Local well-posedness of the higher-order nonlinear Schrödinger equation on the half-line: Single-boundary condition case

We establish local well-posedness in the sense of Hadamard for a certain third-order nonlinear Schrödinger equation with a multiterm linear part and a general power nonlinearity, known as higher-order nonlinear Schrödinger equation, formulated on the half-line $\{x>0\}$. We consider the scenario of associated coefficients such that only one boundary condition is required and hence assume a general nonhomogeneous boundary datum of Dirichlet type at $x=0$. Our functional framework centers around fractional Sobolev spaces $H_x^s({\mathbb{R}}_{+})$ with respect to the spatial variable. We treat both high regularity ( $s>\frac{1}{2}$) and low regularity ( ) solutions: in the former setting, the relevant nonlinearity can be handled via the Banach algebra property; in the latter setting, however, this is no longer the case and, instead, delicate Strichartz estimates must be established. This task is especially challenging in the framework of nonhomogeneous initial-boundary value problems, as it involves proving boundary-type Strichartz estimates that are not common in the study of Cauchy (initial value) problems. The linear analysis, which forms the core of this work, crucially relies on a weak solution formulation defined through the novel solution formulae obtained via the Fokas method (also known as the unified transform) for the associated forced linear problem. In this connection, we note that the higher-order Schrödinger equation comes with an increased level of difficulty due to the presence of more than one spatial derivatives in the linear part of the equation. This feature manifests itself via several complications throughout the analysis, including (i) analyticity issues related to complex square roots, which require careful treatment of branch cuts and deformations of integration contours; (ii) singularities that emerge upon changes of variables in the Fourier analysis arguments; and (iii) complicated oscillatory kernels in the weak solution formula for the linear initial-boundary value problem, which require a subtle analysis of the dispersion in terms of the regularity of the boundary data. The present work provides a first, complete treatment via the Fokas method of a nonhomogeneous initial-boundary value problem for a partial differential equation associated with a multiterm linear differential operator.     

Existence of global solutions to the nonlocal Schrödinger equation on the line

We address the existence of global solutions to the Cauchy problem for the integrable nonlocal nonlinear Schrödinger (nonlocal NLS) equation under the initial data $q_0(x)\in H^{1,1}(\mathbb{R})$ with the $L^1(\mathbb{R})$ small-norm. The nonlocal NLS equation was first introduced by Ablowitz and Musslimani as a new nonlocal reduction of the well-known Ablowitz–Kaup–Newell–Segur system. The main technical difficulty for proving its global well-posedness on the line in $H^1(\mathbb{R})$ is due to the fact that mass and energy conservation laws, being nonlocal, do not preserve any reasonable norm and may be negative. In this paper, we use the inverse scattering transform approach to prove the existence of global solutions in $H^{1,1}(\mathbb{R})$ based on the representation of a Riemann–Hilbert (RH) problem associated with the Cauchy problem of the nonlocal NLS equation. A key of this approach is, by applying the Volttera integral operator and Cauchy integral operator, to establish a Lipschitz bijective map between the solution of the nonlocal NLS equation and reflection coefficients associated with the RH problem. By using the reconstruction formula and estimates on the solution of the time-dependent RH problem, we further affirm the existence of a unique global solution to the Cauchy problem for the nonlocal NLS equation.     

On blowups of vorticity for the homogeneous Euler equation

Blowups of vorticity for the three- and two-dimensional homogeneous Euler equations are studied. Two regimes of approaching a blow-up point, respectively, with variable or fixed time are analyzed. It is shown that in the n-dimensional ( $n=2,3$) generic case the blowups of degrees $1,\cdots ,n$ at the variable time regime and of degrees $1/2,\cdots ,(n+1)/(n+2)$ at the fixed time regime may exist. Particular situations when the vorticity blows while the direction of the vorticity vector is concentrated in one or two directions are realizable.     

Epidemic dynamics and spatial segregation driven by cognitive diffusion and nonlinear incidence

We propose susceptible-infected-susceptible epidemic reaction–diffusion models with cognitive movement and nonlinear incidence $S^q{I}^p$ $(p,q>0)$ in a spatially heterogeneous environment. The cognitive dispersal term takes either random diffusion or symmetric diffusion. Building upon the $L^{\infty }$-estimates of positive solutions under $p,q>0$, we state the asymptotic dynamics for , $q>0$. The numerical results reveal spatial segregation of susceptible and infected populations: (a) the heterogeneous random diffusion can segregate the population and reduce the infection fraction significantly; (b) the segregation phenomenon disappears as the ratio $p/q$ approaches one from below; (c) the disease-free region strengthens the segregation induced by heterogeneous random diffusion; (d) the segregation governed by random diffusion is more sensitive to the incidence mechanism; (e) the distribution of steady states driven by symmetric diffusion is always similar to that by homogeneous diffusion.     

The KP limit of a reduced quantum Euler–Poisson equation

In this paper, we consider the derivation of the Kadomtsev–Petviashvili (KP) equation for cold ion-acoustic wave in the long wavelength limit of a two-dimensional reduced quantum Euler–Poisson system under different scalings for varying directions in the Gardner–Morikawa transform. It is shown that the types of the KP equation depend on the scaled quantum parameter $H>0$. The KP-I is derived for $H>2$, KP-II for , and the dispersiveless KP (dKP) equation for the critical case $H=2$. The rigorous proof for these limits is given in the well-prepared initial data case, and the norm that is chosen to close the proof is anisotropic in the two directions, in accordance with the anisotropic structure of the KP equation as well as the Gardner–Morikawa transform. The results can be generalized in several directions.     

The Riemann problem for a generalized Burgers equation with spatially decaying sound speed. I Large-time asymptotics

In this paper, we consider the classical Riemann problem for a generalized Burgers equation, $$u_t+h_{\alpha }(x)u{u}_x=u_{xx},$$ with a spatially dependent, nonlinear sound speed, $h_{\alpha }(x)\equiv {(1+x^2)}^{-\alpha }$ with $\alpha >0$, which decays algebraically with increasing distance from a fixed spatial origin. When $\alpha =0$, this reduces to the classical Burgers equation. In this first part of a pair of papers, we focus attention on the large-time structure of the associated Riemann problem, and obtain its detailed structure, as $t\to \infty$, via the method of matched asymptotic coordinate expansions (this uses the classical method of matched asymptotic expansions, with the asymptotic parameters being the independent coordinates in the evolution problem; this approach is developed in detail in the monograph of Leach and Needham, as referenced in the text), over all parameter ranges. We identify a significant bifurcation in structure at $\alpha =\frac{1}{2}$.     

Properties of given and detected unbounded solutions to a class of chemotaxis models

This paper deals with unbounded solutions to a class of chemotaxis systems. In particular, for a rather general attraction–repulsion model, with nonlinear productions, diffusion, sensitivities, and logistic term, we detect Lebesgue spaces where given unbounded solutions also blow up in the corresponding norms of those spaces; subsequently, estimates for the blow-up time are established. Finally, for a simplified version of the model, some blow-up criteria are proved. More precisely, we analyze a zero-flux chemotaxis system essentially described as (⋄) The problem is formulated in a bounded and smooth domain Ω of ${\mathbb{R}}^n$, with $n\ge 1$, for some $m_1,m_2,m_3\in \mathbb{R}$, $\chi ,\xi ,\alpha ,\beta ,\lambda ,\mu >0$, $k>1$, and with $T_{max}\in (0,\infty ]$. A sufficiently regular initial data $u_0\ge 0$ is also fixed. Under specific relations involving the above parameters, one of these always requiring some largeness conditions on $m_2+\alpha$,

• (i) we prove that any given solution to ($\diamond$), blowing up at some finite time $T_{max}$ becomes also unbounded in $L^{\mathfrak{p}}(\mathrm{\Omega })$-norm, for all $\mathfrak{p}>\frac{n}{2}(m_2-m_1+\alpha )$;
• (ii) we give lower bounds T (depending on ${\int }_{\mathrm{\Omega }}{u}_0^{\overline{p}}$) of $T_{max}$ for the aforementioned solutions in some $L^{\overline{p}}(\mathrm{\Omega })$-norm, being $\overline{p}=\overline{p}(n,m_1,m_2,m_3,\alpha ,\beta )\ge \mathfrak{p}$;
• (iii) whenever $m_2=m_3$, we establish sufficient conditions on the parameters ensuring that for some u₀ solutions to ($\diamond$) effectively are unbounded at some finite time.

Within the context of blow-up phenomena connected to problem ($\diamond$), this research partially improves the analysis in Wang et al. (J Math Anal Appl. 2023;518(1):126679) and, moreover, contributes to enrich the level of knowledge on the topic.     

Polynomial ergodicity of an SIRS epidemic model with density-dependent demographics

In this paper, a stochastic susceptible-infective-recovered-susceptible (SIRS) model with density-dependent demographics is proposed to study the dynamics of transmission of infectious diseases under stochastic environmental fluctuations. We demonstrate that the position of the basic reproduction number $R_0^s$ with respect to 1 is the threshold between extinction and persistence of the disease under mild extra conditions. That is, under mild extra conditions, when , the disease is eradicated with probability 1; when $R_0^s>1$, the disease is persistent almost surely and the Markov process has a unique stationary distribution and is polynomial ergodic. As an application, we use the 2017 influenza A data from Western Asia to estimate the parameter values of the model and based on that investigate the effect of random noises on the dynamics of the model. Our study reveals that the basic reproduction number $R_0^s$ is negatively correlated with the noise intensity for the infected but positively correlated with that for the susceptible population, which are different from the findings obtained in the existing literature.     

Long-time asymptotics and the radiation condition with time-periodic boundary conditions for linear evolution equations on the half-line and experiment

The asymptotic Dirichlet-to-Neumann (D-N) map is constructed for a class of scalar, constant coefficient, linear, third-order, dispersive equations with asymptotically time/periodic Dirichlet boundary data and zero initial data on the half-line, modeling a wavemaker acting upon an initially quiescent medium. The large time $t$ asymptotics for the special cases of the linear Korteweg-de Vries and linear Benjamin–Bona–Mahony (BBM) equations are obtained. The D-N map is proven to be unique if and only if the radiation condition that selects the unique wave number branch of the dispersion relation for a sinusoidal, time-dependent boundary condition holds: (i) for frequencies in a finite interval, the wave number is real and corresponds to positive group velocity, and (ii) for frequencies outside the interval, the wave number is complex with positive imaginary part. For fixed spatial location $x$, the corresponding asymptotic solution is (i) a traveling wave or (ii) a spatially decaying, time-periodic wave. The linearized BBM asymptotics are found to quantitatively agree with viscous core-annular fluid experiments.     

Uniformly 3-connected graphs

Let $k$ be a positive integer. A graph is said to be uniformly $k$-connected if between any pair of vertices the maximum number of independent paths is exactly $k$. Dawes showed that all minimally 3-connected graphs can be constructed from $K_4$ such that every graph in each intermediate step is also minimally 3-connected. In this paper, we generalize Dawes' result to uniformly 3-connected graphs. We give a constructive characterization of the class of uniformly 3-connected graphs which differs from the characterization provided by Göring et al., where their characterization requires the set of all 3-connected and 3-regular graphs as a starting set, the new characterization requires only the graph $K_4$. Eventually, we obtain a tight bound on the number of edges in uniformly 3-connected graphs.     

Asymptotic approximation of a highly oscillatory integral with application to the canonical catastrophe integrals

We consider the highly oscillatory integral $F(w):={\int }_{-\infty }^{\infty }{e}^{iw(t^{K+2}+e^{i\theta }{t}^p)}g(t)dt$ for large positive values of w, , K and p positive integers with $1\le p\le K$, and $g(t)$ an entire function. The standard saddle point method is complicated and we use here a simplified version of this method introduced by López et al. We derive an asymptotic approximation of this integral when $w\to +\infty$ for general values of K and p in terms of elementary functions, and determine the Stokes lines. For $p\ne 1$, the asymptotic behavior of this integral may be classified in four different regions according to the even/odd character of the couple of parameters K and p; the special case $p=1$ requires a separate analysis. As an important application, we consider the family of canonical catastrophe integrals ${\mathrm{\Psi }}_K(x_1,x_2,\text{…},x_K)$ for large values of one of its variables, say $x_p$, and bounded values of the remaining ones. This family of integrals may be written in the form $F(w)$ for appropriate values of the parameters w, θ and the function $g(t)$. Then, we derive an asymptotic approximation of the family of canonical catastrophe integrals for large $|x_p|$. The approximations are accompanied by several numerical experiments. The asymptotic formulas presented here fill up a gap in the NIST Handbook of Mathematical Functions by Olver et al.