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Global invertible symmetries act unitarily on local observables or states of a quantum system. In this note, we aim to generalise this statement to non-invertible symmetries by considering unitary actions of higher fusion category symmetries $\mathcal{C}$ on twisted sector local operators. We propose that the latter transform in $\ast$-representations of the tube algebra associated to $\mathcal{C}$, which we introduce and classify using the notion of higher $S$-matrices of higher braided fusion categories., Comment: 18 pages

Global internal symmetries act unitarily on local observables or states of a quantum system. In this note, we aim to generalise this statement to extended observables by considering unitary actions of finite global 2-group symmetries $\mathcal{G}$ on line operators. We propose that the latter transform in unitary 2-representations of $\mathcal{G}$, which we classify up to unitary equivalence. Our results recover the known classification of ordinary 2-representations of finite 2-groups, but provide additional data interpreted as a type of reflection anomaly for $\mathcal{G}$., Comment: 36 pages, v2: citations added

We study the following Neumann boundary problem related to the stationary solutions of the Keller-Segel system, a basic model of chemotaxis phenomena: \[ -\Delta_g u +\beta u =\lambda\left(\frac{Ve^u}{\int_{\Sigma} Ve^u d v_g}-\frac{1}{|\Sigma|_g}\right) \text { in } \mathring\Sigma \] with $\partial_{ \nu_g} u=0, \text { on } \partial \Sigma$, where $(\Sigma, g)$ is a compact Riemann surface with the interior $\mathring\Sigma$ and the smooth boundary $\partial \Sigma$. Here, $\lambda, \beta\geq 0$ are non-negative parameters, and $V$ is a smooth non-negative function with a finite zero set. For any given integers $m\geq k\geq 0$, we establish a sufficient condition on $V$ for the existence of a sequence of blow-up solutions as $\lambda$ approaches the critical values $4\pi(m+k)$. Moreover, the study expands to the corresponding singular problem.

This work studies the partial blow-up phenomena for the $SU(3)$ Toda system on compact Riemann surfaces with smooth boundary. We consider the following coupled Liouville system with Neumann boundary conditions: $$ -\Delta_g u_1 = 2\rho_1\left( \frac{V_1 e^{u_1}}{\int_{\Sigma} V_1 e^{u_1} \, dv_g} - \frac 1 {|\Sigma|_g}\right) - \rho_2\left( \frac{V_2 e^{u_2}}{\int_{\Sigma} V_2 e^{u_2} \, dv_g} - \frac{1}{|\Sigma|_g}\right) \text{in} \,\mathring\Sigma$$ and $$ -\Delta_g u_2 = 2\rho_2\left( \frac{V_2 e^{u_2}}{\int_{\Sigma} V_2 e^{u_2} \, dv_g} - \frac{1}{|\Sigma|_g}\right) - \rho_1\left( \frac{V_1 e^{u_1}}{\int_{\Sigma} V_1 e^{u_1} \, dv_g} - \frac{1}{|\Sigma|_g}\right) \text{in} \,\mathring\Sigma$$ with boundary conditions $ \partial_{\nu_g} u_1 = \partial_{\nu_g} u_2 = 0 \text{ on} \, \partial \Sigma,$ where $(\Sigma, g)$ is a compact Riemann surface with the interior $\mathring\Sigma$ and smooth boundary $\partial\Sigma$, $\rho_i$ is a non-negative parameter and $V_i$ is a smooth positive function for $i=1,2$. We construct a family of blow-up solutions via the Lyapunov-Schmidt reduction and variational methods, wherein one component remains uniformly bounded from above, while the other exhibits partial blow-ups at a prescribed number of points, both in the interior and on the boundary. This construction is based on the existence of a non-degeneracy solution of a so-called shadow system. Moreover, we establish the existence of partial blow-up solutions in three cases: (i) for any $\rho_2>0$ sufficiently small; (ii) for generic $V_1, V_2$ and any $\rho_2\in (0,2\pi)$; (iii) for generic $V_1, V_2$, the Euler characteristic $\chi(\Sigma)<1$ and any $\rho_2\in (2\pi,+\infty)\setminus 2\pi \mathbb{N}_+$.

On a compact Riemann surface $(\Sigma, g)$ with a smooth boundary $\partial \Sigma$, we consider the following mean field equations with Neumann boundary conditions: $$ -\Delta_g u = \lambda \left(\frac{Ve^u}{\int_{\Sigma} Ve^u \, dv_g} - \frac{1}{|\Sigma|_g}\right) \text{ in } \Sigma \text{ with } \partial_{\nu_g} u = 0 \text{ on } \partial \Sigma, $$ We find conditions on the potential function $V: \Sigma \to \mathbb{R}^+$ such that solutions exist for the parameter $\lambda$ when it is in a small right (or left) neighborhood of a critical value $4\pi(m+k)$ for $k \leq m \in \mathbb{N}_+$ and blow up as $\lambda$ approaches the critical parameter. The blow-up occurs exactly at $k$ points in the interior of $\Sigma$ and $(m-k)$ points on the boundary $\partial \Sigma$.

In this paper we study the Morse property for functions related to limit functions of mean field equations on a smooth, compact surface $\Sigma$ with boundary $\partial\Sigma$. Given a Riemannian metric $g$ on $\Sigma$ we consider functions of the form \[ f_g(x) := \sum_{i=1}^m\sigma_i^2R^g(x_i)+\sum_{i,j=1\\i\ne j}^m\sigma_i\sigma_jG^g(x_i,x_j)+h(x_1,\ldots,x_m), \] where $\sigma_i \neq 0$ for $i=1,\ldots,m$, $G^g$ is the Green function of the Laplace-Beltrami operator on $(\Sigma,g)$ with Neumann boundary conditions, $R^g$ is the corresponding Robin function, and $h \in \mathcal{C}^{2}(\Sigma^m,\mathbb{R})$ is arbitrary. We prove that for any Riemannian metric $g$, there exists a metric $\widetilde g$ which is arbitrarily close to $g$ and in the conformal class of $g$ such that $f_{\widetilde g}$ is a Morse function. Furthermore we show that, if all $\sigma_i>0$, then the set of Riemannian metrics for which $f_g$ is a Morse function is open and dense in the set of all Riemannian metrics., Comment: 19 pages

The paper addresses an open problem raised in [Bartsch, Molle, Rizzi, Verzini: Normalized solutions of mass supercritical Schr\"odinger equations with potential, Comm. Part. Diff. Equ. 46 (2021), 1729-1756] on the existence of normalized solutions to Schr\"odinger equations with potentials and inhomogeneous nonlinearities. We consider the problem \[ -\Delta u+V(x)u+\lambda u = |u|^{q-2}u+\beta |u|^{p-2}u, \quad \|u\|^2_2=\int|u|^2dx = \alpha, \] both on $\mathbb{R}^N$ as well as on domains $r\Omega$ where $\Omega\subset\mathbb{R}^N$ is an open bounded convex domain and $r>0$ is large. The exponents satisfy $2

This paper addresses the question of how categorical symmetries act on extended operators in quantum field theory. Building on recent results in two dimensions, we introduce higher tube categories and algebras associated to higher fusion category symmetries. We show that twisted sector extended operators transform in higher representations of higher tube algebras and interpret this result from the perspective of the sandwich construction of finite symmetries via the Drinfeld center. Focusing on three dimensions, we discuss a variety of examples to illustrate the general constructions. In the case of invertible symmetries, we show that higher tube algebras are higher analogues of twisted Drinfeld doubles of finite groups, generalising known constructions in two dimensions. Building on this foundation, we discuss non-invertible Ising-like symmetry categories obtained by gauging finite subgroups. We also consider non-invertible topological symmetry lines described by braided fusion categories and discuss connections to the M\"uger center and braided module categories., Comment: 84 pages + appendix, 59 figures

It is known that local operators in quantum field theory transform in representations of ordinary global symmetry groups. The purpose of this paper is to generalise this statement to extended operators such as line and surface defects. We explain that $(n-1)$-dimensional operators transform in $n$-representations of a finite invertible or group-like symmetry and thoroughly explore this statement for $n = 1,2,3$. We therefore propose higher representation theory as the natural framework to describe the action of symmetries on the extended operator content in quantum field theory., Comment: 60 pages + appendix. v2: minor corrections

Let $(M_1,\textit{g}^{(1)})$, $(M_2,\textit{g}^{(2)})$ be closed Riemannian spin manifolds. We study the existence of solutions of the spinorial Yamabe problem on the product $M_1\times M_2$ equipped with a family of metrics $\varepsilon^{-2}\textit{g}^{(1)}\oplus\textit{g}^{(2)}$, $\varepsilon>0$. Via variational methods and blow-up techniques, we prove the existence of solutions which depend only on the factor $M_1$, and which exhibit a spike layer as $\varepsilon\to0$. Moreover, we locate the asymptotic position of the peak points of the solutions in terms of the curvature tensor on $(M_1,\textit{g}^{(1)})$., Comment: 38 Pages. arXiv admin note: text overlap with arXiv:2005.01448

We study the possible blow-up behavior of solutions to the slightly subcritical elliptic problem with Hardy term \[ \left\{ \begin{aligned} -\Delta u-\mu\frac{u}{|x|^2} &= |u|^{2^{\ast}-2-\varepsilon}u &&\quad \text{in } \Omega, \\\ u &= 0&&\quad \text{on } \partial\Omega, \end{aligned} \right. \] in a bounded domain $\Omega\subset\mathbb{R}^N (N\ge7)$ with $0\in\Omega$, as $\mu,\varepsilon\to 0^+$. In \cite{BarGuo-ANS}, we obtained the existence of nodal solutions that blow up positively at the origin and negatively at a different point as $\mu=O(\epsilon^\alpha)$ with $\alpha>\frac{N-4}{N-2}$, $\varepsilon\to 0^+$. Here we prove the existence of nodal bubble tower solutions, i.e.\ superpositions of bubbles of different signs, all blowing up at the origin but with different blow-up order, as $\mu=O(\varepsilon)$, $\varepsilon\to0^+$., Comment: 18 pages. arXiv admin note: text overlap with arXiv:1508.04007

We consider the slightly subcritical elliptic problem with Hardy term $$ \left\{ \begin{aligned} -\Delta u-\mu\frac{u}{|x|^2} &= |u|^{2^{\ast}-2-\epsilon}u &&\quad \text{in } \Omega\subset\mathbb{R}^N, \\\ u &= 0&&\quad \text{on } \partial \Omega, \end{aligned} \right. $$ where $0\in\Omega$ and $\Omega$ is invariant under the subgroup $SO(2)\times\{\pm E_{N-2}\}\subset O(N)$; here $E_n$ denots the $n\times n$ identity matrix. If $\mu=\mu_0\epsilon^\alpha$ with $\mu_0>0$ fixed and $\alpha>\frac{N-4}{N-2}$ the existence of nodal solutions that blow up, as $\epsilon\to0^+$, positively at the origin and negatively at a different point in a general bounded domain has been proved in \cite{BarGuo-ANS}. Solutions with more than two blow-up points have not been found so far. In the present paper we obtain the existence of nodal solutions with a positive blow-up point at the origin and $k=2$ or $k=3$ negative blow-up points placed symmetrically in $\Omega\cap(\mathbb{R}^2\times\{0\})$ around the origin provided a certain function $f_k:\mathbb{R}^+\times\mathbb{R}^+\times I\to\mathbb{R}$ has stable critical points; here $I=\{t>0:(t,0,\dots,0)\in\Omega\}$. If $\Omega=B(0,1)\subset\mathbb{R}^N$ is the unit ball centered at the origin we obtain two solutions for $k=2$ and $N\ge7$, or $k=3$ and $N$ large. The result is optimal in the sense that for $\Omega=B(0,1)$ there cannot exist solutions with a positive blow-up point at the origin and four negative blow-up points placed on the vertices of a square centered at the origin. Surprisingly there do exist solutions on $\Omega=B(0,1)$ with a positive blow-up point at the origin and four blow-up points on the vertices of a square with alternating positive and negative signs. The results of our paper show that the structure of the set of blow-up solutions of the above problem offers fascinating features and is not well understood., Comment: 22 pages

In this paper we continue our investigation of the global categorical symmetries that arise when gauging finite higher groups and their higher subgroups with discrete torsion. The motivation is to provide a common perspective on the construction of non-invertible global symmetries in higher dimensions and a precise description of the associated symmetry categories. We propose that the symmetry categories obtained by gauging higher subgroups may be defined as higher group-theoretical fusion categories, which are built from the projective higher representations of higher groups. As concrete applications we provide a unified description of the symmetry categories of gauge theories in three and four dimensions based on the Lie algebra $\mathfrak{so}(N)$, and a fully categorical description of non-invertible symmetries obtained by gauging a 1-form symmetry with a mixed 't Hooft anomaly. We also discuss the effect of discrete torsion on symmetry categories, based a series of obstructions determined by spectral sequence arguments., Comment: 56 pages + appendix, v2: clarifications and citations added

The paper is concerned with the existence and asymptotic properties of normalized ground states of the following nonlinear Schr\"odinger system with critical exponent: \begin{equation*} \left\{\begin{aligned} &-\delta u+\lambda_1 u=|u|^{2^*-2}u+{\nu\alpha} |u|^{\alpha-2}|v|^\beta u,\quad \text{in }\mathbb{R}^N, &-\delta v+\lambda_2 v=|v|^{2^*-2}v+{\nu\beta} |u|^\alpha |v|^{\beta-2}v,\quad \text{in }\mathbb{R}^N, &\int u^2=a^2,\;\;\; \int v^2=b^2, \end{aligned} \right. \end{equation*} where $N=3,4$, $\alpha,\beta>1$, $2<\alpha+\beta<2^*=\frac{2N}{N-2}$. We prove that a normalized ground state does not exist for $\nu<0$. When $\nu>0$ and $\alpha+\beta\le 2+\frac{4}{N}$, we show that the system has a normalized ground state solution for $0<\nu<\nu_0$, the constant $\nu_0$ will be explicitly given. In the case $\alpha+\beta>2+\frac{4}{N}$ we prove the existence of a threshold $\nu_1\ge 0$ such that a normalized ground state solution exists for $\nu>\nu_1$, and does not exist for $\nu<\nu_1$. We also give conditions for $\nu_1=0$. Finally we obtain the asymptotic behavior of the minimizers as $\nu\to0^+$ or $\nu\to+\infty$.

We prove the existence of critical points of vortex type Hamiltonians \[ H(p_1,\ldots, p_N) = \sum_{{i,j=1},{i\ne j}}^N \Gamma_i\Gamma_jG(p_i,p_j)+\psi(p_1,\dots,p_N) \] on a closed Riemannian surface $(\Sigma,g)$ which is not homeomorphic to the sphere or the projective plane. Here $G$ denotes the Green function of the Laplace-Beltrami operator in $\Sigma$, $\psi:\Sigma^N\to\mathbb{R}$ may be any function of class $C^1$, and $\Gamma_1,\dots,\Gamma_N\in\mathbb{R}\setminus\{0\}$ are the vorticities. The Kirchhoff-Routh Hamiltonian from fluid dynamics corresponds to $\psi = -\sum_{i=1}^N \Gamma_i^2h(p_i,p_i)$ where $h:\Sigma\times\Sigma\to\mathbb{R}$ is the regular part of the Laplace-Beltrami operator. We obtain critical points $p=(p_1,\dots,p_N)$ for arbitrary $N$ and vorticities $(\Gamma_1,\dots,\Gamma_N)$ in $\mathbb{R}^N\setminus V$ where $V$ is an explicitly given algebraic variety of codimension 1., Comment: 18 pages, to appear in Topol. Methods Nonlinear Anal

This paper is concerned with the existence of normalized solutions of the nonlinear Schr\"odinger equation \[ -\Delta u+V(x)u+\lambda u = |u|^{p-2}u \qquad\text{in $\mathbb{R}^N$} \] in the mass supercritical and Sobolev subcritical case $2+\frac{4}{N}

We study semiclassical states of the nonlinear Dirac equation \[ -i\hbar\partial_t\psi = ic\hbar\sum_{k=1}^3\alpha_k\partial_k\psi - mc^2\beta \psi - M(x)\psi + f(|\psi|)\psi,\quad t\in\mathbb{R},\ x\in\mathbb{R}^3, \] where $V$ is a bounded continuous potential function and the nonlinear term $f(|\psi|)\psi$ is superlinear, possibly of critical growth. Our main result deals with standing wave solutions that concentrate near a critical point of the potential. Standard methods applicable to nonlinear Schr\"odinger equations, like Lyapunov-Schmidt reduction or penalization, do not work, not even for the homogeneous nonlinearity $f(s)=s^p$. We develop a variational method for the strongly indefinite functional associated to the problem.

In the present paper, we prove the existence of solutions $(\lambda_1,\lambda_2,u,v)\in\mathbb{R}^2\times H^1(\mathbb{R}^3,\mathbb{R}^2)$ to systems of coupled Schr\"odinger equations $$ \begin{cases} -\Delta u+\lambda_1u=\mu_1 u^3+\beta uv^2\quad &\hbox{in}\;\mathbb{R}^3\\ -\Delta v+\lambda_2v=\mu_2 v^3+\beta u^2v\quad&\hbox{in}\;\mathbb{R}^3\\ u,v>0&\hbox{in}\;\mathbb{R}^3 \end{cases} $$ satisfying the normalization constraint $ \displaystyle\int_{\mathbb{R}^3}u^2=a^2\quad\hbox{and}\;\int_{\mathbb{R}^3}v^2=b^2, $ which appear in binary mixtures of Bose-Einstein condensates or in nonlinear optics. The parameters $\mu_1,\mu_2,\beta>0$ are prescribed as are the masses $a,b>0$. The system has been considered mostly in the fixed frequency case. And when the masses are prescribed, the standard approach to this problem is variational with $\lambda_1,\lambda_2$ appearing as Lagrange multipliers. Here we present a new approach based on bifurcation theory and the continuation method. We obtain the existence of normalized solutions for any given $a,b>0$ for $\beta$ in a large range. We also give a result about the nonexistence of positive solutions. From which one can see that our existence theorem is almost the best. Especially, if $\mu_1=\mu_2$ we prove that normalized solutions exist for all $\beta>0$ and all $a,b>0$., Comment: 27 pages, 1 figure

Reaction rates of chemical reactions under nonequilibrium conditions can be determined through the construction of the normally hyperbolic invariant manifold (NHIM) [and moving dividing surface (DS)] associated with the transition state trajectory. Here, we extend our recent methods by constructing points on the NHIM accurately even for multidimensional cases. We also advance the implementation of machine learning approaches to construct smooth versions of the NHIM from a known high-accuracy set of its points. That is, we expand on our earlier use of neural nets, and introduce the use of Gaussian process regression for the determination of the NHIM. Finally, we compare and contrast all of these methods for a challenging two-dimensional model barrier case so as to illustrate their accuracy and general applicability., Comment: 28 pages, 13 figures, table of contents figure

Let $(M,\textit{g},\sigma)$ be a compact Riemannian spin manifold of dimension $m\geq2$, let $\mathbb{S}(M)$ denote the spinor bundle on $M$, and let $D$ be the Atiyah-Singer Dirac operator acting on spinors $\psi:M\to\mathbb{S}(M)$. We study the existence of solutions of the nonlinear Dirac equation with critical exponent \[ D\psi = \lambda\psi + f(|\psi|)\psi + |\psi|^{\frac2{m-1}}\psi \tag{NLD} \] where $\lambda\in\mathbb{R}$ and $f(|\psi|)\psi$ is a subcritical nonlinearity in the sense that $f(s)=o\big(s^{\frac2{m-1}}\big)$ as $s\to\infty$. A model nonlinearity is $f(s)=\alpha s^{p-2}$ with $20$, even if $\lambda$ is an eigenvalue of $D$. For some classes of nonlinearities $f$ we also obtain solutions of (NLD) for every $\lambda\in\mathbb{R}$, except for non-positive eigenvalues. If $m\not\equiv3$ (mod 4) we obtain solutions of (NLD) for every $\lambda\in\mathbb{R}$, except for a finite number of non-positive eigenvalues. In certain parameter ranges we obtain multiple solutions of (NLD) and (BND), some near the trivial branch, others away from it. The proofs of our results are based on variational methods using the strongly indefinite energy functional associated to (NLD)., Comment: 42 pages

For a bounded domain $\Omega\subset\mathbb{R}^n$ let $H_\Omega:\Omega\times\Omega\to\mathbb{R}$ be the regular part of the Dirichlet Green function for the Laplace operator. Given a fixed arbitrary ${\mathcal C}^2$ function $f:{\mathcal D}\to\mathbb{R}$, defined on an open subset ${\mathcal D}\subset\mathbb{R}^{nN}$, and fixed coefficients $\lambda_1,\dots,\lambda_N\in\mathbb{R}\setminus\{0\}$ we consider the function $f_\Omega:{\mathcal D}\cap\Omega^N\to\mathbb{R}$ defined as \[ f_\Omega(x_1,\dots,x_N) = f(x_1,\dots,x_N) - \sum_{j,k=1}^N \lambda_j\lambda_k H_\Omega(x_j,x_k). \] We prove that $f_\Omega$ is a Morse function for most domains $\Omega$ of class ${\mathcal C}^{m+2,\alpha}$, any $m\ge0$, $0<\alpha<1$. This applies in particular to the Robin function $h:\Omega\to\mathbb{R}$, $h(x)=H_\Omega(x,x)$, and to the Kirchhoff-Routh path function where $\Omega\subset\mathbb{R}^2$, ${\mathcal D}=\{x\in\mathbb{R}^{2N}: \text{$x_j\ne x_k$ for $j\ne k$}\}$, and \[ f(x_1,\dots,x_N) = - \frac{1}{2\pi}\sum_{\genfrac{}{}{0pt}{}{j,k=1}{j\ne k}}^N\lambda_j\lambda_k\log|x_j-x_k|. \], Comment: 14 pages

We prove the existence of infinitely many solutions $\lambda_1, \lambda_2 \in \mathbb{R}$, $u,v \in H^1(\mathbb{R}^3)$, for the nonlinear Schr\"odinger system \[ \begin{cases} -\Delta u - \lambda_1 u = \mu u^3+ \beta u v^2 & \text{in $\mathbb{R}^3$} -\Delta v- \lambda_2 v = \mu v^3 +\beta u^2 v & \text{in $\mathbb{R}^3$} u,v>0 & \text{in $\mathbb{R}^3$} \int_{\mathbb{R}^3} u^2 = a^2 \quad \text{and} \quad \int_{\mathbb{R}^3} v^2 = a^2, \end{cases} \] where $a,\mu>0$ and $\beta \le -\mu$ are prescribed. Our solutions satisfy $u\ne v$ so they do not come from a scalar equation., Comment: 21 pages, substantial difference with respect to the previous version, the section 3 is entirely new

The search for time-harmonic solutions of nonlinear Maxwell equations in the absence of charges and currents leads to the elliptic equation $$\nabla\times\left(\mu(x)^{-1} \nabla\times u\right) - \omega^2\varepsilon(x)u = f(x,u)$$ for the field $u:\Omega\to\mathbb{R}^3$ in a domain $\Omega\subset\mathbb{R}^3$. Here $\varepsilon(x) \in \mathbb{R}^{3\times3}$ is the (linear) permittivity tensor of the material, and $\mu(x) \in \mathbb{R}^{3\times3}$ denotes the magnetic permeability tensor. The nonlinearity $f:\Omega\times\mathbb{R}^3\to\mathbb{R}^3$ comes from the nonlinear polarization. If $f=\nabla_uF$ is a gradient then this equation has a variational structure. The goal of this paper is to give an introduction to the problem and the variational approach, and to survey recent results on ground and bound state solutions. It also contains refinements of known results and some new results.

The paper deals with the existence of nonstationary collision-free periodic solutions of singular first order Hamiltonian systems of $N$-vortex type in a domain $\Omega\subset\mathbb{C}$. These are solutions $z(t)=(z_1(t),\dots,z_N(t))$ of \[ \dot{z}_j(t)=-i\nabla_{z_j} H_\Omega\big(z(t)\big),\quad j=1,\dots,N, \tag{HS} \] where the Hamiltonian $H_\Omega$ has the form \[ H_\Omega(z_1,\dots,z_N) = -\sum_{{j,k=1}\over{j\ne k}}^N \frac{1}{2\pi}\log|z_j-z_k| -\sum_{j,k=1}^N g(z_j,z_k). \] The function $g:\Omega\times\Omega\to\mathbb{R}$ is required to be of class $C^3$ and symmetric, the regular part of a hydrodynamic Green function being our model. The Hamiltonian is unbounded from above and below, and the associated action integral is not defined on an open subset of the space of periodic $H^{1/2}$ functions. Given a closed connected component $\Gamma\subset\partial\Omega$ of class $C^3$ we are interested in periodic solutions of (HS) near $\Gamma$. We present quite general conditions on the behavior of $g$ near $\Gamma$ which imply that there exists a family of periodic solutions $z^{(r)}(t)$, $0

We consider Hamiltonian systems with two degrees of freedom of point vortex type \[ \kappa_j \dot{z}_j = J \nabla_{z_j} H_\Omega(z_1,z_2), \quad j=1,2, \] for $z_1,z_2$ in a domain $\Omega\subset\mathbb{R}^2$. In the classical point vortex context the Hamiltonian $H_\Omega$ is of the form \[ H_\Omega(z_1,z_2) = -\frac{\kappa_1 \kappa_2}{\pi} \log |z_1-z_2| - 2\kappa_1 \kappa_2g(z_1,z_2) - \kappa_1^2 h(z_1) - \kappa_2^2 h(z_2), \] where $g:\Omega\times\Omega\to\mathbb{R}$ is the regular part of a hydrodynamic Green function in $\Omega$, $h:\Omega\to\mathbb{R}$ is the Robin function: $h(z)=g(z,z)$, and $\kappa_1$, $\kappa_2$ are the vortex strengths. We prove the existence of infinitely many periodic solutions with prescribed minimal period that are superpositions of a slow motion of the center of vorticity close to a star-shaped level line of $h$ and of a fast rotation of the two vortices around their center of vorticity. The proofs are based on a recent higher dimensional version of the Poincar\'e-Birkhoff theorem due to Fonda and Ure\~na., Comment: 17 pages

The usual identification of reactive trajectories for the calculation of reaction rates requires very time-consuming simulations, particularly if the environment presents memory effects. In this paper, we develop a new method that permits the identification of reactive trajectories in a system under the action of a stochastic colored driving. This method is based on the perturbative computation of the invariant structures that act as separatrices for reactivity. Furthermore, using this perturbative scheme, we have obtained a formally exact expression for the reaction rate in multidimensional systems coupled to colored noisy environments., Comment: 14 pages, 5 figures

We are concerned with the dynamics of $N$ point vortices $z_1,\dots,z_N\in\Omega\subset\mathbb{R}^2$ in a planar domain. This is described by a Hamiltonian system \[ \Gamma_k\dot{z}_k(t)=J\nabla_{z_k} H\big(z(t)\big),\quad k=1,\dots,N, \] where $\Gamma_1,\dots,\Gamma_N\in\mathbb{R}\setminus\{0\}$ are the vorticities, $J\in\mathbb{R}^{2\times2}$ is the standard symplectic $2\times2$ matrix, and the Hamiltonian $H$ is of $N$-vortex type: \[ H(z_1,\dots,z_N) = -\frac1{2\pi}\sum_{j\ne k}^N \Gamma_j\Gamma_k\log|z_j-z_k| - \sum_{j,k=1}^N\Gamma_j\Gamma_kg(z_j,z_k). \] Here $g:\Omega\times\Omega\to\mathbb{R}$ is an arbitrary symmetric function of class $C^2$, e.g.\ the regular part of a hydrodynamic Green function. Given a nondegenerate critical point $a_0\in\Omega$ of $h(z)=g(z,z)$ and a nondegenerate relative equilibrium $Z(t)\in\mathbb{R}^{2N}$ of the Hamiltonian system in the plane with $g=0$, we prove the existence of a smooth path of periodic solutions $z^{(r)}(t)=\big(z^{(r)}_1(t),\dots,z^{(r)}_N(t)\big)\in\Omega^N$, $0

We prove existence of normalized solutions to \[ \begin{cases} -\Delta u - \lambda_1 u = \mu_1 u^3+ \beta u v^2 & \text{in $\mathbb{R}^3$} -\Delta v- \lambda_2 v = \mu_2 v^3 +\beta u^2 v & \text{in $\mathbb{R}^3$}\int_{\mathbb{R}^3} u^2 = a_1^2 \quad \text{and} \quad \int_{\mathbb{R}^3} v^2 = a_2^2 \end{cases} \] for any $\mu_1,\mu_2,a_1,a_2>0$ and $\beta<0$ prescribed. The approach is based upon the introduction of a natural constraint associated to the problem. Our method can be adapted to the scalar NLS equation with normalization constraint, and leads to alternative and simplified proofs to some results already available in the literature., Comment: Final version, accepted for publication on J. Functional Analysis

The accuracy of rate constants calculated using transition state theory depends crucially on the correct identification of a recrossing--free dividing surface. We show here that it is possible to define such optimal dividing surface in systems with non--Markovian friction. However, a more direct approach to rate calculation is based on invariant manifolds and avoids the use of a dividing surface altogether, Using that method we obtain an explicit expression for the rate of crossing an anharmonic potential barrier. The excellent performance of our method is illustrated with an application to a realistic model for LiNC$\rightleftharpoons$LiCN isomerization., Comment: 5 pages, 3 figures, 1 table

The paper deals with singular first order Hamiltonian systems of the form \[ \Gamma_k\dot{z}_k(t)=J\nabla_{z_k} H\big(z(t)\big),\quad z_k(t) \in \Omega \subset \mathbb{R}^2,\ k=1,\dots,N, \] where $J\in\mathbb{R}^{2\times2}$ defines the standard symplectic structure in $\mathbb{R}^2$, and the Hamiltonian $H$ is of $N$-vortex type: \[ H(z_1,\dots,z_N) = -\frac1{2\pi} \sum_{j\neq k=1}^N \Gamma_j \Gamma_k \log|z_j-z_k| - F(z). \] This is defined on the configuration space $\{(z_1,\ldots,z_N)\in \Omega^{2N}:z_j\neq z_k\text{ for }j\neq k\}$ of $N$ different points in the domain $\Omega\subset\mathbb{R}^2$. The function $F:\Omega^N\to\mathbb{R}$ may have additional singularities near the boundary of $\Omega^N$. We prove the existence of a global continuum of periodic solutions $z(t)=(z_1(t),\dots,z_N(t))\in\Omega^N$ that emanates, after introducing a suitable singular limit scaling, from a relative equilibrium $Z(t)\in\mathbb{R}^{2N}$ of the $N$-vortex problem in the whole plane (where $F=0$). Examples for $Z$ include Thomson's vortex configurations, or equilateral triangle solutions. The domain $\Omega$ need not be simply connected. A special feature is that the associated action integral is not defined on an open subset of the space of $2\pi$-periodic $H^{1/2}$ functions, the natural form domain for first order Hamiltonian systems. This is a consequence of the singular character of the Hamiltonian. Our main tool in the proof is a degree for $S^1$-equivariant gradient maps that we adapt to this class of potential operators., Comment: 26 pages

We find solutions $E:\Omega\to\mathbb{R}^3$ of the problem \begin{eqnarray*} \left\{ \begin{aligned} &\nabla\times(\mu(x)^{-1}\nabla\times E) - \omega^2\epsilon(x) E = \partial_E F(x,E) &&\quad \text{in }\Omega\\%\newline &\nu\times E = 0 &&\quad \text{on }\partial\Omega \end{aligned} \right. \end{eqnarray*} on a bounded Lipschitz domain $\Omega\subset\mathbb{R}^3$ with exterior normal $\nu:\partial\Omega\to\mathbb{R}^3$. Here $\nabla\times$ denotes the curl operator in $\mathbb{R}^3$. The equation describes the propagation of the time-harmonic electric field $\Re\{E(x)e^{i\omega t}\}$ in an anisotropic material with a magnetic permeability tensor $\mu(x)\in\mathbb{R}^{3\times3}$ and a permittivity tensor $\epsilon(x)\in\mathbb{R}^{3\times3}$. The boundary conditions are those for $\Omega$ surrounded by a perfect conductor. It is required that $\mu(x)$ and $\epsilon(x)$ are symmetric and positive definite uniformly for $x\in\Omega$, and that $\mu,\epsilon\in L^{\infty}(\Omega,\mathbb{R}^{3\times 3})$. The nonlinearity $F:\Omega\times\mathbb{R}^3\to\mathbb{R}$ is superquadratic and subcritical in $E$, the model nonlinearity being of Kerr-type: $F(x,E)=|\Gamma(x)E|^p$ for some $2

The paper is concerned with the slightly subcritical elliptic problem with Hardy term \[ \left\{ \begin{aligned} -\Delta u-\mu\frac{u}{|x|^2} &= |u|^{2^{\ast}-2-\epsilon}u &&\quad \text{in } \Omega, \\\ u &= 0&&\quad \text{on } \partial\Omega, \end{aligned} \right. \] in a bounded domain $\Omega\subset\mathbb{R}^N$ with $0\in\Omega$, in dimensions $N\ge7$. We prove the existence of multi-bubble nodal solutions that blow up positively at the origin and negatively at a different point as $\epsilon\to0$ and $\mu=\epsilon^\alpha$ with $\alpha>\frac{N-4}{N-2}$. In the case of $\Omega$ being a ball centered at the origin we can obtain solutions with up to $5$ bubbles of different signs. We also obtain nodal bubble tower solutions, i.e. superpositions of bubbles of different signs, all blowing up at the origin but with different blow-up order. The asymptotic shape of the solutions is determined in detail., Comment: 51 pages

We consider the existence of \emph{normalized} solutions in $H^1(\R^N) \times H^1(\R^N)$ for systems of nonlinear Schr\"odinger equations which appear in models for binary mixtures of ultracold quantum gases. Making a solitary wave ansatz one is led to coupled systems of elliptic equations of the form \[ \left\{ \begin{aligned} -\De u_1 &= \la_1u_1 + f_1(u_1)+\pa_1F(u_1,u_2),\\ -\De u_2 &= \la_2u_2 + f_2(u_2)+\pa_2F(u_1,u_2),\\ u_1,u_2&\in H^1(\R^N),\ N\ge2, \end{aligned} \right. \] and we are looking for solutions satisfying \[ \int_{\R^N}|u_1|^2 = a_1,\quad \int_{\R^N}|u_2|^2 = a_2 \] where $a_1>0$ and $a_2>0$ are prescribed. In the system $\la_1$ and $\la_2$ are unknown and will appear as Lagrange multipliers. We treat the case of homogeneous nonlinearities, i.e.\ $f_i(u_i)=\mu_i|u_i|^{p_i-1}u_i$, $F(u_1,u_2)=\be|u_1|^{r_1}|u_2|^{r_2}$, with positive constants $\be, \mu_i, p_i, r_i$. The exponents are Sobolev subcritical but may be $L^2$-supercritical: $p_1,p_2,r_1+r_2\in]2,2^*[\,\setminus\left\{2+\frac4N\right\}$., Comment: 19 pages

We consider the system of coupled elliptic equations \[ \begin{cases} -\Delta u - \lambda_1 u = \mu_1 u^3+ \beta u v^2 \\ -\Delta v- \lambda_2 v = \mu_2 v^3 +\beta u^2 v \end{cases} \text{in $\mathbb{R}^3$}, \] and study the existence of positive solutions satisfying the additional condition \[ \int_{\mathbb{R}^3} u^2 = a_1^2 \quad \text{and} \quad \int_{\mathbb{R}^3} v^2 = a_2^2. \] Assuming that $a_1,a_2,\mu_1,\mu_2$ are positive fixed quantities, we prove existence results for different ranges of the coupling parameter $\beta>0$. The extension to systems with an arbitrary number of components is discussed, as well as the orbital stability of the corresponding standing waves for the related Schr\"odinger systems.

Transition State Theory is a central cornerstone in reaction dynamics. Its key step is the identification of a dividing surface that is crossed only once by all reactive trajectories. This assumption is often badly violated, especially when the reactive system is coupled to an environment. The calculations made in this way then overestimate the reaction rate and the results depend critically on the choice of the dividing surface. In this Letter, we study the phase space of a stochastically driven system close to an energetic barrier in order to identify the geometric structure unambiguously determining the reactive trajectories, which is then incorporated in a simple rate formula for reactions in condensed phase that is both independent of the dividing surface and exact., Comment: 4 pages, 3 figures

When a chemical reaction is driven by an external field, the transition state that the system must pass through as it changes from reactant to product -for example, an energy barrier- becomes time-dependent. We show that for periodic forcing the rate of barrier crossing can be determined through stability analysis of the non-autonomous transition state. Specifically, strong agreement is observed between the difference in the Floquet exponents describing stability of the transition state trajectory, which defines a recrossing-free dividing surface [G. T. Craven, T. Bartsch, and R. Hernandez, Phys. Rev. E 89, 040801(R) (2014)], and the rates calculated by simulation of ensembles of trajectories. This result opens the possibility to extract rates directly from the intrinsic stability of the transition state, even when it is time-dependent, without requiring a numerically-expensive simulation of the long-time dynamics of a large ensemble of trajectories.

Chemical reaction rates must increasingly be determined in systems that evolve under the control of external stimuli. In these systems, when a reactant population is induced to cross an energy barrier through forcing from a temporally varying external field, the transition state that the reaction must pass through during the transformation from reactant to product is no longer a fixed geometric structure, but is instead time-dependent. For a periodically forced model reaction, we develop a recrossing-free dividing surface that is attached to a transition state trajectory [T. Bartsch, R. Hernandez, and T. Uzer, Phys. Rev. Lett. 95, 058301 (2005)]. We have previously shown that for single-mode sinusoidal driving, the stability of the time-varying transition state directly determines the reaction rate [G. T. Craven, T. Bartsch, and R. Hernandez, J. Chem. Phys. 141, 041106 (2014)]. Here, we extend our previous work to the case of multi-mode driving waveforms. Excellent agreement is observed between the rates predicted by stability analysis and rates obtained through numerical calculation of the reactive flux. We also show that the optimal dividing surface and the resulting reaction rate for a reactive system driven by weak thermal noise can be approximated well using the transition state geometry of the underlying deterministic system. This agreement persists as long as the thermal driving strength is less than the order of that of the periodic driving. The power of this result is its simplicity. The surprising accuracy of the time-dependent noise-free geometry for obtaining transition state theory rates in chemical reactions driven by periodic fields reveals the dynamics without requiring the cost of brute-force calculations.

We study the bifurcation of solutions of semilinear elliptic boundary value problems of the form \begin{align*} \begin{aligned} -\Delta u &= f_\lambda(|x|,u,|\nabla u|) &&\text{in }\Omega, u &= 0 &&\text{on }\partial\Omega, \end{aligned} \end{align*} on an annulus $\Omega\subset\mathbb{R}^N$, with a concave-convex nonlinearity, a special case being the nonlinearity first considered by Ambrosetti, Brezis and Cerami: $f_\lambda(|x|,u,|\nabla u|)=\lambda|u|^{q-2}u + |u|^{p-2}u$ with $1

We consider the nonlinear curl-curl problem $\nabla\times\nabla\times U + V(x) U= \Gamma(x)|U|^{p-1}U$ in $\mathbb{R}^3$ related to the nonlinear Maxwell equations for monochromatic fields. We search for solutions as minimizers (ground states) of the corresponding energy functional defined on subspaces (defocusing case) or natural constraints (focusing case) of $H(\mathrm{curl};\mathbb{R}^3)$. Under a cylindrical symmetry assumption on the functions $V$ and $\Gamma$ the variational problem can be posed in a symmetric subspace of $H(\mathrm{curl};\mathbb{R}^3)$. For a strongly defocusing case $\mathrm{esssup}\, \Gamma <0$ with large negative values of $\Gamma$ at infinity we obtain ground states by the direct minimization method. For the focusing case $\mathrm{essinf}\, \Gamma >0$ the concentration compactness principle produces ground states under the assumption that zero lies outside the spectrum of the linear operator $\nabla \times \nabla \times +V(x)$. Examples of cylindrically symmetric functions $V$ are provided for which this holds., Comment: 32 pages, 1 figure

In this paper, we study local bifurcations of an indefinite elliptic system with multiple components: \begin{equation*} \left\{\begin{array}{ll} -\Delta u_j + au_j = \mu_ju_j^3+\beta\sum_{k\ne j}u_k^2u_j, u_j>0\ \ \hbox{in}\ \Omega, u_j=0 \ \ \hbox{on}\ \partial\Omega,\ j=1,\dots,n. \end{array} \right. \end{equation*} Here $\Omega\subset{\mathbb{R}}^N$ is a smooth and bounded domain, $n\ge3$, $a<-\Lambda_1$ where $\Lambda_1$ is the principal eigenvalue of $(-\Delta, H_0^1(\Omega))$; $\mu_j$ and $\beta$ are real constants. Using the positive and non-degenerate solution of the scalar equation $-\Delta\omega-\omega=-\omega^3$, $\omega\in H_0^1(\Omega)$, we construct a synchronized solution branch $\mathcal{T}_\omega$. Then we find a sequence of local bifurcations with respect to $\mathcal{T}_\omega$, and we find global bifurcation branches of partially synchronized solutions., Comment: 16 pages, 2 figures

We prove the existence of critical points of the $N$-vortex Hamiltonian $H_\Omega (x_1,\ldots, x_N) =\sum\limits^N_{i=1}\Gamma^2_i h(x_i) + \sum\limits_{i,j=1\atop j\not= k}^N \Gamma_i\Gamma_jG(x_i,x_j)+2\sum\limits_{i=1}^N\Gamma_i\psi_0(x_i)$ in a bounded domain $\Omega\subset{\mathbb R}^2$ which may be simply or multiply connected. Here $G$ denotes the Green function for the Dirichlet Laplace operator in $\Omega$, more generally a hydrodynamic Green function, and $h$ the Robin function. Moreover $\psi_0\in C^1(\overline\Omega)$ is a harmonic function on $\Omega$. We obtain new critical points $x=(x_1,\dots,x_N)$ for $N=3$ or $N=4$ under conditions on the vorticities $\Gamma_i\in{\mathbb R}\setminus\{0\}$. These critical points correspond to point vortex equilibria of the Euler equation in vorticity form. The case $\Gamma_i=(-1)^i$ of counter-rotating vortices with identical vortex strength is included. The point vortex equilibria can be desingularized to obtain smooth steady state solutions of the Euler equations for an ideal fluid. The velocity of these steady states will be irrotational except for $N$ vorticFity blobs near $x_1,\dots,x_N$.

Chemical reactions subjected to time-varying external forces cannot generally be described through a fixed bottleneck near the transition state barrier or dividing surface. A naive dividing surface attached to the instantaneous, but moving, barrier top also fails to be recrossing-free. We construct a moving dividing surface in phase space over a transition state trajectory. This surface is recrossing-free for both Hamiltonian and dissipative dynamics. This is confirmed even for strongly anharmonic barriers using simulation. The power of transition state theory is thereby applicable to chemical reactions and other activated processes even when the bottlenecks are time-dependent and move across space.

We investigate the existence of collision-free nonconstant periodic solutions of the $N$-vortex problem in domains $\Omega\subset\mathbb{C}$. These are solutions $z(t)=(z_1(t),\dots,z_N(t))$ of the first order Hamiltonian system \[ \dot{z}_k(t)=-i\nabla_{z_k} H_\Omega\big(z(t)\big),\quad k=1,\dots,N, \] where the Hamiltonian $H_\Omega$ has the form \[ H_\Omega(z_1,\dots,z_N) = \frac1{2\pi}\sum_{{j,k=1}\atop{j\ne k}}^N \log\frac1{|z_j-z_k|} - F(z). \] The function $F:\Omega^N\to\mathbb{R}$ depends on the regular part of the hydrodynamic Green's function and is unbounded from above. The Hamiltonian is unbounded from above and below, it is singular, not integrable, energy surfaces are not compact and not known to be of contact type. We prove the existence of a family of periodic solutions $z^r(t)$, $0

We find solutions $E:\Omega\to\mathbb{R}^3$ of the problem \[ \left\{\begin{aligned} &\nabla\times(\nabla\times E) + \lambda E = \partial_E F(x,E) &&\quad \text{in}\Omega\\ &\nu\times E = 0 &&\quad \text{on}\partial\Omega \end{aligned} \right. \] on a simply connected, smooth, bounded domain $\Omega\subset\mathbb{R}^3$ with connected boundary and exterior normal $\nu:\partial\Omega\to\mathbb{R}^3$. Here $\nabla\times$ denotes the curl operator in $\mathbb{R}^3$, the nonlinearity $F:\Omega\times\mathbb{R}^3\to\mathbb{R}$ is superquadratic and subcritical in $E$. The model nonlinearity is of the form $F(x,E)=\Gamma(x)|E|^p$ for $\Gamma\in L^\infty(\Omega)$ positive, some $2