Papers

In this paper, we derive the rotation-two-component $b$-family shallow-water wave system from the Green–Naghdi type equations incorporating the effects of the Coriolis force and constant vorticity. The system admits, among its special cases, the two-component Degasperis–Procesi (2DP) system as a reduction, thereby confirming its genuine role as a rotational extension of the $b$-family. We then focus on a specific reduction, the rotation-two-component Degasperis-Procesi system, whose particular interest lies in the fact that it combines two independent sources of difficulty: unlike the two-component Camassa-Holm type system, it possesses no conserved $H^1\times L^2$-type quantity, and in contrast to the standard 2DP system, it contains a higher-order nonlinear coupling term induced by the Coriolis force. To overcome this double difficulty, we introduce an auxiliary variable and establish a local $L^\infty$-estimate for $u$ without using any conservation law, based on which we derive two distinct blow-up criteria. When the Coriolis force vanishes, our results recover known criteria for the 2DP system and also provide new criteria for both the 2DP system and Degasperis-Procesi equation. The results extend naturally to the periodic setting and to relevant special cases of the rotational $b$-family system.

We study the Neumann boundary value problem $$ -\operatorname{div}\left(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right) = \lambda a(x)g(u) \quad \text{in } \Omega, \qquad \frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\cdot \mathbf n = 0 \quad \text{on } \partial\Omega, $$ where $\Omega \subset \mathbb R^N$ is a bounded convex domain, $a\in L^\infty(\Omega)$ is an indefinite weight with $\int_\Omega a < 0$, and $g$ is a nonlinearity. Under suitable assumptions on $g$, we prove the existence of two positive solutions for sufficiently large $\lambda>0$, using Szulkin's theory for nonsmooth functionals: a global minimizer $u_\lambda^{(l)}$ with negative energy, and a mountain-pass critical point $u_\lambda^{(s)}$ with positive energy. Furthermore, we study the asymptotic behaviour of both solutions as $\lambda \to +\infty$ in the model case $g(u) = |u|^{p-2} u$. We show that the mountain-pass energy level decays at the explicit rate $c_\lambda = O(\lambda^{-2/(p-2)})$, and that $u_\lambda^{(s)} \to 0$ in $C(\overline\Omega)$; moreover, we prove that $u_\lambda^{(l)} \to u_\infty$ uniformly, where $u_\infty$ solves a constrained maximization problem. The limiting profile $u_\infty$ saturates the geometric constraint, $\|\nabla u_\infty\|_{L^\infty(\Omega)}=1$, and on every connected open set where the gradient constraint is inactive and $a\neq 0$, the function $u_\infty$ is constant.

We study an inverse source problem for a two-mode nonlinear Helmholtz system motivated by second-harmonic generation. The datum is the full first Fréchet derivative of the nonlinear Dirichlet-to-Neumann map at one fixed, common, small boundary state, which need not be zero. For real-valued small data, a known susceptibility bounded away from zero, and sources supported in a fixed compact subset of the domain, we prove uniqueness and a conditional single-logarithmic stability estimate. The first boundary variation is the Dirichlet-to-Neumann map of a symmetric $2\times2$ matrix Schrödinger operator. Absorbing the two known Helmholtz energies into its matrix potential permits the use of standard zero-energy complex geometrical optics solutions with a common null phase geometry. A bilinear Alessandrini identity then gives a uniform Fourier estimate for the two entries containing the background fields. Zero extension below the exponent $3/2$, a low/high frequency splitting, and interpolation between $H^{-2}$ and the a priori $H^s$ source bound yield the stated stability modulus.

For any \(n\geq 1\), let \((M,\theta)\) be a compact strictly pseudoconvex hypersurface type CR manifold with dimension \(2n+1\), \(Q=2n+2\) be its homogeneous dimension. We prove the sharp Folland–Stein Sobolev inequality on \(M\): if \(Q^*=2Q/(Q-2)\), then there exists a constant \(A>0\), depending only on \((M,\theta)\), such that \[ \|u\|_{L^{Q^*}(M)}^2 \le K(n)^2\|\nabla_b u\|_{L^2(M)}^2 + A\|u\|_{L^2(M)}^2, \qquad u\in S^{1,2}(M). \] Here \(K(n)\) is the sharp constant in the Folland–Stein Sobolev inequality on the Heisenberg group \(\mathbb H^n\).

In this paper we investigate the optimal dimension-dependent estimates of the second-order discrete Riesz transforms \[ R_{\mathrm{dis}}^{(jk)}f(n) = c_d\sum_{m\in\mathbb Z^d\setminus\{0\}} \frac{m_jm_k}{|m|^{d+2}}f(n-m), \qquad c_d=\frac{\Gamma\left(\frac{d+2}{2}\right)}{\pi^{d/2}}. \] For $j\neq k$ and every fixed $1<p<\infty$, we prove that \[ \|R_{\mathrm{dis}}^{(jk)}\|_{\ell^p\to\ell^p} = c_d\left[ \frac{2}{2^{d/2}} + \left(\frac83+o(1)\right)\frac{d}{3^{d/2}} \right] \] and \[ \frac{2c_d}{2^{d/2}} \leq \|R_{\mathrm{dis}}^{(jk)}\|_{\ell^1\to\ell^{1,\infty}} \leq c_d\left[ \frac{2}{2^{d/2}} + \left(\frac83+o(1)\right)\frac{d}{3^{d/2}} \right]. \] Since $ c_d\sim \sqrt{\pi d}\left(\frac{d}{2\pi e}\right)^{d/2} $ by Stirling's formula, thl $\ell^p$ estimates give a negative answer to the conjecture proposed by Bañuelos and Kim in \cite{BK2}. The diagonal case exhibits quite a different phenomenon: for every $j$ and every $1<p<\infty$, $R_{\mathrm{dis}}^{(jj)}$ is neither bounded from $\ell^p(\mathbb Z^d)$ to $\ell^p(\mathbb Z^d)$ nor of weak type $(1,1)$. Cancellation is restored for the operators $R_{\mathrm{dis}}^{(jj-kk)} =R_{\mathrm{dis}}^{(jj)}-R_{\mathrm{dis}}^{(kk)}$. For every fixed $1<p<\infty$, \[ \|R_{\mathrm{dis}}^{(jj-kk)}\|_{\ell^p\to\ell^p} = 4c_d\left[ 1+(1+o(1))\frac{d}{2^{d/2}} \right], \] and \[ 4c_d \leq \|R_{\mathrm{dis}}^{(jj-kk)}\|_{\ell^1\to\ell^{1,\infty}} \leq 4c_d\left[ 1+(1+o(1))\frac{d}{2^{d/2}} \right]. \]

Let $N>p>1$, $s>-p$, $0<\mu<\bar\mu:=\big(\frac{N-p}{p}\big)^{p}$ and $q>p-1$. We determine the behaviour at infinity of every positive solution of the quasilinear Hardy–Emden–Fowler equation \[ -\big(r^{N-1}|v'|^{p-2}v'\big)'=\mu\,r^{N-1-p}v^{p-1}+r^{N-1+s}v^{q}, \qquad r>R_{0}. \] Three rates compete: the self-similar rate $r^{-\gamma_{*}}$ with $\gamma_{*}=\frac{p+s}{q-p+1}$, and the two Hardy rates $r^{-\nu_{\pm}}$, where $\nu_{\pm}$ are the roots of $h(\nu):=\nu^{p-1}(N-p-(p-1)\nu)=\mu$. Writing $p^{*}_{s}=\frac{p(N+s)}{N-p}$ and $q_{S}=p-1+\frac{p+s}{\nu_{+}}$ for the Hardy–Sobolev and Serrin exponents, we obtain a complete picture in four regimes. If $q>p^{*}_{s}-1$, every solution realises exactly one of the three rates, with a logarithmic correction in the borderline case $h(\gamma_{*})=\mu$, and we compute the leading constants. If $q=p^{*}_{s}-1$ the system becomes integrable: besides the singular self-similar solution and a heteroclinic “bubble” there is a one-parameter family of Fowler–Delaunay solutions, for which $r^{\gamma_{*}}v(r)$ is a non-constant periodic function of $\ln r$. If $q_{S}<q<p^{*}_{s}-1$ only two orbits survive, and if $q\le q_{S}$ there is no positive solution at all in an exterior domain. This is the quasilinear, weighted counterpart of the picture obtained by Franca and Sfecci for $p=2$. We work with the slope variables $x=-rv'/v$, $Y=r^{p+s}v^{q-p+1}$, in which the equation becomes a planar system that is real-analytic on the half-plane $\{x>0\}$ for every $1<p<N$, so that no restriction of the type $p\le2$ is needed; the analysis then rests on a universal barrier $x<\nu_{+}$, on an explicit Dulac function which degenerates into a first integral exactly at $q=p^{*}_{s}-1$, and on an explicit Lyapunov function.

In this paper we study the FitzHugh-Nagumo system posed on the half-line with a sinusoidal forcing term acting at the origin through the Neumann boun\-da\-ry condition. By applying an averaging method, we show that, if the frequency of the oscillating forcing is high enough, then the system can be approximated by a combination of a highly-oscillatory term and the solution of a simpler system. This approach allows us to establish stability results under persistent time-varying boundary conditions. Along with the averaging method, we use an adaptation of the contracting rectangles method, tailored to handle parabolic equations with spatially and temporally varying coefficients.

We establish a density property for the obstacle problem of a linearly elastic elliptic membrane shell required to remain confined in a prescribed half-space. For a bounded Lipschitz reference domain, and under the sole assumption that the reference configuration lies at a strictly positive distance from the confining plane, we prove that smooth admissible displacements for which the confinement constraint is inactive (satisfied with a strictly positive margin) are dense in the set of all admissible displacements. This removes the additional condition on the unit normal to the mid-surface required in previous work, and the approximation scheme is elementary and adaptable to other unilateral obstacle problems.

We study the least bending energy $m_L$ of a closed embedded wire of prescribed length $L$ confined to the closed unit disk. It is classical that the bending energy of any closed curve in the disk is at least its length, with equality only for multiply covered unit circles. Such a circle is the energy limit of a stack of disjoint embedded circles, but not of a single embedded loop, and the size of the resulting defect $m_L-L$ has remained open. We prove that \[ m_L \;=\; L+c_1\,L^{4/9}+O\bigl(L^{1/3}\bigr)\qquad\text{as }L\to\infty , \] with a sharp constant $c_1=5.223049\ldots$ given in closed form by two one-dimensional model problems. In particular $m_L-L\asymp L^{4/9}$, which improves the previously known upper bound of order $\sqrt{L}$ and provides the first lower bound beyond the trivial one. This paper was produced with substantial assistance from large language models; Section 7 sets out in detail what they contributed and what I verified.

In this article, we study the second-order fluctuation of the solutions of one-dimensional polynomial stochastic partial differential equations (SPDEs) of the form \begin{equation*} (\partial_t - \Delta) \Phi_{\varepsilon} = -P(\Phi_{\varepsilon}) + \xi_{\varepsilon}, \end{equation*} where $P$ is a polynomial of degree greater than or equal to $2$, $\xi_\varepsilon$ is the white-noise after being convoluted (in space) by the heat kernel $K_\varepsilon = e^{\varepsilon \Delta}$. More precisely, taking advantage of the local solutions of pointwise well-posedness of limits $\Phi= \lim_{\varepsilon \to 0}\Phi_{\varepsilon}$, we characterise the limit of $\Phi^{err}=\lim_{\varepsilon \to 0} \varepsilon^{-1}(\Phi_{\varepsilon}-\Phi)$ as the solution of a more irregular stochastic partial differential equation. This is performed by applying the Da Prato–Debussche decomposition in the non-linear equation and characterising the limit of each of the terms. We also discuss possible characterisations of second-order fluctuations performed for higher-dimensional SPDEs in the weakly-coupled regime.

We investigates the transonic shock problem for steady relativistic Euler equations in hemispherical shells. We show the existence and uniqueness of spherically symmetric transonic shock solutions via the shooting method and the monotonicity between the exit pressure and shock position. Furthermore, without any restrictions on background transonic shocks, we prove the existence and stability of transonic shocks under three dimensional perturbations of the exit pressure. The key techniques involve the decoupling the hyperbolic and elliptic components in steady relativistic Euler equations using the deformation tensor and vorticity. To address the coordinate singularities, the “spherical projection coordinates" combining the spherical coordinates and the stereographic projection is employed. Subsequently, the Rankine-Hugoniot conditions are appropriately reformulated to determine the shock front and derive the boundary conditions on the shock front for a first-order nonlocal deformation-curl system.

We study the Vlasov equation with general power-law (Riesz-type) potentials with exponent \(\alpha\). We prove global well-posedness in every dimension \(d\ge4\), for both attractive and repulsive interactions, throughout the range \(0<\alpha<3\), for arbitrary nonnegative compactly supported bounded initial data. No smallness condition is imposed on the initial data. For the attractive problem, the virial identity gives finite-time breakdown for negative-energy solutions when \(\alpha\ge3\). The proof of global well-posedness combines the analysis of characteristics, estimates in Lagrangian coordinates, and a crucial application of a compensated integrability inequality due to Denis Serre.

We consider a geometric mechanochemical model of vesicles which couples the Helfrich flow for the shape of a lipid bilayer vesicle membrane $X$ with a reaction-diffusion equations for a single “morphogen” $\phi$ on $X$. The Helfrich flow is the $L^2$ gradient flow of the elastic bending energy $E(X)=\int_X (H-c_0)^2 dS$ of $X$, typically supplemented by area or volume constraints, or both. The morphogen $\phi$ adsorbs/desorbs at places of high/low mean curvature $H$, i.e., the kinetics of $\phi$ depend on $H$, and conversely $\phi$ modifies the spontaneous curvature $c_0$ on $X$. The flow is no longer gradient, and hence allows for more complicated dynamics, including time periodic orbits, e.g., “breathing and moving” vesicle shapes. We show how to compute bifurcation diagrams for such solution branches via numerical continuation and bifurcation methods. We mostly focus on “planar” vesicles (1D closed curves) but also give an outlook on 3D vesicles (2D closed membranes).

In this paper, we consider $2\times2$ hyperbolic systems of conservation laws in one spatial dimension. We use the computational ansatz introduced in the hyperbolic theory by the author and Székelyhidi to study the constitutive set corresponding to the PDE. The rank-one convex geometry of this set relates to non-uniqueness and the existence of low-regularity solutions. Through a computer-assisted search, we find a pressure law $p$ such that the $p$-system with this pressure law, and its natural strictly convex entropy, verifies all of the conditions necessary for the large data $L^2$ stability and the technique of “$a$-contraction with shifts” and thus we have uniqueness of certain Riemann solutions in the class of solutions verifying the Strong Trace Property. At the same time, the constitutive set contains a $T_6$ configuration and we use it to construct non-unique solutions without the Strong Trace Property. This resolves the question of sharpness of strong traces. We also present proofs which show nonexistence of $T_\infty$ structures for all genuinely nonlinear systems and nonexistence of $T_N$ structures for all $N$ for the $p$-system with $p''>0$, thus blocking these routes towards convex integration.

We give a self-contained exposition of metric differentiability for Lipschitz mappings from sets in Euclidean spaces into arbitrary metric spaces, together with the corresponding area and co-area formulas. The novelty is a new metric implicit function theorem for mappings into metric spaces, which is then used to give a direct and geometric proof of the co-area formula for Lipschitz mappings into metric spaces.

We prove the Trudinger inequality with exponent $\frac{s}{s-1}$ for $M^{1,s}$ Sobolev spaces on metric measure spaces under the sole measure growth assumption $\mu(B(x,r))\ge br^s$. This improves the previously known exponential integrability with power one. Our argument also yields sharp asymptotic bounds for the Sobolev constants as $p\uparrow s$. On doubling spaces, we further remove the connectedness assumption from the Trudinger inequality associated with a $(1,p)$-Poincaré inequality when $p<s$. An Ahlfors regular counterexample shows that this extension fails at the critical exponent $p=s$.

We provide a pathwise proof of global-in-time well-posedness for the generalised KPZ equation in the full subcritical regime by an adaptation of the strategy recently applied to the generalised Parabolic Anderson Model in [ES26]. Since this strategy relies crucially on the assumption that control of the supremum norm is sufficient to continue the solution, the main additional ingredient required is a treatment of the initial layer for gKPZ with merely $L^\infty$ initial data, rather than the more usual setting of $C^\theta$ initial data with $\theta > 0$. Our approach is based on an expansion around a deterministic profile followed by the introduction of an integrating factor in order to remove the terms which have critical scaling at time $0$. For pedagogical purposes, we first demonstrate the proof techniques in the case of the standard $(1+1)$-dimensional KPZ equation before turning to the more computationally involved case of generalised KPZ.

We consider a minimal pressure-based model of heterogeneous cell populations consisting of proliferative and non-proliferative cells with different mobilities. The model is formulated as a system of reaction–cross–diffusion equations describing the spatio-temporal dynamics of the cell densities. The model is known to admit one-dimensional travelling wave solutions with strictly segregated components: non-proliferative cells occupy a finite region at the leading edge, while proliferative cells remain at the rear. However, the speed, parameter dependence, and stability of these waves remain poorly understood. In this work, we derive an almost explicit variational bound on the wave speed by reformulating the problem as a free-boundary problem for a generalised porous–Fisher equation. The estimates we obtain apply to general pressure laws and growth kinetics, agree closely with the results of numerical simulations, and become sharp in the incompressible limit, where we formally recover a fully explicit characterisation of the wave speed. We then analyse the stability of the waves to show that segregated waves are stable only when non-proliferative cells are more mobile than proliferative cells. Finally, motivated by numerical observations of finger-like protrusions, we investigate the stability of incompressible segregated circular waves through asymptotic shape-perturbation analysis. This yields explicit expressions for the pressure, interface velocity, and growth rates of angular modes, thereby making evident the destabilisation mechanisms that may lead to the emergence of fingering instability. Interestingly, we find that, in contrast with the one-dimensional case, the stability of such circular waves is not determined solely by the relative value of the mobility coefficients, and thus instabilities may arise irrespective of which cell type has the larger mobility.

We prove the existence of arbitrarily high-dimensional families of minimal surfaces of any prescribed genus and conformal structure in even-dimensional round spheres. More precisely, let $n\geq2$ and let $\Sigma$ be any closed Riemann surface of genus $g$. We construct a sequence of degrees $d_\ell\to+\infty$ such that, for every $\ell$, there exists a complex manifold of complex dimension $2d_\ell+n^2(1-g)$ consisting of linearly full branched superminimal immersions of $\Sigma$ into $\mathbb S^{2n}$ of degree $d_\ell$. In particular, this yields parametrized families of linearly full branched minimal immersions whose dimensions tend to infinity.

In this paper we consider two-dimensional water waves, under the action of gravity and surface tension. We prove a controllability result for irrotational waves in a fluid domain with finite depth and general bottom topography. The result holds for an open and dense set of bottom topographies in $H^{s+1/2}(\mathbb{T})$ (where $s$ is sufficiently large) that do not touch the free surface. We point out that the bottom topographies that we allow for are not necessarily small perturbations of the flat bottom case: this leads to many technical difficulties, since the eigenvalues of the Dirichlet-Neumann operator at a still free surface with general bottom topography are not explicit, nor are they necessarily close (for low frequencies) to the eigenvalues of the corresponding operator for the flat bottom case. In turn, this leads to a more involved argument to prove Ingham-type estimates, which are needed to prove observability, and motivates the restriction mentioned above on the admissible bottom topographies. We also prove a controllability result for waves with constant vorticity in a fluid domain with flat bottom topography.

In this paper, we establish the well-posedness and exponential stability of a class of semilinear neutral abstract evolution equations with constant time delay. Two damping mechanisms are considered: frictional damping of the form $CC^*u_t$ and viscoelastic damping with memory. Under suitable assumptions, we prove global well-posedness and exponential decay of solutions for sufficiently small initial data. This result is of considerable importance, as it provides a unified framework for the analysis of a wide class of neutral systems arising in structural mechanics, seismic isolation, and control theory. The abstract results are illustrated by some concrete examples. The analysis relies on a combination of semigroup theory, energy estimates, Duhamel's formula, and Gronwall-type arguments.

We show that singularities of the three-dimensional cubic complex Ginzburg-Landau equation developing from smooth initial data can induce non-uniqueness of solutions after the time of blowup. The solutions we study start as backward self-similar solutions, and the source of the non-uniqueness is an unstable eigenvalue of the linearization around the forward self-similar profile generated by the singularity. We establish the existence of this eigenvalue, and hence the non-uniqueness, by a computer-assisted proof for specific values of the parameters in the equation. The work is in part motivated by similarities to conjectured behavior for the Navier-Stokes equation, with which the complex Ginzburg-Landau equation shares several important properties, such as an energy inequality and the scaling symmetry.

We prove a growth gap for the gradient of entire anisotropic minimal graphs. For each $n\geq2$ and each smooth uniformly elliptic parametric integrand $\Phi$ on $\mathbb{R}^{n+1}$, there is an exponent $\alpha=\alpha(n,\Phi)>0$ such that every smooth nonaffine entire $\Phi$-minimal graph $u:\mathbb{R}^n\to\mathbb{R}$ satisfies $\sup_{B_R^n}|Du|\geq cR^\alpha$ for all $R\geq R_0$, for some constants $c>0$ and $R_0<\infty$ depending on the solution. In particular, gradient growth $o(R^\alpha)$ forces flatness, resolving a conjecture of Mooney and the author. The result holds in every dimension, without any assumption that $\Phi$ is close to the Euclidean area integrand.

Let $K\subset\R^m$ be the closure of a bounded domain whose boundary is a compact embedded hypersurface, and let \( \overline M=\R^m\setminus\operatorname{int}K \) be the allowed target. We prove that if $\partial K$ is of class $C^{1,1}$, then every $\overline M$-valued local minimizer of the Dirichlet energy $E(u;B):=\int_B|Du|^2\dd x$ is locally $W^{2,\infty}$, and hence locally $C^{1,1}$, on its continuity set. For every $0<\alpha<1$, we also construct a convex body with $C^{1,\alpha}$ boundary and an everywhere continuous global minimizer that is not $C^{1,1}$. Thus $C^{1,1}$ is the sharp target regularity threshold in the Hölder scale.

We consider an inverse problem for a semilinear wave equation with time-independent damping, linear potential, and nonlinear potential coefficients in a bounded domain of $\mathbb{R}^n$ for $n\geq 2$. The main objective is to establish stability estimates for the simultaneous recovery of these coefficients from the associated Dirichlet-to-Neumann map. Our approach combines second-order linearization with suitably constructed geometric optics and asymptotic solutions. We establish Hölder-type stability estimates for the recovery of each of the three coefficients appearing in the semilinear wave equation under suitable a priori bounds on these coefficients. To the best of our knowledge, this is the first stability result for simultaneous determination of time-independent damping, linear and nonlinear potentials in a semiliner wave equation.

We consider the weakly damped stochastic nonlinear Schrödinger (NLS) equation on the real line, driven by a noise that is white in time and smooth in space. Assuming that the noise is sufficiently non-degenerate, we prove that the equation has a unique stationary measure in the class of probability measures concentrated on $H^2$, and establish polynomial mixing in the dual-Lipschitz metric over $H^1$. We do not impose any restriction on the size of the damping. The proof is based on a coupling argument, whose key ingredient is a Foiaş-Prodi-type estimate in the $H^1$-norm, derived by means of a Lyapunov functional adapted to the linearized NLS dynamics. To compensate for the loss of compactness, we combine this estimate with a truncated Poincaré inequality and a space-time weight function quantifying the spatial decay of solutions.

This paper is concerned with exact high-order generalized polarization tensor (GPT) cancellation in two-dimensional noncircular multilayers with finite positive isotropic coating conductivities. We first prove that, for every admissible cyclically symmetric core–shell geometry, there exists a unique positive coating conductivity for which the structure is weakly neutral to every uniform incident field. For higher orders, we establish locally unique coating conductivities that cancel the complete \(K\)-contracted GPTs (K-CGPTs) block on a broad class of fixed noncircular multilayers with cyclic symmetry. This includes homothetic multilayers generated by strictly star-shaped symmetric domains, without assuming that the geometry is close to concentric disks. We also replace the near-background condition by a small-core condition covering finite, insulating, and perfectly conducting cores. By varying selected interface modes together with the coating conductivities near a radial GPT-vanishing structure, we further construct exact families with neither rotational nor reflection symmetry. For three coatings, the relevant nondegeneracy conditions and an explicit chiral example are rigorously certified. Finally, we show that the resulting cancellation yields enhanced near-cloaking for every entire harmonic incident field. A set of certified numerical examples fully corroborates the theoretical predictions established in this work.

For every subsonic speed $c\in(0,\sqrt2)$, we prove the existence of a finite-energy traveling wave for the planar Gross–Pitaevskii equation. This resolves the longstanding problem of the existence of prescribed-speed traveling-wave solutions in two dimensions, explicitly stated as open by Mariş (Ann. of Math., 2013) and Bellazzini and Ruiz (Amer. J. Math., 2023). The proof relies essentially on the energy estimate \begin{equation*} E(\psi)\le C_J\bigl(I_c(\psi)+\ind(\psi)\bigr), \qquad c\in J, \end{equation*} where $E$ is the energy, $I_c$ the action at speed $c$, $\ind$ the real Morse index, $J$ is any compact interval contained in $(0,\sqrt2)$, and $C_J$ is a positive constant depending only on $J$. We also prove finite-bubble compactness, including splitting of the energy, action, potential energy, and momentum, and attainment of the action among nonconstant waves of Morse index at most one.

In this paper, we prove pathwise uniform convergence at rates up to $1/2$ for tamed exponential Euler schemes for semilinear hyperbolic stochastic evolution equations with superlinearly growing nonlinearities and multiplicative noise. We take the term hyperbolic to mean that the leading operator generates a contractive $C_0$-semigroup but no parabolic smoothing occurs. Under local Lipschitz, polynomial growth, coercivity, and monotonicity conditions on the nonlinearities, we establish pathwise uniform strong error estimates of the form \begin{equation*} \Big(\mathbb{E}\max_{0\le j \le N} \|U(t_j)-U^j\|_X^p\Big)^{1/p} \lesssim \sqrt{k} \end{equation*} on a Hilbert space $X$ for $p\in [2,\infty)$. Here, $U$ is the mild solution and $U^j$ is the tamed exponential Euler approximation at time $t_j=jk$ with step size $k>0$. This extends previous convergence results for non-parabolic SPDEs from globally to locally Lipschitz nonlinearities, allowing both drift and diffusion to grow polynomially. In a stochastic Kato framework, we further establish local and global well-posedness as well as uniform a priori estimates for the mild solution and its approximation. Applications to nonlinear stochastic transport, Airy, wave-type, and dissipatively damped nonlinear Schrödinger equations are included, covering different nonlinearities-stopped and fractionally tamed schemes. For the Klein-Gordon equation with cubic velocity damping, this complements previous results obtained for additive noise.

We study modified scattering for the two-dimensional defocusing nonlinear Schrödinger equation (NLS) with the gauge-invariant, scattering-critical nonlinearity $|u|u$. We prove that the remodulated interaction representation has a local $L^2$ limit for general data in $\Sigma=H^1\cap\F^{-1}H^1$. For radial data, we prove convergence in $L^p(\R^2)$ for every $2<p<\infty$.