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P. Topalov

Publications and source records attributed to P. Topalov.

At least 19 recordsLinked to original sources
arXiv2026

Integrable metrics with potentials on the Hardy space and a class of PDEs

We introduce a new class of quadratic in the momenta Hamiltonians with potential on Banach spaces of infinite dimension which possess an infinite family of conserved quantities in involution with respect to a naturally defined Poisson structure. We show that the Hamiltonian flows of the constructed integrals are locally well defined and that they are related to a hierarchy of evolution PDEs via nonlinear analytic transformations of the phase space, called $Q$-transforms. Our framework offers a new geometric interpretation of hierarchies of integrable PDEs, linking infinite-dimensional phase spaces directly to the classical geometric phenomenon of geodesic equivalence.

math.DS↗
arXiv2024

The viscosity limit of fluid flows with growth/decay conditions at infinity

We prove that the Navier-Stokes equation is well-posed in function spaces on $\mathbb{R}^d$, $d\ge 2$, that contain vector fields of order $O(|x|^\kappa)$ as $|x|\to\infty$ with $\kappa<1/2$. The corresponding solutions depend continuously on the viscosity parameter $\nu\ge 0$ and converge to the solutions of the Euler equation as $\nu\to 0+$. Our proof is based on the properties of the conjugated heat flow on weighted Sobolev spaces and on a new variant of the Lie-Trotter product formula for nonlinear semigroups.

math.AP↗
arXiv2022

Spatial asymptotic expansions in the Navier-Stokes equation

We prove that the Navier-Stokes equation for a viscous incompressible fluid in $\mathbb{R}^d$ is locally well-posed in spaces of functions allowing spatial asymptotic expansions with log terms as $|x|\to\infty$ of any a priori given order. The solution depends analytically on the initial data and time so that for any $0<\vartheta<π/2$ it can be holomorphically extended in time to a conic sector in $\mathbb{C}$ with angle $2\vartheta$ at zero. We discuss the approximation of solutions by their asymptotic parts.

math.AP↗
arXiv2021

On the analyticity of the nonlinear Fourier transform of the Benjamin-Ono equation on $\mathbb{T}$

We prove that the nonlinear Fourier transform of the Benjamin-Ono equation on $\mathbb{T}$, also referred to as Birkhoff map, is a real analytic diffeomorphism from the scale of Sobolev spaces $H^{s}_{0}(\mathbb{T},\mathbb{R})$, $s > -1/2$, to the scale of weighted $\ell^2-$sequence spaces, $\mathfrak{h}^{s +1/2}_{r,0}(\mathbb{N},\mathbb{C})$, $s >-1/2$. As an application we show that for any $-1/2<s<0$, the flow map of the Benjamin-Ono equation $\mathcal{S}_0^t : H^{s}_{0}(\mathbb{T},\mathbb{R})\to H^{s}_{0}(\mathbb{T},\mathbb{R})$ is {\em nowhere locally uniformly continuous} in $H^{s}_{0}(\mathbb{T},\mathbb{R})$.

math.AP↗
arXiv2021

On the analytic Birkhoff normal form of the Benjamin-Ono equation and applications

In this paper we prove that the Benjamin-Ono equation admits an analytic Birkhoff normal form in an open neighborhood of zero in $H^{s}_{0}(\T, \R)$ for any $s>-1/2$ where $H^{s}_{0}(\T, \R)$ denotes the subspace of the Sobolev space $H^{s}(\T, \R)$ of elements with mean $0$. As an application we show that for any $-1/2<s<0$, the flow map of the Benjamin-Ono equation $\mathcal{S}_0^t : H^{s}_{0}(\T, \R)\to H^{s}_{0}(\T, \R)$ is nowhere locally uniformly continuous in a neighborhood of zero in $H^{s}_{0}(\T, \R)$.

math.AP↗
arXiv2020

Sharp well-posedness results of the Benjamin-Ono equation in $H^{s}(\mathbb{T},\mathbb{R})$ and qualitative properties of its solution

We prove that the Benjamin--Ono equation on the torus is globally in time well-posed in the Sobolev space $H^{s}(\mathbb{T},\mathbb{R})$ for any $s > - 1/2$ and ill-posed for $s \le - 1/2$. Hence the critical Sobolev exponent $s_c=-1/2$ of the Benjamin--Ono equation is the threshold for well-posedness on the torus. The obtained solutions are almost periodic in time. Furthermore, we prove that the traveling wave solutions of the Benjamin-Ono equation on the torus are orbitally stable in $H^{s}(\mathbb{T},\mathbb{R})$ for any $ s > - 1/2$. Novel conservation laws and a nonlinear Fourier transform on $H^{s}(\mathbb{T},\mathbb{R})$ with $s > - 1/2$ are key ingredients into the proofs of these results.

math.AP↗
arXiv2019

From K.A.M. Tori to Isospectral Invariants and Spectral Rigidity of Billiard Tables

This article is a part of a project investigating the relationship between the dynamics of completely integrable or close to completely integrable billiard tables, the integral geometry on them, and the spectrum of the corresponding Laplace-Beltrami operators. It is concerned with new isospectral invariants and with the spectral rigidity problem for families of Laplace-Beltrami operators with Dirichlet, Neumann or Robin boundary conditions, associated with C^1 families of billiard tables. We introduce a notion of weak isospectrality for such deformations. The main dynamical assumption on the initial billiard table is that the corresponding billiard ball map or an iterate of it has a Kronecker invariant torus with a Diophantine frequency and that the corresponding Birkhoff Normal Form is nondegenerate in Kolmogorov sense. Then we obtain C^1 families of Kronecker tori with Diophantine frequencies. If the family of the Laplace-Beltrami operators satisfies the weak isospectral condition, we prove that the average action on the tori and the Birkhoff Normal Form of the billiard ball maps remain the same along the perturbation. As an application we obtain infinitesimal spectral rigidity for Liouville billiard tables in dimensions two and three. Applications are obtained also for strictly convex billiard tables of dimension two as well as in the case when the initial billiard table admits an elliptic periodic billiard trajectory. Spectral rigidity of billard tables close elliptical billiard tables is obtained. The results are based on a construction of C^1 families of quasi-modes associated with the Kronecker tori and on suitable KAM theorems for C^1 families of Hamiltonians.

math.SP↗
arXiv2015

Scattering-like phenomena of the periodic defocusing NLS equation

In this paper we prove approximation properties of the solutions of the defoucsing NLS equation on the circle by nearly linear flows. In addition we show that spatially periodic solutions of the defocusing NLS equation evolving in fractional Sobolev spaces $H^s$ with $s\geq 1$ remain bounded for all times.

math.AP↗
arXiv2014

Interpolation of nonlinear maps

Let $(X_0, X_1)$ and $(Y_0, Y_1)$ be complex Banach couples and assume that $X_1\subseteq X_0$ with norms satisfying $\|x\|_{X_0} \le c\|x\|_{X_1}$ for some $c > 0$. For any $0<θ<1$, denote by $X_θ= [X_0, X_1]_θ$ and $Y_θ= [Y_0, Y_1]_θ$ the complex interpolation spaces and by $B(r, X_θ)$, $0 \le θ\le 1,$ the open ball of radius $r>0$ in $X_θ$, centered at zero. Then for any analytic map $Φ: B(r, X_0) \to Y_0+ Y_1$ such that $Φ: B(r, X_0)\to Y_0$ and $Φ: B(c^{-1}r, X_1)\to Y_1$ are continuous and bounded by constants $M_0$ and $M_1$, respectively, the restriction of $Φ$ to $B(c^{-θ}r, X_θ)$, $0 < θ< 1,$ is shown to be a map with values in $Y_θ$ which is analytic and bounded by $M_0^{1-θ} M_1^θ$.

math.FA↗
arXiv2013

Qualitative features of periodic solutions of KdV

In this paper we prove new qualitative features of solutions of KdV on the circle. The first result says that the Fourier coefficients of a solution of KdV in Sobolev space $H^N,\, N\geq 0$, admit a WKB type expansion up to first order with strongly oscillating phase factors defined in terms of the KdV frequencies. The second result provides estimates for the approximation of such a solution by trigonometric polynomials of sufficiently large degree.

math.AP↗
arXiv2012

Generic non-selfadjoint Zakharov-Shabat operators

In this paper we develop tools to study families of non-selfadjoint operators $L(φ), φ\in P$, characterized by the property that the spectrum of $L(φ)$ is (partially) simple. As a case study we consider the Zakharov-Shabat operators $L(φ)$ appearing in the Lax pair of the focusing NLS on the circle. The main result says that the set of potentials $φ$ of Sobolev class $H^N, N \geq 0$, so that all small eigenvalues of $L(φ)$ are simple, is path connected and dense.

math.SP↗
arXiv2010

Invariants of isospectral deformations and spectral rigidity

We introduce a notion of weak isospectrality for continuous deformations. Consider the Laplace-Beltrami operator on a compact Riemannian manifold with boundary with Robin boundary conditions. Given a Kronecker invariant torus $Λ$ of the billiard ball map with a vector of rotation satisfying a Diophantine condition we prove that certain integrals on $Λ$ involving the function in the Robin boundary conditions remain constant under weak isospectral deformations. To this end we construct continuous families of quasimodes associated with $Λ$. We obtain also isospectral invariants of the Laplacian with a real-valued potential on a compact manifold for continuous deformations of the potential. As an application we prove spectral rigidity in the case of Liouville billiard tables of dimension two.

math.SP↗