We study the logarithmic Schrödinger equation with the sign-changing potential function. It is known that the corresponding functional is not well defined in H1(RN), but by imposing some condition on V(x), we can show that the functional is well defined in a subspace of H1(RN). Then, the existence and multiplicity of solutions is obtained by using variational methods. We remark that the existence of

We estimate the L∞ norm of the first derivative of the solution to the Chern–Simons gauged O(3) sigma model in R1+1 by using local energy conservation and observing the transport equation along the characteristic line. As an application of the bound, we improve the growth rate of the H2 norm of the solution from an exponential to a polynomial one. We also apply a similar idea to the Chern–Simons–Higgs

In this paper, we derive some new blow up criterion for the 2D full compressible Navier–Stokes equations in terms of the density and the temperature in critical spaces. This is an improvement of corresponding results obtained by Wang and Zhang [Appl. Math. Comput. 232, 719–726 (2014)].

In this paper, we investigate the generalized Camassa–Holm equation with k + 1 degree nonlinearities. The global existence is established, provided that 0 < b < k and the initial data y0=(1−∂x2)u0 do not change sign. Long time behavior for the support of momentum density is obtained, and more precisely, we deduce the limit of the support of momentum density as t goes to +∞.

In this work, we prove the local well-posedness of strong solutions to the isentropic compressible magnetohydrodynamics system with vacuum in a bounded domain Ω⊂R3.

In this paper, for degenerate n-degree Fisher-type equations, we discuss the stability of their traveling front solutions with noncritical speeds. In fact, when the initial perturbations around these noncritical traveling front solutions are in some weighted Banach spaces, we have proved that these solutions are globally exponentially stable in the form of (1+t)13e−νt for ν ∈ (0, 1) via L1-energy estimates

In the present work, we obtain the existence and multiplicity of nontrivial solutions for the Choquard logarithmic equation (−Δ)psu+a|u|p−2u+λ(ln|⋅|*|u|p)|u|p−2u=f(u)inRN, where N = sp, s ∈ (0, 1), p > 2, a > 0, λ > 0, and f:R→R is a continuous nonlinearity with exponential critical and subcritical growth. We guarantee the existence of a nontrivial solution at the mountain pass level and a nontrivial

In this work, we analyze the flow filtration process of slightly compressible fluids in porous media containing fractures with complex geometries. We model the coupled fracture-porous media system where the linear Darcy flow is considered in porous media and the nonlinear Forchheimer equation is used inside the fracture. We develop a model to examine the flow inside fractures with complex geometries

We consider the nonlinear 2 × 2 Gurevich–Zybin system as a model for the one-dimensional dynamics of dark matter. In this setting, we evaluate the movement of a particle of mass subjected to a possible jump discontinuity in the hydrodynamic speed. This problem is solved by extending the concept of usual solution to the framework of a product of distributions. A brief survey of the ideas and formulas

We investigate mean dynamics and invariant measures for a multi-stochastic discrete sine-Gordon equation driven by random viscosity, stochastic forces, and infinite-dimensional nonlinear noise. We first show the existence of a unique solution when the random viscosity is bounded and the nonlinear diffusion of noise is locally Lipschitz continuous, which leads to the existence of a mean random dynamical

We show that non-obtuse trapezoids are uniquely determined by their Dirichlet Laplace spectrum. This extends our previous result [Hezari et al., Ann. Henri Poincare 18(12), 3759–3792 (2017)], which was only concerned with the Neumann Laplace spectrum.

We consider polygon and simplex equations, of which the simplest nontrivial examples are pentagon (5-gon) and Yang–Baxter (2-simplex), respectively. We examine the general structure of (2n + 1)-gon and 2n-simplex equations in direct sums of vector spaces. Then, we provide a construction for their solutions, parameterized by elements of the Grassmannian Gr(n + 1, 2n + 1).

We revisit the free field construction of the deformed W-superalgebras Wq,t(sl(2|1)) by Ding and Feigin, Contemp. Math. 248, 83–108 (1998), where the basic W-current and screening currents have been found. In this paper, we introduce higher W-currents and obtain a closed set of quadratic relations among them. These relations are independent of the choice of Dynkin diagrams for the superalgebra sl(2|1)

In this paper, we study the conjecture of Benson and Ratcliff, which deals with the class of nilpotent Lie algebras of a one-dimensional center. We show that this conjecture is true for any nilpotent Lie algebra g with dimg≤5, but it fails for the dimensions greater or equal to 6. To this end, we produce counter-examples to the Benson–Ratcliff conjecture in all dimensions n ≥ 6. Finally, we show that

We present a fresh look at the methods introduced by Boccato, Brennecke, Cenatiempo, and Schlein [Commun. Math. Phys. 359(3), 975–1026 (2018); Acta Math. 222(2), 219–335 (2019); Commun. Math. Phys. 376, 1311 (2020)] concerning the trapped Bose gas and give a conceptually very simple and concise proof of Bose–Einstein condensation in the Gross–Pitaevskii limit for small interaction potentials.

We use the characteristic function of the Wigner function (its double Fourier transform) to give solution to any generic open quantum linear systems (systems whose Hamiltonian is at most quadratic). The solution is carried out in terms of the application of the transition matrix of the dynamical evolution in the Fourier space. We address two cases: the time-independent coefficients for which we give

The accidental degeneracy appearing in cycloacenes as triplets and quadruplets is explained with the concept of segmentation, introduced here with the aim of describing the effective disconnection of π orbitals on these organic compounds. For periodic systems with time reversal symmetry, the emergent nodal domains are shown to divide the atomic chains into simpler carbon structures analog to benzene

A family of non-Hermitian real and tridiagonal-matrix candidates H(N)(λ)=H0(N)+λW(N)(λ) for a hiddenly Hermitian (a.k.a. quasi-Hermitian) quantum Hamiltonian is proposed and studied. Fairly weak assumptions are imposed upon the unperturbed matrix [the square-well-simulating spectrum of H0(N) is not assumed equidistant)] and upon its maximally non-Hermitian N-parametric antisymmetric-matrix perturbations

We focus on two types of coherent states, the coherent states of multi-graviton states and the coherent states of giant graviton states, in the context of gauge/gravity correspondence. We conveniently use a phase shift operator and its actions on the superpositions of these coherent states. We find N-state Schrödinger cat states, which approach the one-row Young tableau states, with fidelity between

A double field theory algebroid (DFT algebroid) is a special case of the metric (or Vaisman) algebroid, shown to be relevant in understanding the symmetries of double field theory. In particular, a DFT algebroid is a structure defined on a vector bundle over doubled spacetime equipped with the C-bracket of double field theory. In this paper, we give the definition of a DFT algebroid as a curved L∞-algebra

We generalize the symmetry superalgebras of isometries and geometric Killing spinors on a manifold to include all the hidden symmetries of the manifold generated by Killing spinors in all dimensions. We show that bilinears of geometric Killing spinors produce special Killing–Yano forms and special conformal Killing–Yano forms. After defining the Lie algebra structure of hidden symmetries generated

We show that the closed-form monotone solution linking two different vacuum states, for the exterior Yang–Mills wave equation over the extremal Reissner–Nordström spacetime found in the recent work of Bizoń and Kahl [Classical Quantum Gravity 33, 175013 (2016)], is the unique solution to an associated domain-wall type energy minimization problem. We also derive the accompanying interior Yang–Mills

A teleparallel geometry is an n-dimensional manifold equipped with a frame basis and an independent spin connection. For such a geometry, the curvature tensor vanishes and the torsion tensor is non-zero. A straightforward approach to characterizing teleparallel geometries is to compute scalar polynomial invariants constructed from the torsion tensor and its covariant derivatives. An open question has

We study emergent dynamics of the Lohe Hermitian sphere (LHS) model, which can be derived from the Lohe tensor model [S.-Y. Ha and H. Park, SIAM J. Appl. Dyn. Syst. 13, 1312–1342 (2020)] as a complex counterpart of the Lohe sphere model. The LHS model describes aggregate dynamics of point particles on the Hermitian sphere HSd lying in Cd+1, and the coupling terms in the LHS model consist of two terms

We prove a large deviation principle (LDP) and a fluctuation theorem for the entropy production rate (EPR) of the following d dimensional stochastic differential equation dXt=AXtdt+QdBt, where A is a real normal stable matrix, Q is positive definite, and the matrices A and Q commute. The rate function for the EPR takes the following explicit form: I(x)=x1+ℓ0(x)−12+12∑k=1dαk2−βk2ℓ0(x)+αk for x ≥ 0 and

We consider the four-dimensional hyperchaotic system ẋ=a(y−x), ẏ=bx+u−y−xz, ż=xy−cz, and u̇=−du−jx+exz, where a, b, c, d, j, and e are real parameters. This system extends the famous Lorenz system to four dimensions and was introduced in Zhou et al., Int. J. Bifurcation Chaos Appl. Sci. Eng. 27, 1750021 (2017). We characterize the values of the parameters for which their equilibrium points are zero-Hopf

A geometric environment for the study of non-holonomic Lagrangian systems is developed. A definition of admissible displacement valid in the presence of arbitrary non-linear kinetic constraints is proposed. The meaning of ideality for non-strictly mechanical systems is analyzed. The concepts of geometric and/or dynamical symmetry of a constrained system are discussed and embodied in a subsequent non-holonomic

To further confirm the causality and stability of a second-order hyperbolic system of partial differential equations that models the relativistic dynamics of barotropic fluids with viscosity and heat conduction [H. Freistühler and B. Temple, J. Math. Phys. 59, 063101 (2018)], this paper studies the Fourier–Laplace modes of this system and shows that all such modes, relative to arbitrary Lorentz frames

We study the time evolution of an incompressible fluid with an axial symmetry without swirl when the vorticity is sharply concentrated on N annuli of radii ≈r0 and thickness ɛ. We prove that when r0 = |log ɛ|α, α > 2, the vorticity field of the fluid converges as ɛ → 0 to the point-vortex model, at least for a small but positive time. This result generalizes a previous paper that assumed a power law

We consider the equilibrium points of the electrostatic potential of three mutually repelling point charges with Coulomb interaction placed at the vertices of a given triangle T. It is proven that for each point P inside the triangle T, there exists a unique collection of positive point charges, called stationary charges for P in T, such that P is a critical point of the electrostatic potential of

Several sums of Neumann series with Bessel and trigonometric functions are evaluated, as finite sums of trigonometric functions. They arise from a generalization of the Neumann expansion of the eigenstates of the Laplacian in regular polygons. A simple accurate approximation of J0(x) is found on the interval [0, 2].

We study symmetry properties and the possibility of exact integration of Klein–Gordon equations in external electromagnetic fields on 3D de Sitter background dS3. We present an algorithm for constructing the first-order symmetry algebra and describe its structure in terms of Lie algebra extensions. Based on the well-known classification of the subalgebras of the algebra so(1,3), we classify all electromagnetic

We prove the existence of ballistic transport for a Schrödinger operator with a generic quasi-periodic potential in any dimension d > 1.

In this article, we construct the Riemann solutions of the compressible Euler equations in non-isentropic modified Chaplygin gas. In order to compare the Riemann solutions of modified and pure Chaplygin gas, we extend the solvable region of the Riemann solution of the pure Chaplygin gas state and discover the formation of the δ-shock waves. By studying the limiting behavior, we find that the limit

In this work, we investigate the Cauchy problem for a generalized Riemann-type hydrodynamical equation. The local well-posedness of the equation in Besov spaces is derived by using Littlewood–Paley decomposition and transport equation theory. Then, we show that a finite maximal life span for a solution necessarily implies wave breaking for this solution and give a condition on the initial data to ensure

This paper is concerned with an attraction–repulsion Navier–Stokes system with signal-dependent motility and sensitivity in a two-dimensional smooth bounded domain under zero Neumann boundary conditions for n, c, v and the homogeneous Dirichlet boundary condition for u. This system describes the evolution of cells that react on two different chemical signals in a liquid surrounding environment and

In this paper, we consider a two-species chemotaxis system with a logistic source. We present the global existence of generalized solutions for the two-species chemotaxis system under appropriate regularity assumptions on the initial data. This result partially generalizes and improves previously known ones.

In this paper, the two-component Fornberg–Whitham system is studied. We first investigate the local well-posedness in classical Sobolev space and establish a blow-up criterion by a local-in-time priori estimate, and then, we give some sufficient conditions on the initial data to lead to wave breaking. Finally, we discuss analytically the existence of periodic traveling waves by the bifurcation theorem

We consider the stochastic nonlinear Schrödinger equations on a half-line under Dirichlet brown-noise boundary conditions. We establish the global existence and uniqueness of solutions to the initial-boundary value problem with values in H1. We are also interested in the regularity behavior of solutions near the origin, where the boundary data are highly irregular.

In this paper, we combine a two-step iterative shooting argument and a fixed-point theorem approach to establish the existence of the solution of the dark monopole equations in a recently formulated non-Abelian gauge field theory by Deglmann and Kneipp and discuss its qualitative properties. In addition, the energy-minimizing solution of the dark monopole equations also satisfies the monotonicity,

In this paper, we study Keller–Segel type chemotaxis systems with power-like nonlinear sensitivity, production of signals, and switching chemotaxis mechanism. We establish explicit relations to ensure local- and global-in-time boundedness of classical solutions. In the chemo-attractive setting, our results cover and unify separate cases and they are critical to the quite known blow-up results in the

In this paper, we consider the magnetic Ginzburg–Landau equation with external potentials μV(x) for the type II superconductors. We prove, by reduction arguments, that under suitable conditions on V(x), the magnetic Ginzburg–Landau equation with external potentials in R2 has infinitely many multi-vortex solutions for μ > 0 being sufficiently small.

This work concerns the Kirchhoff equation −(a + b∫Ω|∇u|2dx)Δu + u = |u|p−2u, where a, b > 0, Ω⊆R3 is an exterior domain with a smooth boundary and 4 < p < 6. By establishing a global compactness result, we prove that the equation has at least one positive solution. Our result is an extension of the work by Benci and Cerami [Arch. Ration. Mech. Anal. 99(4), 283–300 (1987)].

In this paper, we prove the global existence and almost exponential decay of solutions for a Boussinesq approximation Bénard convection fluid. Compared with our previous paper [B. Guo, B. Xie, and L. Zeng, J. Differ. Equations 267, 2261–2283 (2019)], here the effect of the surface tension is neglected on the free surface. The analysis needs more higher regularity than the surface tension case.

We consider the longitudinal dynamical two-point function of the XXZ quantum spin chain in the antiferromagnetic massive regime. It has a series representation based on the form factors of the quantum transfer matrix of the model. The nth summand of the series is a multiple integral accounting for all n-particle–n-hole excitations of the quantum transfer matrix. In previous works, the expressions for

In the presence of a confining potential V, the eigenfunctions of a continuous Schrödinger operator −Δ + V decay exponentially with the rate governed by the part of V, which is above the corresponding eigenvalue; this can be quantified by a method of Agmon. Analogous localization properties can also be established for the eigenvectors of a discrete Schrödinger matrix. This note shows, perhaps surprisingly

The virtues of resolvent algebras, compared to other approaches for the treatment of canonical quantum systems, are exemplified by infinite systems of non-relativistic bosons. Within this framework, equilibrium states of trapped and untrapped bosons are defined on a fixed C*-algebra for all physically meaningful values of the temperature and chemical potential. Moreover, the algebra provides the tools

We look at periodic Jacobi matrices on trees. We provide upper and lower bounds on the gap of such operators analogous to the well-known gap in the spectrum of the Laplacian on the upper half-plane with a hyperbolic metric. We make some conjectures about antibound states and make an interesting observation for the so-called rg-model where the underlying graph has r red and g green vertices and where

In this paper, we establish the Heisenberg’s uncertainty inequality associated with the Caputo derivative of order q ∈ (1, ∞) in the generalized Bargmann–Fock space. We also determine exactly when the equality occurs in the uncertainty inequality. It is done by estimating the growth of the eigenvalues of the commutator [Dq,zq]. We prove that the sequence of eigenvalues of [Dq,zq] tends to ∞ when the

The transfer matrix M of a short-range potential may be expressed in terms of the time-evolution operator for an effective two-level quantum system with a time-dependent non-Hermitian Hamiltonian. This leads to a dynamical formulation of stationary scattering. We explore the utility of this formulation in the study of the low-energy behavior of the scattering data. In particular, for the exponentially

We investigate the transition probabilities for the “flavor” eigenstates in the two-level quantum system, which is described by a non-Hermitian Hamiltonian with the parity and time-reversal (PT) symmetry. Particularly, we concentrate on the so-called PT-broken phase, where two eigenvalues of the non-Hermitian Hamiltonian turn out to be a complex-conjugate pair. In this case, we find that the transition

In this work, it is shown that there is an inherent nonlinear evolution in the dynamics of the so-called generalized coherent states. To show this, the immersion of a classical manifold into the Hilbert space of quantum mechanics is employed. Then, one may parameterize the time dependence of the wave function through the variation of parameters in the classical manifold. Therefore, the immersion allows

We consider a system of N bosons where the particles experience a short-range two-body interaction given by N−1vN(x) = N3β−1v(Nβx), where v∈Cc∞(R3), without a definite sign on v. We extend the results of Grillakis and Machedon [Commun. Math. Phys. 324(2), 601–636 (2013)] and Kuz [Differ. Integral Equations 30(7/8), 587–630 (2017)] regarding the second-order correction to the mean-field evolution of

This paper will explore the restriction of the coherent information to the positive definite density matrices in the special case where the quantum channels are strictly positive linear maps. The space of positive definite density matrices is equipped with an embedded submanifold structure of the real vector space of Hermitian matrices. These ensure that the n-shot coherent information is differentiable

Which matrix norms induce proper measures for quantifying quantum coherence? We study this problem for two important classes of norms and show that (i) coherence measures cannot be induced by any unitary similarity invariant norm and (ii) the ℓq,p-norm induces a coherence measure if and only if q = 1 and 1 ≤ p ≤ 2, thus giving a new class of coherence measures with simple closed forms that are easy

We describe how to extend the notion of infinitesimal Markovian divisibility from quantum channels to general linear maps and compact and convex sets of generators. We give a general approach toward proving necessary criteria for (infinitesimal) Markovian divisibility. With it, we prove two necessary criteria for infinitesimal divisibility of quantum channels in any finite dimension d: an upper bound

We consider discrete one-dimensional nonlinear equations and present the procedure of lifting them to Z-graded graphs. We identify conditions that allow one to lift one-dimensional solutions to solutions on graphs. In particular, we prove the existence of solitons for static potentials on graded fractal graphs. We also show that even for a simple example of a topologically interesting graph, the corresponding

We introduce the notion of compatibility dimension for a set of quantum measurements: it is the largest dimension of a Hilbert space on which the given measurements are compatible. In the Schrödinger picture, this notion corresponds to testing compatibility with ensembles of quantum states supported on a subspace, using the incompatibility witnesses of Carmeli, Heinosaari, and Toigo [Phys. Rev. A 98

We show that charge-quantization of the M-theory C-field in J-twisted Cohomotopy implies the emergence of a higher Sp(1)-gauge field on single heterotic M5-branes, which exhibits a worldvolume Stringc2-structure.

In this paper, we investigate the ϕ4 model with cutoffs. By introducing a spatial cutoff and a momentum cutoff, the total Hamiltonian is a self-adjoint operator on a boson Fock space. Under regularity conditions of the momentum cutoff, we obtain the first order expansion of a non-degenerate ground state energy of the total Hamiltonian.