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.     

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.     

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.     

Uniform boundedness and eventual Hölder continuity to a cancer invasion model with remodeling of ECM and nonlinear diffusion

This paper is concerned with a class of cancer invasion model with nonlinear diffusion and remodeling of ECM. The primary difficulty that arises here is that due to nonlinear diffusion, the good coupling structure between the diffusion term and haptotactic term is destroyed, rendering the effective methods used in the linear diffusion model no longer applicable. Fortunately, we have found a new effective combination between the diffusion term and the haptotactic term, which allows the diffusion term to dominate the haptotactic term, thus preliminarily improving the regularity of the weak solution. Based on these results, we can prove part of long-time asymptotic behavior of the solution, thereby finally proving the uniform boundedness of the weak solution. Subsequently, by improving the convergence of cancer cells u from $L^p$-norm to $L^{\infty }$-norm, it is also proved that after a long time, the weak solution will eventually be Hölder continuous for some slow diffusion cases.     

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.     

Three-dimensional lattice ground states for Riesz and Lennard-Jones–type energies

The Riesz potential $f_s(r)=r^{-s}$ is known to be an important building block of many interactions, including Lennard-Jones–type potentials $f_{n,m}^{\mathrm{LJ}}(r):=a{r}^{-n}-b{r}^{-m}$, $n>m$ that are widely used in molecular simulations. In this paper, we investigate analytically and numerically the minimizers among three-dimensional lattices of Riesz and Lennard-Jones energies. We discuss the minimality of the body-centered-cubic (BCC) lattice, face-centered-cubic (FCC) lattice, simple hexagonal (SH) lattices, and hexagonal close-packing (HCP) structure, globally and at fixed density. In the Riesz case, new evidence of the global minimality at fixed density of the BCC lattice is shown for and the HCP lattice is computed to have higher energy than the FCC (for $s>3/2$) and BCC (for ) lattices. In the Lennard-Jones case with exponents , the ground state among lattices is confirmed to be an FCC lattice whereas an HCP phase occurs once added to the investigated structures. Furthermore, phase transitions of type “FCC-SH” and “FCC-HCP-SH” (when the HCP lattice is added) as the inverse density V increases are observed for a large spectrum of exponents $(n,m)$. In the SH phase, the variation of the ratio Δ between the interlayer distance d and the lattice parameter a is studied as V increases. In the critical region of exponents , the SH phase with an extreme value of the anisotropy parameter Δ dominates. If one limits oneself to rigid lattices, the BCC-FCC-HCP phase diagram is found. For , the BCC lattice is the only energy minimizer. Choosing , the FCC and SH latices become minimizers.     

Boundary behavior of the solution to the linear Korteweg-De Vries equation on the half line

In this paper, we consider the solution to the linear Korteweg-De Vries (KdV) equation, both homogeneous and forced, on the quadrant $\{x\in {\mathbb{R}}^{+},t\in {\mathbb{R}}^{+}\}$ via the unified transform method of Fokas and we provide a complete rigorous study of the integrals of the formula provided by the method, especially focusing on the explicit verification of the considered initial-boundary-value problems (IBVPs), with generic data, as well as on the uniform convergence of all its derivatives, as $(x,t)$ approaches the boundary of the quadrant, and their rapid decay as $x\to \infty$.     

Self-contained two-layer shallow-water theory of strong internal bores

We show that interfacial gravity waves comprising strong hydraulic jumps (bores) can be described by a two-layer hydrostatic shallow-water (SW) approximation without invoking additional front conditions. The theory is based on a new SW momentum equation which is derived in locally conservative form containing a free parameter α. This parameter, which defines the relative contribution of each layer to the pressure at the interface, affects only hydraulic jumps but not continuous waves. The Rankine–Hugoniot jump conditions for the momentum and mass conservation equations are found to be mathematically equivalent to the classical front conditions, which were previously thought to be outside the scope of SW approximation. Dimensional arguments suggest that α depends on the density ratio. For nearly equal densities, both layers are expected to affect interfacial pressure with approximately equal weight coefficients, which corresponds to $\alpha \approx 0$. The front propagation velocity for $\alpha =0$ agrees well with experimental and numerical results in a wide range of bore strengths. A remarkably better agreement with high-accuracy numerical results is achieved by $\alpha =\sqrt{5}-2$, which yields the largest height that a stable gravity current can have.     

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 $\alpha =\frac{1}{2}$.     

Hyperbolicity study of the modulation equations for the Benjamin–Bona–Mahony equation

A complete study of the modulation equations for the Benjamin–Bona–Mahony equation is performed. In particular, the boundary between the hyperbolic and elliptic regions of the modulation equations is found. When the wave amplitude is small, this boundary is approximately defined by $k=\sqrt{3}$, where k is the wave number. This particular value corresponds to the inflection point of the linear dispersion relation for the BBM equation. Numerical results are presented showing the appearance of the Benjamin–Feir instability when the periodic solutions are inside the ellipticity region.     

Multiple orthogonal polynomials associated with the exponential integral

We introduce a new family of multiple orthogonal polynomials satisfying orthogonality conditions with respect to two weights $(w_1,w_2)$ on the positive real line, with $w_1(x)=x^{\alpha }{e}^{-x}$ the gamma density and $w_2(x)=x^{\alpha }{E}_{\nu +1}(x)$ a density related to the exponential integral $E_{\nu +1}$. We give explicit formulas for the type I functions and type II polynomials, their Mellin transform, Rodrigues formulas, hypergeometric series, and recurrence relations. We determine the asymptotic distribution of the (scaled) zeros of the type II multiple orthogonal polynomials and make a connection to random matrix theory. Finally, we also consider two related families of mixed-type multiple orthogonal polynomials.     

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.     

A characterization of well-posedness for the second order abstract Cauchy problems with finite delay

In this work, we introduce a strongly continuous one-parameter family of bounded linear operators that completely describes the well-posedness of a second order abstract differential delay equation in the initial history space $L^p([-r,0];X)$, $r>0$. This family, which satisfies a specific functional equation is applied to characterize the mild solution of the considered second order delay equation.     

Nonlinear inviscid damping and shear-buoyancy instability in the two-dimensional Boussinesq equations

We investigate the long-time properties of the two-dimensional inviscid Boussinesq equations near a stably stratified Couette flow, for an initial Gevrey perturbation of size ε. Under the classical Miles-Howard stability condition on the Richardson number, we prove that the system experiences a shear-buoyancy instability: the density variation and velocity undergo an $O(t^{-1/2})$ inviscid damping while the vorticity and density gradient grow as $O(t^{1/2})$. The result holds at least until the natural, nonlinear timescale $t\approx {\epsilon }^{-2}$. Notice that the density behaves very differently from a passive scalar, as can be seen from the inviscid damping and slower gradient growth. The proof relies on several ingredients: (A) a suitable symmetrization that makes the linear terms amenable to energy methods and takes into account the classical Miles-Howard spectral stability condition; (B) a variation of the Fourier time-dependent energy method introduced for the inviscid, homogeneous Couette flow problem developed on a toy model adapted to the Boussinesq equations, that is, tracking the potential nonlinear echo chains in the symmetrized variables despite the vorticity growth.     

Nonintegrability of the Painlevé IV equation in the Liouville–Arnold sense and Stokes phenomena

In this paper we study the integrability of the Hamiltonian system associated with the fourth Painlevé equation. We prove that one two-parametric family of this Hamiltonian system is not integrable in the sense of the Liouville–Arnold theorem. Computing explicitly the Stokes matrices and the formal invariants of the second variational equations, we deduce that the connected component of the unit element of the corresponding differential Galois group is not Abelian. Thus the Morales–Ramis–Simó theory leads to a nonintegrable result. Moreover, combining the new result with our previous one we establish that for all values of the parameters for which the $P_{IV}$ equation has a particular rational solution the corresponding Hamiltonian system is not integrable by meromorphic first integrals which are rational in t.