In this paper, we establish general results for the asymptotic behaviour of solutions of dynamical systems in Banach spaces. We show that if the initial datum possesses a certain decay, then the corresponding solution emanating from the Cauchy problem studied inherits the same behaviour at any further time for which it exists. Our results are applied to a wide class of linear and non-linear models. In particular, we use our main results to show persistence properties for classical linear and non-linear ordinary differential equations (ODEs), the Benjamin–Bona–Mahony (BBM) equation, and the generalized Boussinesq equation.

As noted by Landau and Lifshitz in their work Course of Theoretical Physics (Vol. 6, Pergamon Press, Oxford, 2nd edition, 1987), “if the relaxation time of these processes is long, a considerable dissipation of energy occurs when the fluid is compressed or expanded,” then the volume viscosity coefficient becomes relatively large, making the shear viscosity coefficient much smaller in comparison. In this article, we take this phenomenon into consideration and investigate isentropic compressible magnetohydrodynamic flow with zero shear viscosity in a periodic domain. We prove the global existence of smooth solutions when the initial data are close to a background magnetic field. In addition, stability and large-time decay rates are also obtained. Mathematically, the zero shear viscosity makes the elliptic operator “$\mu \Delta +\nu \nabla \text{div}\, $” lose its uniform ellipticity, rendering the classical dissipation mechanisms for velocity inapplicable. The stabilizing effect of the background magnetic field plays a crucial role in our analysis.

This work investigates the dynamics of positive classical solutions to a diffusive susceptible-exposed-infected-recovered-susceptible epidemic model with a mass-action incidence mechanism in spatially heterogeneous environments. Under minimal assumptions on the initial data, the global existence of classical solutions is established. Moreover, the eventual boundedness of these solutions is proved when either the spatial domain has dimension five or lower or the susceptible and exposed subpopulations share the same diffusion rate. Next, we define the basic reproduction number, $\mathcal{R}_0$, and demonstrate that the disease-free equilibrium is globally stable when $\mathcal{R}_0$ is sufficiently small. However, due to the complex interaction between population movement and spatial variation in transmission rates, we find that the disease may persist even when $\mathcal{R}_0$ is slightly less than one. In such cases, we show that the system admits at least two endemic equilibrium (EE) solutions, an outcome not observed under the frequency-dependent incidence mechanism. These results highlight the significant influence of the transmission mechanism on disease dynamics. Furthermore, we examine the spatial profiles of the EE solutions when diffusion rates are small. Our analysis suggests that limiting the movement of the susceptible population can significantly reduce disease prevalence, provided that the total population remains below a specific threshold. In contrast, restricting the movement of the infected, exposed, or recovered populations alone may not eradicate the disease. Overall, our findings provide important insights into the spatial dynamics of infectious diseases and may offer guidance for developing and implementing effective containment strategies.

This paper employs the Riemann-Hilbert problem and nonlinear steepest descent method of Deift-Zhou to provide a comprehensive analysis of the asymptotic behavior of the genus two Korteweg-de Vries soliton gases. It is demonstrated that the genus two soliton gas is related to the two-phase Riemann-Theta function as $x \to +\infty $, and approaches zero as $x \to -\infty $. Additionally, the long-time asymptotic behavior of this genus two soliton gas can be categorized into five distinct regions in the x-t plane, which from left to right are quiescent region, modulated one-phase wave, unmodulated one-phase wave, modulated two-phase wave, and unmodulated two-phase wave. Moreover, an innovative method is introduced to solve the model problem associated with the high-genus Riemann surface, leading to the determination of the leading terms, which is also related to the multiphase Riemann-Theta function. A general discussion on the case of arbitrary genus N soliton gas is also presented.

We analyse the Maxwell’s spectrum on thin tubular neighbourhoods of embedded surfaces of $\mathbb R^3$. We show that the Maxwell’s eigenvalues converge to the Laplacian eigenvalues of the surface as the thin parameter tends to zero. To achieve this, we reformulate the problem in terms of the spectrum of the Hodge Laplacian with relative conditions acting on co-closed differential $1$-forms. The result leads to new examples of domains where the Faber–Krahn inequality for Maxwell’s eigenvalues fails, examples of domains with any number of arbitrarily small eigenvalues, and underlines the failure of spectral stability under singular perturbations changing the topology of the domain. Additionally, we explicitly produce Maxwell’s eigenfunctions on product domains with the product metric, extending previous constructions valid in the Euclidean case.

The goal of this work is to revisit the eigenfunction-expansion-based perturbation theory of the defocusing nonlinear Schrödinger equation on a nonzero background, and develop it to correctly predict the slow-time evolution of the dark soliton parameters, as well as the radiation shelf emerging on the soliton sides. Proof of the closure of the squared eigenfunctions is provided, and the complete set of eigenfunctions of the linearisation operator is used to expand the first-order perturbation solution. Our closure/completeness relation accounts for the singularities of the scattering data at the branch points of the continuous spectrum, which leads to the correct discrete eigenfunctions. Using the one-soliton closure relation and its correct discrete eigenmodes, the slow-time evolution equations of the soliton parameters are determined. Moreover, the first-order correction integral to the dark soliton is shown to contain a pole due to singularities of the scattering data at the branch points. Analysis of this integral leads to predictions for the shelves, as well as a formula for the slow-time evolution of the soliton’s phase, which in turn allows one to determine the slow-time dependence of the soliton centre. All the results are corroborated by direct numerical simulations and compared with earlier results.

In this paper, we consider the time-dependent Born–Oppenheimer approximation (BOA) of a classical quantum molecule involving a possibly large number of nuclei and electrons, described by a Schrödinger equation. In the spirit of Born and Oppenheimer’s original idea, we study quantitatively the approximation of the molecular evolution. We obtain an iterable approximation of the molecular evolution to arbitrary order, and we derive an effective equation for the reduced dynamics involving the nuclei equivalent to the original Schrödinger equation and containing no electron variables. We estimate the coefficients of the new equation and find tractable approximations for the molecular dynamics going beyond the one corresponding to the original Born and Oppenheimer approximation.

We show sharp well-posedness with analytic data-to-solution mapping in the semilinear regime for dispersion-generalized KP-I equations on $\mathbb {R}^2$ and $\mathbb {R} \times \mathbb {T}$. On $\mathbb {R}^2$ we cover the full subcritical range, whereas on $\mathbb {R} \times \mathbb {T}$ the sharp well-posedness is strictly subcritical. We rely on linear and bilinear Strichartz estimates which are proved using decoupling techniques and square function estimates. Nonlinear Loomis-Whitney inequalities are a further ingredient. These are presently proved for Borel measures with growth condition reflecting the different geometries of the plane $\mathbb {R}^2$, the cylinder $\mathbb {R} \times \mathbb {T}$, and the torus $\mathbb {T}^2$. Finally, we point out that on tori $\mathbb {T}^2_\gamma $, KP-I equations are never semilinear.

The transient response of an ice shelf to an incident wave packet from the open ocean is studied with a model that allows for extensional waves in the ice shelf, in addition to the standard flexural waves. Results are given for strains imposed on the ice shelf by the incident packet, over a range of peak periods in the swell regime and a range of packet widths. In spite of large differences in speeds of the extensional and flexural waves, it is shown that there is generally an interval of time during which they interact, and the coherent phases of the interactions generate the greatest ice shelf strain magnitudes. The findings indicate that incorporating extensional waves into models is potentially important for predicting the response of Antarctic ice shelves to swell, in support of previous findings based on frequency-domain analysis.

We consider the Maxwell–Schrödinger equations in the Coulomb gauge describing the interaction of extended fermions with their self-generated electromagnetic field. They heuristically emerge as mean-field equations from nonrelativistic quantum electrodynamics in a mean-field limit of many fermions. In the semiclassical regime, we establish the convergence of the Maxwell–Schrödinger equations for extended charges toward the nonrelativistic Vlasov–Maxwell dynamics and provide explicit estimates on the accuracy of the approximation. To this end, we build a well-posedness and regularity theory for the Maxwell–Schrödinger equations and for the Vlasov–Maxwell system for extended charges.

The KP-I equation arises as a weakly nonlinear model equation for gravity-capillary waves with Bond number $\beta \gt 1/3$, also called strong surface tension. This equation has recently been shown to have a family of nondegenerate, symmetric ‘fully localized’ or ‘lump’ solitary waves which decay to zero in all spatial directions. The full-dispersion KP-I equation is obtained by retaining the exact dispersion relation in the modelling from the water-wave problem. In this paper, we show that the FDKP-I equation also has a family of symmetric fully localized solitary waves which are obtained by casting it as a perturbation of the KP-I equation and applying a suitable variant of the implicit-function theorem.

We consider quasilinear Schrödinger equations in $\mathbb {R}^N$ of the form $-\Delta u+V(x)u-u\Delta (u^2)=g(u)$, where the potential V is allowed to be sign-changing and the nonlinearity g is sublinear at zero. Except for being subcritical, no additional condition is imposed on $g(u)$ for $|u|$ large. We obtain a sequence of solutions with negative energy and converging to zero via Clark’s theorem. We also obtain a similar result for fourth-order quasilinear Schrödinger equations in $\mathbb {R}^N$ of the form $\Delta ^2u-\Delta u+ V(x)u-u\Delta (u^2)=g(u)$.

In this paper, we prove the codimension-one nonlinear asymptotic stability of the extremal Reissner–Nordström family of black holes in the spherically symmetric Einstein–Maxwell-neutral scalar field model, up to and including the event horizon. More precisely, we show that there exists a teleologically defined, codimension-one “submanifold” ${\mathfrak{M}}_{\mathrm{stab}}$ of the moduli space of spherically symmetric characteristic data for the Einstein–Maxwell-scalar field system lying close to the extremal Reissner–Nordström family, such that any data in ${\mathfrak{M}}_{\mathrm{stab}}$ evolve into a solution with the following properties as time goes to infinity: (i) the metric decays to a member of the extremal Reissner–Nordström family uniformly up to the event horizon, (ii) the scalar field decays to zero pointwise and in an appropriate energy norm, (iii) the first translation-invariant ingoing null derivative of the scalar field is approximately constant on the event horizon $\mathcal H^+$, (iv) for “generic” data, the second translation-invariant ingoing null derivative of the scalar field grows linearly along the event horizon. Due to the coupling of the scalar field to the geometry via the Einstein equations, suitable components of the Ricci tensor exhibit nondecay and growth phenomena along the event horizon. Points (i) and (ii) above reflect the “stability” of the extremal Reissner–Nordström family and points (iii) and (iv) verify the presence of the celebrated Aretakis instability [11] for the linear wave equation on extremal Reissner–Nordström black holes in the full nonlinear Einstein–Maxwell-scalar field model.

We establish sufficient conditions for the existence of ground states of the following normalized nonlinear Schrödinger–Newton system with a point interaction:

\begin{equation*}\begin{cases}- \Delta_\alpha u + \omega u=w u+\beta u |u|^{p - 2}&\text{on} ~ \mathbb{R}^2;\\- \Delta w=2 \pi |u|^2&\text{on} ~ \mathbb{R}^2;\\\|u\|_{L^2}^2 = c,\end{cases}\end{equation*}

where $p \gt 2$; $\alpha, \beta \in \mathbb{R}$; $c \gt 0$ and $- \Delta_\alpha$ denotes the Laplacian of point interaction with s-wave scattering length $(- 2 \pi \alpha)^{- 1}$, the unknowns being $u \colon \mathbb{R}^2 \to \mathbb{C}$, $w \colon \mathbb{R}^2 \to \lbrack0, \infty\lbrack$ and the Lagrange multiplier $\omega \in \mathbb{R}$. Additionally, we show that critical points of the corresponding constrained energy functional are naturally associated with standing waves of the evolution problem

\begin{equation*}\mathrm{i} \psi' (t)=- \Delta_\alpha \psi (t)-(\log |\cdot| \ast |\psi (t)|^2)\psi (t)-\beta\psi (t)|\psi (t)|^{p - 2}.\end{equation*}

We consider the existence of normalized solutions to non-linear Schrödinger equations on non-compact metric graphs in the L2-supercritical regime. For sufficiently small prescribed mass (L2 norm), we prove existence of positive solutions on two classes of graphs: periodic graphs and non-compact graphs with finitely many edges and suitable topological assumptions. Our approach is based on mountain pass techniques. A key point to overcome the serious lack of compactness is to show that all solutions with small mass have positive energy. To complement our analysis, we prove that this is no longer true, in general, for large masses. To the best of our knowledge, these are the first results with an L2-supercritical non-linearity extended on the whole graph and unravelling the role of topology in the existence of solutions.

Substantially extending previous results of the authors for smooth solutions in the viscous case, we develop linear damping estimates for periodic roll-wave solutions of the inviscid Saint-Venant equations and related systems of hyperbolic balance laws. Such damping estimates, consisting of $H^s$ energy estimates yielding exponential slaving of high-derivative to low-derivative norms, have served as crucial ingredients in nonlinear stability analyses of traveling waves in hyperbolic or partially parabolic systems, both in obtaining high-frequency resolvent estimates and in closing a nonlinear iteration for which available linearized stability estimates apparently lose regularity. Here, we establish for systems of size $n\leq 6$ a Lyapunov-type theorem stating that such energy estimates are available whenever strict high-frequency spectral stability holds; for dimensions $7$ and higher, there may be in general a gap between high-frequency spectral stability and existence of the type of energy estimate that we develop here. A key ingredient is a dimension-dependent linear algebraic lemma reminiscent of Lyapunov’s Lemma for ODE that is to our knowledge new.

We extend a classical model of continuous opinion formation to explicitly include an age-structured population. We begin by considering a stochastic differential equation model which incorporates ageing dynamics and birth/death processes, in a bounded confidence type opinion formation model. We then derive and analyse the corresponding mean field partial differential equation and compare the complex dynamics on the microscopic and macroscopic levels using numerical simulations. We rigorously prove the existence of stationary states in the mean field model, but also demonstrate that these stationary states are not necessarily unique. Finally, we establish connections between this and other existing models in various scenarios.

We consider the Marguerre–von Kármán equations that model the deformation of a thin, nonlinearly elastic, shallow shell, subjected to a specific class of boundary conditions of von Kármán’s type. Next, we reduce these equations to a single equation with a cubic operator following Berger’s classical method, whose second member depends on the function defining the middle surface of the shallow shell and the resultant of the vertical forces acting on the shallow shell. We also prove the existence and uniqueness of a weak solution to the reduced equation. Then, we prove the existence theorem for the optimal control problem governed by Marguerre–von Kármán equations, with a control variable on the resultant of the vertical forces. Using the Fréchet differentiability of the state function with respect to the control variable, we prove the uniqueness of the optimal control and derive the necessary optimality condition. As a result, this work addresses the more general geometry of Marguerre–von Kármán shallow shells to study the quadratic cost optimal control problems governed by these equations.

This paper is devoted to a comprehensive analysis of a family of solutions of the focusing nonlinear Schrödinger equation called general rogue waves of infinite order. These solutions have recently been shown to describe various limit processes involving large-amplitude waves, and they have also appeared in some physical models not directly connected with nonlinear Schrödinger equations. We establish the following key property of these solutions: they are all in $L^2(\mathbb{R})$ with respect to the spatial variable but they exhibit anomalously slow temporal decay. In this paper, we define general rogue waves of infinite order, establish their basic exact and asymptotic properties, and provide computational tools for calculating them accurately.

The Cahn–Hilliard model with reaction terms can lead to situations in which no coarsening is taking place and, in contrast, growth and division of droplets occur which all do not grow larger than a certain size. This phenomenon has been suggested as a model for protocells, and a model based on the modified Cahn–Hilliard equation has been formulated. We introduce this equation and show the existence and uniqueness of solutions. Then, formally matched asymptotic expansions are used to identify a sharp interface limit using a scaling of the reaction term, which becomes singular when the interfacial thickness tends to zero. We compute planar solutions and study their stability under non-planar perturbations. Numerical computations for the suggested model are used to validate the sharp interface asymptotics. In addition, the numerical simulations show that the reaction terms lead to diverse phenomena such as growth and division of droplets in the obtained solutions, as well as the formation of shell-like structures.