolution limitations that remain near the blow-up time.

math.AP↗

arXiv2026

Numerical study of the 2D Kaup-Broer-Kuperschmidt Boussinesq system

In this work we consider the well posed version of the Kaup-Broer-Kuperschmidt system in two dimensions. We numerically construct soliton type solutions and show that they are unstable both against dispersion and singularity formation. Further, we study line solitons and their stability, as well as generally localised initial data. In either case we fail to find stable structures.

math.AP↗

arXiv2026

Multi-domain spectral approach for Zakharov-Kuznetsov equations in 3D with cylindrical symmetry

We present a novel numerical framework for studying nonlinear dispersive equations in higher-dimensional settings, specifically designed for solutions featuring traveling waves along a preferred axis (or field-aligned traveling waves). Using the three-dimensional generalized Zakharov-Kuznetsov (gZK) equation as a model, we convert it into cylindrical coordinates and implement a domain decomposition strategy. By partitioning the computational domain into distinct regions based on expected solution behavior, we significantly reduce computational complexity while maintaining the high resolution necessary for capturing small-scale dynamics. Another key innovation of our method is the ability to efficiently handle fractional nonlinearities, specifically, the critical power $p = 7/3$ in 3D, which typically introduces significant computational overhead and numerical instabilities that compromise simulation accuracy. Using this framework, we are able to investigate the dynamics of solutions (with cylindrical symmetry) close to the ground state soliton and show that for the 3D critical ZK equation, the ground state serves as the sharp threshold for global vs. finite time existence of solutions. Our method successfully tracks the profiles of these singular solutions, providing new insights into the dynamics of wave collapse in three-dimensional magnetized media.

math.NA↗

arXiv2026

Dynamics of solutions in the 1d bi-harmonic nonlinear Schrödinger equation

We consider the one dimensional 4th order, or bi-harmonic, nonlinear Schrödinger (NLS) equation, namely, $i u_t - Δ^2 u - 2a Δu + |u|^α u = 0, ~ x,a \in \R$, $α>0$, and investigate the dynamics of its solutions for various powers of $α$, including the ground state solutions and their perturbations, leading to scattering or blow-up dichotomy when $a \leq 0$, or to a trichotomy when $a>0$. Ground state solutions are numerically constructed, and their stability is studied, finding that the ground state solutions may form two branches, stable and unstable, which dictates the long-term behavior of solutions. Perturbations of the ground states on the unstable branch either lead to dispersion or the jump to a stable ground state. In the critical and supercritical cases, blow-up in finite time is also investigated, and it is conjectured that the blow-up happens with a scale-invariant profile (when $a=0$) regardless of the value of $a$ of the lower dispersion. The blow-up rate is also explored.

math.AP↗

arXiv2023

Numerical study of the transverse stability of line solitons of the Zakharov-Kuznetsov equations

We present a detailed numerical study of the stability under periodic perturbations of line solitons of two-dimensional, generalized Zakharov-Kuznetsov equations with various power nonlinearities. In the $L^{2}$-subcritical case, in accordance with a theorem due to Yamazaki we find a critical speed, below which the line soliton is stable. For higher velocities, the numerical results indicate an instability against the formation of lumps, solitons localized in both spatial directions. In the $L^2$-critical and supercritical cases but subcritical for the 1D generalized Korteweg-de Vries equation), the line solitons are shown to be numerically stable for small velocities, and strongly unstable for large velocities, with a blow-up observed in finite time.

math.AP↗

arXiv2022

Large $|k|$ behavior for the reflection coefficient for Davey-Stewartson II equations

The study of complex geometric optics solutions to a system of d-bar equations appearing in the context of electrical impedance tomography and the scattering theory of the integrable Davey-Stewartson II equations for large values of the spectral parameter $k$ in \cite{KlSjSt20} is extended to the reflection coefficient. For the case of potentials $q$ with compact support on some domain $Ω$ with smooth strictly convex boundary, improved asymptotic relations are provided.

math.AP↗

arXiv2018

Numerical Approach to Painlevé Transcendents on Unbounded Domains

A multidomain spectral approach for Painlevé transcendents on unbounded domains is presented. This method is designed to study solutions determined uniquely by a, possibly divergent, asymptotic series valid near infinity in a sector and approximates the solution on straight lines lying entirely within said sector without the need of evaluating truncations of the series at any finite point. The accuracy of the method is illustrated for the example of the tritronquée solution to the Painlevé I equation.

math.CA↗

arXiv2015

'Riemann Equations' in Bidifferential Calculus

We consider equations that formally resemble a matrix Riemann (or Hopf) equation in the framework of bidifferential calculus. With different choices of a first-order bidifferential calculus, we obtain a variety of equations, including a semi-discrete and a fully discrete version of the matrix Riemann equation. A corresponding universal solution-generating method then either yields a (continuous or discrete) Cole-Hopf transformation, or leaves us with the problem of solving Riemann equations (hence an application of the hodograph method). If the bidifferential calculus extends to second order, solutions of a system of `Riemann equations' are also solutions of an equation that arises, on the universal level of bidifferential calculus, as an integrability condition. Depending on the choice of bidifferential calculus, the latter can represent a number of prominent integrable equations, like self-dual Yang-Mills, as well as matrix versions of the two-dimensional Toda lattice, Hirota's bilinear difference equation, (2+1)-dimensional NLS, KP and Davey-Stewartson equations. For all of them, a recent (non-isospectral) binary Darboux transformation result in bidifferential calculus applies, which can be specialized to generate solutions of the associated `Riemann equations'. For the latter, we clarify the relation between these specialized binary Darboux transformations and the aforementioned solution-generating method. From (arbitrary size) matrix versions of the `Riemann equations' associated with an integrable equation, possessing a bidifferential calculus formulation, multi-soliton-type solutions of the latter can be generated. This includes `breaking' multi-soliton-type solutions of the self-dual Yang-Mills and the (2+1)-dimensional NLS equation, which are parametrized by solutions of Riemann equations.

nlin.SI↗