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Michał Kowalczyk

Publications and source records attributed to Michał Kowalczyk.

At least 19 recordsLinked to original sources
arXiv2025

Asymptotic analysis of small energy breathers for the nonlinear Klein-Gordon equation

For a class of nonlinear Klein-Gordon equations, we prove that in the small energy limit, any sequence of breathers decomposes into a finite sum of decoupled, periodically modulated canonical solitons. Each of these solitons is asymptotically equal to an explicit sine-Gordon breather and the distance between them grows to infinity as the energy decreases to 0. Finally we prove that none of these breathers is centered in a bounded set provided that a certain non resonance condition holds.

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arXiv2025

Yang-Mills instanton on a four dimensional wormhole: asymptotic stability in the energy space

In this paper we consider an $SU(2)$ Yang-Mills field propagating in the $4+1$ dimensional wormhole spacetime. Assuming the spherically symmetric magnetic ansatz the problem reduces to a one dimensional non linear wave equation. This equation posses a degree one solution (instanton) which is odd in space. We consider small, odd perturbations of the instanton and show that it is conditionally asymptotically stable in the odd energy space.

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arXiv2025

Energy minimizers in a periodic phase transition model of light-matter interaction in nematic liquid crystals

In this paper we complete the study of global minimizers of a forced, non autonomous, one dimensional, phase transition model, initiated in [8]. Motivated by the recent findings in [9], revealing new configurations of topological structures in light, we consider a forcing term having two periods. We show that depending on the strength of the forcing, at most two thresholds that determine the structure of the minimizers (kinks) are attained. These kinks are now a combination of the previous types encountered in [8], and they may have at most three zeros. The existence of these complex types of phase transition follows from a periodic one dimensional model of matter-light interaction in nematic liquid crystal based on a thin sample limit of the Oseen-Frank energy. We show that the qualitative behaviour of global minimizers is consistent with the original model.

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arXiv2025

A new proof of a Liouville theorem for the one dimensional Gross-Pitaevskii equation

The asymptotic stability of the black and dark solitons of the one-dimensional Gross-Pitaevskii equation was proved by Béthuel, Gravejat and Smets (Ann. Sci. Éc. Norm. Supér. 48 (2015)) and Gravejat and Smets (Proc. Lond. Math. Soc. 111 (2015)), using a rigidity property in the vicinity of solitons. We provide an alternate proof of the Liouville theorems in the above articles using a factorization identity for the linearized operator which trivializes the spectral analysis.

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arXiv2024

Variational Approach for the Singular Perturbation Domain Wall System

In this article we study a coupled system of differential equations with Allen-Cahn type non-linearity. Motivated by physical phenomena one of the unknowns in the system is accompanied by a singular perturbation parameter $ε^2$ . By employing variational techniques, we establish the existence of solutions for all values of $ε$ and get results on their qualitative properties, including regularity. Additionally, we analyse the behaviour of solutions as $ε {\to} 0$, demonstrating their pointwise convergence to the solution of the problem for $ε = 0$. We establish the uniqueness of this solution modulo translations. Additionally, in the final section, through an appropriate change of scale, we relate this problem and the second Painlevé equation.

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arXiv2022

Generation of vortices in the Ginzburg-Landau heat flow

We consider the Ginzburg-Landau heat flow on the two-dimensional flat torus, starting from an initial data with a finite number of nondegenerate zeros -- but possibly very high initial energy. We show that the initial zeros are conserved and the flow rapidly enters a logarithmic energy regime, from which the evolution of vortices can be described by the works of Bethuel, Orlandi and Smets.

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arXiv2022

Phase separating solutions for two component systems in general planar domains

In this paper we consider a two component system of coupled non linear Schrödinger equations modeling the phase separation in the binary mixture of Bose-Einstein condensates and other related problems. Assuming the existence of solutions in the limit of large interspecies scattering length $β$ the system reduces to a couple of scalar problems on subdomains of pure phases. Here we show that given a solution to the limiting problem under some additional non degeneracy assumptions there exists a family of solutions parametrized by the parameter $β\gg 1$.

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arXiv2021

Entire vortex solutions of negative degree for the anisotropic Ginzburg-Landau system

The anisotropic Ginzburg-Landau system \[ Δu+δ\, \nabla (\mathrm{div}\: u) +δ\, \mathrm{curl}^*(\mathrm{curl}\: u)=(|u|^2-1) u, \] for $u\colon\mathbb R^2\to\mathbb R^2$ and $δ\in (-1,1)$, models the formation of vortices in liquid crystals. We prove the existence of entire solutions such that $|u(x)|\to 1$ and $u$ has a prescribed topological degree $d\leq -1$ as $|x|\to\infty$, for small values of the anisotropy parameter $|δ| < δ_0(d)$. Unlike the isotropic case $δ=0$, this cannot be reduced to a one-dimensional radial equation. We obtain these solutions by minimizing the anisotropic Ginzburg-Landau energy in an appropriate class of equivariant maps, with respect to a finite symmetry subgroup.

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arXiv2021

Sine-Gordon on a wormhole

In an attempt to understand the soliton resolution conjecture, we consider the Sine-Gordon equation on a spherically symmetric wormhole spacetime. We show that within each topological sector (indexed by a positive integer degree $n$) there exists a unique linearly stable soliton, which we call the $n$-kink. We give numerical evidence that the $n$-kink is a global attractor in the evolution of any smooth, finite energy solutions of degree $n$. When the radius of the wormhole throat $a$ is large enough, the convergence to the $n$-kink is shown to be governed by internal modes that slowly decay due to the resonant transfer of energy to radiation. We compute the exact asymptotics of this relaxation process for the $1$-kink using the Soffer-Weinstein weakly nonlinear perturbation theory.

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arXiv2020

A sufficient condition for asymptotic stability of kinks in general (1+1)-scalar field models

We study stability properties of kinks for the (1+1)-dimensional nonlinear scalar field theory models \begin{equation*} \partial_t^2ϕ-\partial_x^2ϕ+ W'(ϕ) = 0, \quad (t,x)\in\mathbb{R}\times\mathbb{R}. \end{equation*} The orbital stability of kinks under general assumptions on the potential $W$ is a consequence of energy arguments. Our main result is the derivation of a simple and explicit sufficient condition on the potential $W$ for the asymptotic stability of a given kink. This condition applies to any static or moving kink, in particular no symmetry assumption is required. Last, motivated by the Physics literature, we present applications of the criterion to the $P(ϕ)_2$ theories and the double sine-Gordon theory.

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arXiv2020

Dynamics of strongly interacting kink-antikink pairs for scalar fields on a line

This paper concerns classical nonlinear scalar field models on the real line. If the potential is a symmetric double-well, such a model admits static solutions called kinks and antikinks, which are perhaps the simplest examples of topological solitons. We study pure kink-antikink pairs, which are solutions that converge in one infinite time direction to a superposition of one kink and one antikink, without radiation. Our main result is a complete classification of all kink-antikink pairs in the strongly interacting regime, which means the speeds of the kinks tend asymptotically to zero. We show that up to translation there is exactly one such solution, and we give a precise description of the dynamics of the kink separation.

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arXiv2019

The connecting solution of the Painlevé phase transition model

The second Painlevé O.D.E. $y''-xy-2y^3=0$, $x\in \mathbb{R},$ is known to play an important role in the theory of integrable systems, random matrices, Bose-Einstein condensates and other problems. The generalized second Painlevé equation $Δy -x_1 y - 2 y^3=0$, $(x_1,x_2)\in \mathbb{R}^2$, is obtained by multiplying by $-x_1$ the linear term $u$ of the Allen-Cahn equation $Δu =u^3-u$. It involves a non autonomous potential $H(x_1,y)$ which is bistable for every fixed $x_1<0$, and thus describes as the Allen-Cahn equation a phase transition model. The scope of this paper is to construct a solution $y$ connecting along the vertical direction $x_2$, the two branches of minima of $H$ parametrized by $x_1$. This solution plays a similar role that the heteroclinic orbit for the Allen-Cahn equation. It is the the first to our knowledge solution of the Painlevé P.D.E. both relevant from the applications point of view (liquid crystals), and mathematically interesting.

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arXiv2017

Symmetry breaking and restoration in the Ginzburg-Landau model of nematic liquid crystals

In this paper we study qualitative properties of global minimizers of the Ginzburg-Landau energy which describes light-matter interaction in the theory of nematic liquid crystals near the Friedrichs transition. This model is depends on two parameters: $ε>0$ which is small and represents the coherence scale of the system and $a\geq 0$ which represents the intensity of the applied laser light. In particular we are interested in the phenomenon of symmetry breaking as $a$ and $ε$ vary. We show that when $a=0$ the global minimizer is radially symmetric and unique and that its symmetry is instantly broken as $a>0$ and then restored for sufficiently large values of $a$. Symmetry breaking is associated with the presence of a new type of topological defect which we named the shadow vortex. The symmetry breaking scenario is a rigorous confirmation of experimental and numerical results obtained in our earlier work.

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arXiv2016

Theory of light-matter interaction in nematic liquid crystals and the second Painlevé equation

We study global minimizers of an energy functional arising as a thin sample limit in the theory of light-matter interaction in nematic liquid crystals. We show that depending on the parameters various defects are predicted by the model. In particular we show existence of a new type of topological defect which we call the {\it shadow kink}. Its local profile is described by the second Painlevé equation. As part of our analysis we find new solutions to this equation thus generalizing the well known result of Hastings and McLeod.

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arXiv2016

Nonexistence of small, odd breathers for a class of nonlinear wave equations

In this note, we show that for a large class of nonlinear wave equations with odd nonlinearities, any globally defined odd solution which is small in the energy space decays to $0$ in the local energy norm. In particular, this result shows nonexistence of small, odd breathers for some classical nonlinear Klein Gordon equations such as the sine Gordon equation and $ϕ^4$ and $ϕ^6$ models. It also partially answers a question of Soffer and Weinstein in \cite[p. 19]{MR1681113} about nonexistence of breathers for the cubic NLKG in dimension one.

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arXiv2015

Kink dynamics in the $ϕ^4$ model: asymptotic stability for odd perturbations in the energy space

We consider a classical equation known as the $ϕ^4$ model in one space dimension. The kink, defined by $H(x)=\tanh(x/{\sqrt{2}})$, is an explicit stationary solution of this model. From a result of Henry, Perez and Wreszinski it is known that the kink is orbitally stable with respect to small perturbations of the initial data in the energy space. In this paper we show asymptotic stability of the kink for odd perturbations in the energy space. The proof is based on Virial-type estimates partly inspired from previous works of Martel and Merle on asymptotic stability of solitons for the generalized Korteweg-de Vries equations. However, this approach has to be adapted to additional difficulties, pointed out by Soffer and Weinstein in the case of general Klein-Gordon equations with potential: the interactions of the so-called internal oscillation mode with the radiation, and the different rates of decay of these two components of the solution in large time.

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