arXiv subjects

Piotr Kokocki

Publications and source records attributed to Piotr Kokocki.

17 recordsLinked to original sources

arXiv2026

Standing waves for Schrodinger equations with Kato class potentials and $L^\infty$-bounded nonlinearities

We establish the existence of standing waves for a nonlinear Schrodinger equation with potentials belonging to the Kato class and an $L^\infty$-bounded nonlinearity, whose Lipschitz constant is smaller than the distance from zero to the essential spectrum of the linear part. We consider both the nonresonant and resonant cases. Our approach is based on the Conley index theory applied to study invariant sets of the associated parabolic semiflow. Using properties of the Schrodinger semigroup with Kato-class potentials, which follow from its Feynman-Kac representation, we derive a priori estimates for bounded solutions in the $L^\infty$ and $L^2$ norms, as well as regularity bounds in Sobolev spaces. As a consequence, we obtain conditions ensuring the existence of connecting orbits between stationary solutions, which in turn yield the existence of nontrivial standing waves.

math.AP↗

arXiv2025

Uniqueness problem for Prandtl spirals

In this paper, we study the problem of uniqueness of a divergence-free velocity field with vorticity given by the Prandtl spiral. We show that if the class of admissible velocities is restricted to those satisfying the velocity matching condition and an appropriate decay condition at the origin of the spiral, then the velocity field is uniquely determined. We subsequently extend the result to the case of fields with vorticity composed of unions of concentric logarithmic spirals. As a by-product, we derive an alternative way of deriving formula for the velocity corresponding to the Prandtl spirals. The proof relies on an approach that is of independent interest. We construct an explicit conformal map from the exterior of a logarithmic spiral onto a strip. This transformation reduces the problem to establishing the uniqueness of a holomorphic function defined on the strip, under non-standard boundary and decay conditions.

math.AP↗

arXiv2023

Standing Waves for Schrödinger Equations with Kato-Rellich potentials

We show the existence of standing waves for the nonlinear Schrödinger equation with Kato-Rellich type potential. We consider both resonant with the nonlinearity satisfying one of Landesman-Lazer type or sign conditions and non-resonant case where the linearization at infinity has zero kernel. The approach relies on the geometric and topological analysis of the parabolic semiflow associated to the involved elliptic problem. Tail estimates techniques and spectral theory of unbounded linear operators are used to exploit subtle compactness properties necessary for use of the Conley index theory due to Rybakowski.

math.AP↗

arXiv2023

Linear instability of symmetric logarithmic spiral vortex sheets

We consider Alexander spirals with $M\geq 3$ branches, that is symmetric logarithmic spiral vortex sheets. We show that such vortex sheets are linearly unstable in the $L^\infty$ (Kelvin-Helmholtz) sense, as solutions to the Birkhoff-Rott equation. To this end we consider Fourier modes in a logarithmic variable to identify unstable solutions with polynomial growth in time.

math.AP↗

arXiv2021

Double spiral singularities for a flow of regular planar curves

In this paper we study the singularity formation for the geometric flow of complex curves $$z_t = -z_{xxx} + \frac{3}{2}øz_{x} z_{xx}^2,$$ that was derived [R. E. Goldstein and D. M. Petrich, {\em Phys. Rev. Lett.}, 69 (1992), pp. 555--558] while considering the vortex patch dynamics for the incompressible 2D Euler equation. We prove that arbitrary curve, consisting of two rotating logarithmic spirals, is a finite time singularity developed by a smooth solution of the flow. We provide exact construction of the solution in the terms of appropriate Painlevé II transcendents and furthermore we establish its asymptotic expansion in the vicinity of the singularity.

math.AP↗

arXiv2020

On global solutions of defocusing mKdV equation with specific initial data of critical regularity

We study the asymptotic behavior of the Ablowitz-Segur solutions for the second Painlevé equation using the Riemann-Hilbert approach and methods based on asymptotic expansions of classical special functions. Recent results show that the matrix-valued function satisfying the associated Riemann-Hilbert problem can be represented by means of a local parametrix around the origin, whose existence can be proved by a vanishing lemma. The aim of this paper is to construct the explicit form of this parametrix and apply it to obtain improved asymptotic relations for the real and purely imaginary Ablowitz-Segur solutions of the inhomogeneous Painlevé II equation.

math.CA↗

arXiv2020

On global dynamics of reaction--diffusion systems at resonance

In this paper we use the homotopy invariants methods to study the global dynamics of the reaction-diffusion systems that are at resonance at infinity. Considering degrees of the resonance for the nonlinear perturbation we establish Landesman-Lazer type conditions and use them to express the Rybakowski-Conley index of the invariant set consisting of all bounded solutions. Obtained results are applied to study the existence of solutions connecting stationary points for the system of nonlinear heat equations.

math.AP↗

arXiv2019

Total integrals of Ablowitz-Segur solutions for the inhomogeneous Painlevé II equation

In this paper, we establish a formula determining the value of the Cauchy integrals of the real and purely imaginary Ablowitz-Segur solutions for the inhomogeneous second Painlevé equation. Our approach relies on the analysis of the corresponding Riemann-Hilbert problem and the construction of an appropriate parametrix in a neighborhood of the origin. Obtained integral formulas are consistent with already known analogous results for Ablowitz-Segur solutions of homogeneous Painlevé II equation.

math.CA↗

arXiv2015

Averaging principle and periodic solutions for nonlinear evolution equations at resonance

We study the existence of $T$-periodic solutions $(T > 0)$ for the first order differential equations being at resonance at infinity, where the right hand side is the perturbations of a sectorial operator. Our aim is to prove an index formula expressing the topological degree of the associated translation along trajectories operator on appropriately large ball, in terms of special geometrical assumptions imposed on the nonlinearity. We also prove that the geometrical assumptions are generalization of well known Landesman-Lazer and strong resonance conditions. Obtained index formula is used to derive the criteria determining the existence of $T$-periodic solutions for the heat equation being at resonance at infinity.

math.AP↗

arXiv2015

Connecting orbits for nonlinear differential equations at resonance

We study the existence of orbits connecting stationary points for the first order differential equations being at resonance at infinity, where the right hand side is the perturbations of a sectorial operator. Our aim is to prove an index formula expressing the Conley index of associated semiflow with respect to appropriately large ball, in terms of special geometrical assumptions imposed on the nonlinearity. We also prove that the geometrical assumptions are generalization of well known in literature Landesman-Lazer and strong resonance conditions. Obtained index formula will be used to derive the criteria determining the existence of orbits connecting stationary points for the heat equation being at resonance at infinity.

math.AP↗

arXiv2015

Invariant sets and connecting orbits for nonlinear evolution equations at resonance

We study the problem of existence of orbits connecting stationary points for the nonlinear heat and strongly damped wave equations being at resonance at infinity. The main difficulty lies in the fact that the problems may have no solutions for general nonlinearity. To address this question we introduce geometrical assumptions for the nonlinear term and use them to prove index formulas expressing the Conley index of associated semiflows. We also prove that the geometrical assumptions are generalizations of the well known Landesman- Lazer and strong resonance conditions. Obtained index formulas are used to derive criteria determining the existence of orbits connecting stationary points.

math.DS↗

arXiv2015

Effect of resonance on the existence of periodic solutions for strongly damped wave equation

We are interested in the differential equation $\ddot u(t) = -A u(t) - c A \dot u(t) + λu(t) + F(t,u(t))$, where $c > 0$ is a damping factor, $A$ is a sectorial operator and $F$ is a continuous map. We consider the situation where the equation is at resonance at infinity, which means that $λ$ is an eigenvalue of $A$ and $F$ is a bounded map. We introduce new geometrical conditions for the nonlinearity $F$ and use topological degree methods to find $T$-periodic solutions for this equation as fixed points of Poincaré operator.

math.AP↗

arXiv2015

Homotopy invariants methods in the global dynamics of strongly damped wave equation

We are interested in the following differential equation $\ddot u(t) = -A u(t) - c A \dot u(t) + λu(t) + F(u(t))$ where $c > 0$ is a damping factor, $A$ is a sectorial operator and $F$ is a continuous map. We consider the situation where the equation is at resonance at infinity, which means that $λ$ is an eigenvalue of $A$ and $F$ is a bounded map. We provide geometrical conditions for the nonlinearity $F$ and determine the Conley index of the set $K_\infty$, that is the union of the bounded orbits of this equation.

math.DS↗

arXiv2015

Krasnosel'skii type formula and translation along trajectories method on the scale of fractional spaces

We provide global continuation principle of periodic solutions for the equation $\dot u = - Au + F(t,u)$, where $A:D(A)\to X$ is a sectorial operator on a Banach space $X$ and $F:[0,+\infty)\times X^α\to X$ is a nonlinear map defined on fractional space $X^α$. The approach that we use in this paper is based upon the theory of topological invariants that applies in the situation when Poincaré operator associated with the equation is endowed with some form of compactness.

math.DS↗

arXiv2015

Periodic solutions for nonlinear hyperbolic evolution systems

We shall deal with the periodic problem for nonlinear perturbations of abstract hyperbolic evolution equations generating an evolution system of contractions. We prove an averaging principle for the translation along trajectories operator associated to the nonlinear evolution system, expressed in terms of the topological degree. The abstract results shall be applied to the damped hyperbolic partial differential equation.

math.DS↗

arXiv2015

Krasnosel'skii type formula and translation along trajectories method for evolution equations

The Krasnosel'skii type degree formula for the equation $\dot u = - Au + F(u)$ where $A:D(A)\to E$ is a linear operator on a separable Banach space $E$ such that $-A$ is a generator of a $C_0$ semigroup of bounsed linear operators of $E$ and $F:E\to E$ is a locally Lipschitz $k$-set contraction, is provided. Precisely, it is shown that if $V$ is an open bounded subset of $E$ such that $0\not\in (-A+F)(\partial V \cap D(A))$, then the topological degree of $-A+F$ with respect to $V$ is equal to the fixed point index of the operator of translation along trajectories for sufficiently small positive time. The obtained degree formula is crucial for the method of translation along trajctories. It is applied to the nonautonomous periodic problem and an average principle is derived. As an application a first order system of partial differential equations is considered.

math.DS↗

arXiv2015

Periodic solutions for nonlinear evolution equations at resonance

We are concerned with periodic problems for nonlinear evolution equations at resonance of the form $\dot u(t) = - A u(t) + F (t,u(t))$, where a densely defined linear operator $A\colon D(A)\to X$ on a Banach space $X$ is such that $-A$ generates a compact $C_0$ semigroup and $F\colon [0,+\infty)\times X \to X$ is a nonlinear perturbation. Imposing appropriate Landesman--Lazer type conditions on the nonlinear term $F$, we prove a formula expressing the fixed point index of the associated translation along trajectories operator, in the terms of a time averaging of $F$ restricted to $\mathrm{Ker} \, A$. By the formula, we show that the translation operator has a nonzero fixed point index and, in consequence, we conclude that the equation admits a periodic solution.

math.AP↗