arXiv subjects

Yan Rybalko

Publications and source records attributed to Yan Rybalko.

16 recordsLinked to original sources

arXiv2026

Riemann-Hilbert approach for the nonlocal modified Korteweg-de Vries equation with a step-like oscillating background

This work focuses on the Cauchy problem for the nonlocal modified Korteweg-de Vries equation $$ u_t(x,t)+6u(x,t)u(-x,-t)u_x(x,t)+u_{xxx}(x,t)=0, $$ with the oscillating step-like boundary conditions: $u(x,t)\to 0$ as $x\to-\infty$ and $u(x,t)\backsimeq A\cos(2Bx+8B^3t)$ as $x\to\infty$, where $A,B>0$ are arbitrary constants. The main goal is to develop the Riemann-Hilbert formalism for this problem, paying a particular attention to the case of the ``pure oscillating step'' initial data, that is $u(x,0)=0$ for $x<0$ and $u(x,0)=A\cos(2Bx)$ for $x\geq0$. Also, we derive three new families of two-soliton solutions, which correspond to the values of $A$ and $B$ satisfying $B<\frac{A}{4}$, $B>\frac{A}{4}$, and $B=\frac{A}{4}$.

math.AP↗

arXiv2025

Generic regularity and Lipschitz metric for a two-component Novikov system

We investigate the Cauchy problem for a two-component generalization of the Novikov equation with cubic nonlinearity -- an integrable system whose solutions may develop strong nonlinear phenomena such as gradient blow-up and interactions between peakon-like structures. Our study has two main objectives: first, to analyze the generic regularity of global conservative solutions; and second, to construct a new metric that guarantees the Lipschitz continuity of the flow. Building on the geometric framework developed by Bressan and Chen for quasilinear second-order wave equations, we prove that the solution retains $C^k$ regularity away from a finite number of piecewise $C^{k-1}$ characteristic curves. Furthermore, we provide a description of the solution behavior in the vicinity of these curves. By introducing a Finsler norm on tangent vectors in the space of solutions, expressed in the transformed Bressan-Constantin variables, we introduce a Lipschitz metric representing the minimal energy transportation cost between two solutions.

math.AP↗

arXiv2025

The integrable nonlocal nonlinear Schr\"odinger equation with oscillatory boundary conditions: long-time asymptotics

We consider the Cauchy problem for the integrable nonlocal nonlinear Schr\"odinger equation \[ \I q_{t}(x,t)+q_{xx}(x,t)+2 q^{2}(x,t)\bar{q}(-x,t)=0, \] subject to the step-like initial data: $q(x,0)\to0$ as $x\to-\infty$ and $q(x,0)\simeq Ae^{2\I Bx}$ as $x\to\infty$, where $A>0$ and $B\in\mathbb{R}$. The goal is to study the long-time asymptotic behavior of the solution of this problem assuming that $q(x,0)$ is close, in a certain spectral sense, to the ``step-like'' function $q_{0,R}(x)= \begin{cases} 0, &x\leq R,\\ Ae^{2\I Bx}, &x>R, \end{cases}$ with $R>0$. A special attention is paid to how $B\ne0$ affects the asymptotics.

math.AP↗

arXiv2025

Global semigroup of conservative weak solutions of the two-component Novikov equation

We study the Cauchy problem for the two-component Novikov system with initial data $u_0, v_0$ in $H^1(\mathbb{R})$ such that the product $(\partial_x u_0)\partial_x v_0$ belongs to $L^2(\mathbb{R})$. We construct a global semigroup of conservative weak solutions. We also discuss the potential concentration phenomena of $(\partial_x u)^2dx$, $(\partial_x v)^2dx$, and $\left((\partial_x u)^2(\partial_x v)^2\right)dx$, which contribute to wave-breaking and may occur for a set of time with nonzero measure. Finally, we establish the continuity of the data-to-solution map in the uniform norm.

math.AP↗

arXiv2024

On the well-posedness of the Cauchy problem for the two-component peakon system in $C^k\cap W^{k,1}$

This study focuses on the Cauchy problem associated with the two-component peakon system featuring a cubic nonlinearity, constrained to the class $(m,n)\in C^{k}(\mathbb{R}) \cap W^{k,1}(\mathbb{R})$ with $k\in\mathbb{N}\cup\{0\}$.This system extends the celebrated Fokas-Olver-Rosenau-Qiao equation, and the following nonlocal (two-place) counterpart proposed by Lou and Qiao: $$ \partial_t m(t,x)= \partial_x[m(t,x)(u(t,x)-\partial_xu(t,x)) (u(-t,-x)+\partial_x(u(-t,-x)))], $$ where $m(t,x)=\left(1-\partial_{x}^2\right)u(t,x)$. Employing an approach based on Lagrangian coordinates, we establish the local existence, uniqueness, and Lipschitz continuity of the data-to-solution map in the class $C^k\cap W^{k,1}$. Moreover, we derive criteria for blow-up of the local solution in this class.

math.AP↗

arXiv2023

H\"older continuity of functions in the fractional Sobolev spaces: 1-dimensional case

This paper deals with the embedding of the Sobolev spaces of fractional order into the space of H\"older continuous functions. More precisely, we show that the function $f\in H^s(\mathbb{R})$ with $\frac{1}{2}<s<1$ is H\"older continuous with the exponent $s-\frac{1}{2}$. This is a particular case of the much stronger embedding theorems (see Section 2.8.1 in \textit{H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland Pub. Co., Amsterdam, 1978.}), but here we give an elementary proof for $H^{s}(\mathbb{R})$.

math.AP↗

arXiv2023

On the well-posedness of a nonlocal (two-place) FORQ equation via a two-component peakon system

We investigate the Cauchy problem for a nonlocal (two-place) FORQ equation. By interpreting this equation as a special case of a two-component peakon system (exhibiting a cubic nonlinearity), we convert the Cauchy problem into a system of ordinary differential equations in a Banach space. Using this approach, we are able to demonstrate local well-posedness in the Sobolev space $H^{s}$ where $s > 5/2$. We also establish the continuity properties for the data-to-solution map for a range of Sobolev spaces. Finally, we briefly explore the relationship between the two-component system and the bi-Hamiltonian AKNS hierarchy.

math.AP↗

arXiv2023

Global conservative solutions of the nonlocal NLS equation beyond blow-up

We consider the Cauchy problem for the integrable nonlocal nonlinear Schrödinger (NNLS) equation $ \I\partial_t q(x,t)+\partial_{x}^2q(x,t)+2σq^{2}(x,t)\overline{q(-x,t)}=0 $ with initial data $q(x,0)\in H^{1,1}(\mathbb{R})$. It is known that the NNLS equation is integrable and it has soliton solutions, which can have isolated finite time blow-up points. The main aim of this work is to propose a suitable concept for continuation of weak $H^{1,1}$ local solutions of the general Cauchy problem (particularly, those admitting long-time soliton resolution) beyond possible singularities. Our main tool is the inverse scattering transform method in the form of the Riemann-Hilbert problem combined with the PDE existence theory for nonlinear dispersive equations.

math.AP↗

arXiv2022

Focusing nonlocal nonlinear Schrödinger equation with asymmetric boundary conditions: large-time behavior

We consider the focusing integrable nonlocal nonlinear Schrödinger equation \[\mathrm{i}q_{t}(x,t)+q_{xx}(x,t)+2q^{2}(x,t)\bar{q}(-x,t)=0\] with asymmetric nonzero boundary conditions: $q(x,t)\to\pm A\mathrm{e}^{-2\mathrm{i}A^2t}$ as $x\to\pm\infty$, where $A>0$ is an arbitrary constant. The goal of this work is to study the asymptotics of the solution of the initial value problem for this equation as $t\to+\infty$. For a class of initial values we show that there exist three qualitatively different asymptotic zones in the $(x,t)$ plane. Namely, there are regions where the parameters are modulated (being dependent on the ratio $x/t$) and a central region, where the parameters are unmodulated. This asymptotic picture is reminiscent of that for the defocusing classical nonlinear Schrödinger equation, but with some important differences. In particular, the absolute value of the solution in all three regions depends on details of the initial data.

math.AP↗

arXiv2021

Asymptotic stage of modulation instability for the nonlocal nonlinear Schrödinger equation

We study the initial value problem for the integrable nonlocal nonlinear Schrödinger (NNLS) equation \[ iq_{t}(x,t)+q_{xx}(x,t)+2 q^{2}(x,t)\bar{q}(-x,t)=0 \] with symmetric boundary conditions: $q(x,t)\to Ae^{2iA^2t}$ as $x\to\pm\infty$, where $A>0$ is an arbitrary constant. We describe the asymptotic stage of modulation instability for the NNLS equation by computing the large-time asymptotics of the solution $q(x,t)$ of this initial value problem. We shown that it exhibits a non-universal, in a sense, behavior: the asymptotics of $|q(x,t)|$ depends on details of the initial data $q(x,0)$. This is in a sharp contrast with the local classical NLS equation, where the long-time asymptotics of the solution depends on the initial value through the phase parameters only. The main tool used in this work is the inverse scattering transform method applied in the form of the matrix Riemann-Hilbert problem. The Riemann-Hilbert problem associated with the original initial value problem is analyzed asymptotically by the nonlinear steepest decent method.

math.AP↗

arXiv2020

Curved wedges in the long-time asymptotics for the integrable nonlocal nonlinear Schrödinger equation

We consider the Cauchy problem for the integrable nonlocal nonlinear Schrödinger (NNLS) equation $iq_{t}(x,t)+q_{xx}(x,t)+2 q^{2}(x,t)\bar{q}(-x,t)=0, \, x\in\mathbb{R},\,t>0,$ with a step-like boundary values: $q(x,t)\to 0$ as $x\to-\infty$ and $q(x,t)\to A$ as $x\to\infty$ for all $t\geq0$, where $A>0$ is a constant. The long-time asymptotics of the solution $q(x,t)$ of this problem along the rays $x/t=C\ne 0$ is presented in \cite{RS2}. In the present paper, we extend the asymptotics into a region that is asymptotically closer to the ray $x=0$ than these rays with any nonzero constant $C$. We specify a one-parameter family of wedges in the $x,t$-plane, with curved boundaries, characterized by qualitatively different asymptotic behavior of $q(x,t)$, and present the main asymptotic terms for each wedge. Particularly, for wedges with $x<0$, we show that the solution decays as $t^{p}\sqrt{\ln t}$ with $p<0$ depending on the wedge. For wedges with $x>0$, we show that the asymptotics has an oscillating nature, with the phase functions specific for each wedge and depending on a slow variable parametrizing the wedges. The main tool used in this work is an adaptation of the nonlinear steepest decent method to the case when the stationary phase point of the phase function in the jump of the associated Riemann-Hilbert problem merges with a point which is singular for the corresponding spectral functions.

math.AP↗

arXiv2020

Defocusing nonlocal nonlinear Schrödinger equation with step-like boundary conditions: long-time behavior for shifted initial data

The present paper deals with the long-time asymptotic analysis of the initial value problem for the integrable defocusing nonlocal nonlinear Schrödinger equation $ iq_{t}(x,t)+q_{xx}(x,t)-2 q^{2}(x,t)\bar{q}(-x,t)=0 $ with a step-like initial data: $q(x,0)\to 0$ as $x\to -\infty$ and $q(x,0)\to A$ as $x\to +\infty$. Since the equation is not translation invariant, the solution of this problem is sensitive to shifts of the initial data. We consider a family of problems, parametrized by $R>0$, with the initial data that can be viewed as perturbations of the "shifted step function" $q_{R,A}(x)$: $q_{R,A}(x)=0$ for $x R$, where $A>0$ and $R>0$ are arbitrary constants. We show that the asymptotics is qualitatively different in sectors of the $(x,t)$ plane, the number of which depends on the relationship between $A$ and $R$: for a fixed $A$, the bigger $R$, the larger number of sectors. Moreover, the sectors can be collected into 2 alternate groups: in the sectors of the first group, the solution decays to 0 while in the sectors of the second group, the solution approaches a constant (varying with the direction $x/t=const$).

math.AP↗

arXiv2020

Long-time asymptotics for the integrable nonlocal nonlinear Schrödinger equation with step-like initial data

We study the Cauchy problem for the integrable nonlocal nonlinear Schrödinger (NNLS) equation \[ iq_{t}(x,t)+q_{xx}(x,t)+2 q^{2}(x,t)\bar{q}(-x,t)=0 \] with a step-like initial data: $q(x,0)=q_0(x)$, where $q_0(x)=o(1)$ as $x\to-\infty$ and $q_0(x)=A+o(1)$ as $x\to\infty$, with an arbitrary positive constant $A>0$. The main aim is to study the long-time behavior of the solution of this problem. We show that the asymptotics has qualitatively different form in the quarter-planes of the half-plane $-\infty 0$: (i) for $x<0$, the solution approaches a slowly decaying, modulated wave of the Zakharov-Manakov type; (ii) for $x>0$, the solution approaches the "modulated constant". The main tool is the representation of the solution of the Cauchy problem in terms of the solution of an associated matrix Riemann-Hilbert (RH) problem and the consequent asymptotic analysis of this RH problem.

math.AP↗

arXiv2019

Long-time asymptotics for the integrable nonlocal focusing nonlinear Schrödinger equation for a family of step-like initial data

We study the Cauchy problem for the integrable nonlocal focusing nonlinear Schrödinger (NNLS) equation $ iq_{t}(x,t)+q_{xx}(x,t)+2 q^{2}(x,t)\bar{q}(-x,t)=0 $ with the step-like initial data close to the ``shifted step function'' $χ_R(x)=AH(x-R)$, where $H(x)$ is the Heaviside step function, and $A>0$ and $R>0$ are arbitrary constants. Our main aim is to study the large-$t$ behavior of the solution of this problem. We show that for $R\in\left(\frac{(2n-1)π}{2A},\frac{(2n+1)π}{2A}\right)$, $n=1,2,\dots$, the $(x,t)$ plane splits into $4n+2$ sectors exhibiting different asymptotic behavior. Namely, there are $2n+1$ sectors where the solution decays to $0$, whereas in the other $2n+1$ sectors (alternating with the sectors with decay), the solution approaches (different) constants along each ray $x/t=const$. Our main technical tool is the representation of the solution of the Cauchy problem in terms of the solution of an associated matrix Riemann-Hilbert problem and its subsequent asymptotic analysis following the ideas of nonlinear steepest descent method.

math.AP↗

arXiv2018

Long-time asymptotics for the integrable nonlocal nonlinear Schrödinger equation

We study the initial value problem for the integrable nonlocal nonlinear Schrödinger (NNLS) equation \[ iq_{t}(x,t)+q_{xx}(x,t)+2σq^{2}(x,t)\bar{q}(-x,t)=0 \] with decaying (as $x\to\pm\infty$) boundary conditions. The main aim is to describe the long-time behavior of the solution of this problem. To do this, we adapt the nonlinear steepest-decent method \cite{DZ} to the study of the Riemann-Hilbert problem associated with the NNLS equation. Our main result is that, in contrast to the local NLS equation, where the main asymptotic term (in the solitonless case) decays to $0$ as $O(t^{-1/2})$ along any ray $x/t=const$, the power decay rate in the case of the NNLS depends, in general, on $x/t$, and can be expressed in terms of the spectral functions associated with the initial data.

math.AP↗

arXiv2018

Initial value problem for the time-dependent linear Schrödinger equation with a point singular potential by the uniform transform method

We study an initial value problem for the one-dimensional non-stationary linear Schrödinger equation with a point singular potential. In our approach, the problem is considered as a system of coupled initial-boundary value (IBV) problems on two half-lines, to which we apply the unified approach to IBV problems for linear and integrable nonlinear equations, also known as the Fokas unified transform method. Following the ideas of this method, we obtain the integral representation of the solution of the initial value problem. Since the unified approach is known as providing efficient solutions to both linear and nonlinear problems, the present paper can be viewed as a step in solving the initial value problem for the non-stationary {\em nonlinear} Schrödinger equation with a point singular potential.

math.AP↗