Suppose that $G=(V,E)$ is a finite graph with the vertex set $V$ and the edge set $E$. Let $\unicode[STIX]{x1D6E5}$ be the usual graph Laplacian. Consider the nonlinear Schrödinger equation of the form

$$\begin{eqnarray}-\unicode[STIX]{x1D6E5}u-\unicode[STIX]{x1D6FC}u=f(x,u),\quad u\in W^{1,2}(V),\end{eqnarray}$$

$G$

$f(x,u):V\times \mathbb{R}\rightarrow \mathbb{R}$

$\unicode[STIX]{x1D6FC}$

$G$

$G$

$CD(m,\unicode[STIX]{x1D709})$

$G$

$\unicode[STIX]{x1D6FC}<2\unicode[STIX]{x1D706}_{1}^{2}/m(\unicode[STIX]{x1D706}_{1}-\unicode[STIX]{x1D709})$

$\unicode[STIX]{x1D706}_{1}$

Residential burglary is a social problem in every major urban area. As such, progress has been to develop quantitative, informative and applicable models for this type of crime: (1) the Deterministic-time-step (DTS) model [Short, D’Orsogna, Pasour, Tita, Brantingham, Bertozzi & Chayes (2008) Math. Models Methods Appl. Sci. 18, 1249–1267], a pioneering agent-based statistical model of residential burglary criminal behaviour, with deterministic time steps assumed for arrivals of events in which the residential burglary aggregate pattern formation is quantitatively studied for the first time; (2) the SSRB model (agent-based stochastic-statistical model of residential burglary crime) [Wang, Zhang, Bertozzi & Short (2019) Active Particles, Vol. 2, Springer Nature Switzerland AG, in press], in which the stochastic component of the model is theoretically analysed by introduction of a Poisson clock with time steps turned into exponentially distributed random variables. To incorporate independence of agents, in this work, five types of Poisson clocks are taken into consideration. Poisson clocks (I), (II) and (III) govern independent agent actions of burglary behaviour, and Poisson clocks (IV) and (V) govern interactions of agents with the environment. All the Poisson clocks are independent. The time increments are independently exponentially distributed, which are more suitable to model individual actions of agents. Applying the method of merging and splitting of Poisson processes, the independent Poisson clocks can be treated as one, making the analysis and simulation similar to the SSRB model. A Martingale formula is derived, which consists of a deterministic and a stochastic component. A scaling property of the Martingale formulation with varying burglar population is found, which provides a theory to the finite size effects. The theory is supported by quantitative numerical simulations using the pattern-formation quantifying statistics. Results presented here will be transformative for both elements of application and analysis of agent-based models for residential burglary or in other domains.

We consider the minimizing problem for the energy functional with prescribed mass constraint related to the fractional non-linear Schrödinger equation with periodic potentials. Using the concentration-compactness principle, we show a complete classification for the existence and non-existence of minimizers for the problem. In the mass-critical case, under a suitable assumption of the potential, we give a detailed description of blow-up behaviour of minimizers once the mass tends to a critical value.

We use a method developed by Strauss to obtain global well-posedness results in the mild sense and existence of asymptotic states for the small data Cauchy problem in modulation spaces ${M}^s_{p,q}(\mathbb{R}^d)$, where q = 1 and $s\geq0$ or $q\in(1,\infty]$ and $s>\frac{d}{q'}$ for a nonlinear Schrödinger equation with higher order anisotropic dispersion and algebraic nonlinearities.

We study the steady states and dynamics of a thin-film-type equation with non-conserved mass in one dimension. The evolution equation is a non-linear fourth-order degenerate parabolic partial differential equation (PDE) motivated by a model of volatile viscous fluid films allowing for condensation or evaporation. We show that by changing the sign of the non-conserved flux and breaking from a gradient flow structure, the problem can exhibit novel behaviours including having two distinct classes of co-existing steady-state solutions. Detailed analysis of the bifurcation structure for these steady states and their stability reveals several possibilities for the dynamics. For some parameter regimes, solutions can lead to finite-time rupture singularities. Interestingly, we also show that a finite-amplitude limit cycle can occur as a singular perturbation in the nearly conserved limit.

In lubrication problems, which concern thin film flow, cavitation has been considered as a fundamental element to correctly describe the characteristics of lubricated mechanisms. Here, the well-posedness of a cavitation model that can explain the interaction between viscous effects and micro-bubbles of gas is studied. This cavitation model consists of a coupled problem between the compressible Reynolds partial differential equation (PDE) (that describes the flow) and the Rayleigh–Plesset ordinary differential equation (that describes micro-bubbles evolution). A simplified form without bubbles convection is studied here. This coupled model seems never to be studied before from its mathematical aspects. Local times existence results are proved and stability theorems are obtained based on the continuity of the spectrum for bounded linear operators. Numerical results are presented to illustrate these theoretical results.

Paraconformal or GL(2, ℝ) geometry on an n-dimensional manifold M is defined by a field of rational normal curves of degree n – 1 in the projectivised cotangent bundle ℙT*M. Such geometry is known to arise on solution spaces of ODEs with vanishing Wünschmann (Doubrov–Wilczynski) invariants. In this paper we discuss yet another natural source of GL(2, ℝ) structures, namely dispersionless integrable hierarchies of PDEs such as the dispersionless Kadomtsev–Petviashvili (dKP) hierarchy. In the latter context, GL(2, ℝ) structures coincide with the characteristic variety (principal symbol) of the hierarchy.

Dispersionless hierarchies provide explicit examples of particularly interesting classes of involutive GL(2, ℝ) structures studied in the literature. Thus, we obtain torsion-free GL(2, ℝ) structures of Bryant [5] that appeared in the context of exotic holonomy in dimension four, as well as totally geodesic GL(2, ℝ) structures of Krynski [33]. The latter possess a compatible affine connection (with torsion) and a two-parameter family of totally geodesic α-manifolds (coming from the dispersionless Lax equations), which makes them a natural generalisation of the Einstein–Weyl geometry.

Our main result states that involutive GL(2, ℝ) structures are governed by a dispersionless integrable system whose general local solution depends on 2n – 4 arbitrary functions of 3 variables. This establishes integrability of the system of Wünschmann conditions.

We consider Fokker–Planck equations with tilted periodic potential in the subcritical regime and characterise the spatio-temporal dynamics of the partial masses in the limit of vanishing diffusion. Our convergence proof relies on suitably defined substitute masses and bounds the approximation error using the energy-dissipation relation of the underlying Wasserstein gradient structure. In the appendix, we also discuss the case of an asymmetric double-well potential and derive the corresponding limit dynamics in an elementary way.

We prove the existence of multi-soliton and kink-multi-soliton solutions of the Euler–Korteweg system in dimension one. Such solutions behave asymptotically in time like several travelling waves far away from each other. A kink is a travelling wave with different limits at ±∞. The main assumption is the linear stability of the solitons, and we prove that this assumption is satisfied at least in the transonic limit. The proof relies on a classical approach based on energy estimates and a compactness argument.

We study the stochastic cubic nonlinear Schrödinger equation (SNLS) with an additive noise on the one-dimensional torus. In particular, we prove local well-posedness of the (renormalized) SNLS when the noise is almost space–time white noise. We also discuss a notion of criticality in this stochastic context, comparing the situation with the stochastic cubic heat equation (also known as the stochastic quantization equation).

We prove the existence of the global attractor in ${\dot{H}}^{s}$, $s>11/12$ for the weakly damped and forced mKdV on the one-dimensional torus. The existence of global attractor below the energy space has not been known, though the global well-posedness below the energy space has been established. We directly apply the $I$-method to the damped and forced mKdV, because the Miura transformation does not work for the mKdV with damping and forcing terms. We need to make a close investigation into the trilinear estimates involving resonant frequencies, which are different from the bilinear estimates corresponding to the KdV.

The flow past an obstacle is a fundamental object in fluid mechanics. In 1967 Finn and Smith proved the unique existence of stationary solutions, called the physically reasonable solutions, to the Navier–Stokes equations in a two-dimensional exterior domain modeling this type of flows when the Reynolds number is sufficiently small. The asymptotic behavior of their solution at spatial infinity has been studied in detail and well understood by now, while its stability has remained open due to the difficulty specific to the two-dimensionality. In this paper, we prove that the physically reasonable solutions constructed by Finn and Smith are asymptotically stable with respect to small and well-localized initial perturbations.

We analyse the vorticity production of lake-scale circulation in wind-induced shallow flows using a linear elliptic partial differential equation. The linear equation is derived from the vorticity form of the shallow-water equation using a linear bed friction formula. The features of the wind-induced steady-state flow are analysed in a circular basin with topography as a concave paraboloid, having a quadratic pile in the middle of the basin. In our study, the size of the pile varies by a size parameter. The vorticity production due to the gradient in the topography (and the distance of the boundary) makes the streamlines parallel to topographical contours, and beyond a critical size parameter, it results in a secondary vortex pair. We compare qualitatively and quantitatively the steady-state circulation patterns and vortex evolution of the flow fields calculated by our linear vorticity model and the full, nonlinear shallow-water equations. From these results, we hypothesize that the steady-state topographical vorticity production in lake-scale wind-induced circulations can be described by the equilibrium of the wind friction field and the bed friction field. Moreover, the latter can also be considered as a linear function of the velocity vector field, and hence the problem can be described by a linear equation.

In this paper, we introduce a dynamical urban planning model. This leads to the study of a system of nonlinear equations coupled through multi-marginal optimal transport problems. A first example consists in solving two equations coupled through the solution to the Monge–Ampère equation. We show that theWasserstein gradient flow theory provides a very good framework to solve these highly nonlinear systems. At the end, a uniqueness result is presented in dimension one based on convexity arguments.

The unified transform method (UTM) or Fokas method for analyzing initial-boundary value (IBV) problems provides an important generalization of the inverse scattering transform (IST) method for analyzing initial value problems. In comparison with the IST, a major difficulty of the implementation of the UTM, in general, is the involvement of unknown boundary values. In this paper we analyze the IBV problem for the massive Thirring model in the quarter plane, assuming that the initial and boundary data belong to the Schwartz class. We show that for this integrable model, the UTM is as effective as the IST method: Riemann-Hilbert problems we formulated for such a problem have explicit (x, t)-dependence and depend only on the given initial and boundary values; they do not involve additional unknown boundary values.

We consider the compressible Navier–Stokes system on time-dependent domains with prescribed motion of the boundary. For both the no-slip boundary conditions as well as slip boundary conditions we prove local-in-time existence of strong solutions. These results are obtained using a transformation of the problem to a fixed domain and an existence theorem for Navier–Stokes like systems with lower order terms and perturbed boundary conditions. We also show the weak–strong uniqueness principle for slip boundary conditions which remained so far open question.

We investigate the Cauchy problem of the viscous liquid-gas two-phase flow model in ℝ3. Under the assumption that the initial data is close to the constant equilibrium state in the framework of Sobolev space H2(ℝ3), the Cauchy problem is shown to be globally well-posed by an energy method. If additionally, for 1 ⩽ p < 6/5, Lp-norm of the initial perturbation is bounded, the optimal convergence rates of the solutions in Lq-norm with 2 ⩽ q ⩽ 6 and optimal convergence rates of their spatial derivatives in L2-norm are also obtained by combining spectral analysis with energy methods.

This paper studies the regularity results of classical solutions to the two-dimensional critical Oldroyd-B model in the corotational case. The critical case refers to the full Laplacian dissipation in the velocity or the full Laplacian dissipation in the non-Newtonian part of the stress tensor. Whether or not their classical solutions develop finite time singularities is a difficult problem and remains open. The object of this paper is two-fold. Firstly, we establish the global regularity result to the case when the critical case occurs in the velocity and a logarithmic dissipation occurs in the non-Newtonian part of the stress tensor. Secondly, when the critical case occurs in the non-Newtonian part of the stress tensor, we first present many interesting global a priori bounds, then establish a conditional global regularity in terms of the non-Newtonian part of the stress tensor. This criterion comes naturally from our approach to obtain a global L∞-bound for the vorticity ω.

We study a class of flat bundles, of finite rank $N$, which arise naturally from the Donaldson–Thomas theory of a Calabi–Yau threefold $X$ via the notion of a variation of BPS structure. We prove that in a large $N$ limit their flat sections converge to the solutions to certain infinite-dimensional Riemann–Hilbert problems recently found by Bridgeland. In particular this implies an expression for the positive degree, genus 0 Gopakumar–Vafa contribution to the Gromov–Witten partition function of $X$ in terms of solutions to confluent hypergeometric differential equations.

We are interested in standing waves of a modified Schrödinger equation coupled with the Chern–Simons gauge theory. By applying a constraint minimization of Nehari-Pohozaev type, we prove the existence of radial ground state solutions. We also investigate the nonexistence for nontrivial solutions.