This paper focuses on the generalized Forchheimer flows of isentropic gas in a porous medium, described by a system of two nonlinear degenerate partial differential equations of first order. We prove the existence and uniqueness of the Dirichlet problem for the stationary problem. The technique of semi-discretization in time is used to prove the existence for the time-dependent problem.

All the possible Riemann solutions are constructed in fully explicit forms for the one-dimensional inviscid liquid–gas two-phase isentropic flow model of drift-flux type. The concentration and cavitation phenomena are observed and investigated in the Riemann solutions when the pressure in the mixture momentum equation tends to zero. It is proved rigorously that the Riemann solution consisting of a

In this paper, a doubly nonlinear parabolic problem involving nonstandard growth conditions under homogeneous Dirichlet boundary conditions is studied. To be a little precise, first, some energy estimates and inequalities are exploited to obtain energy decay estimates of the solution. Then, it is proved that the rest field u(x, t) = 0 is asymptotically stable in terms of natural energy associated with

Symmetry- and conservation law-preserving finite difference discretizations are obtained for linear and nonlinear one-dimensional wave equations on five- and nine-point stencils using the theory of Lie point symmetries of difference equations and the discrete direct multiplier method of conservation law construction. In particular, for the linear wave equation, an explicit five-point scheme is presented

We study the long-time asymptotics of prototypical non-linear diffusion equations. Specifically, we consider the case of a non-degenerate diffusivity function that is a (non-negative) polynomial of the dependent variable of the problem. We motivate these types of equations using Einstein’s random walk paradigm, leading to a partial differential equation in non-divergence form. On the other hand, using

The initial-boundary value problem (ibvp) for the mth order Korteweg–de Vries (KdVm) equation on the half-line is studied by extending a novel approach recently developed for the well-posedness of KdV on the half-line, which is based on the solution formula produced via the Fokas unified transform method for the associated forced linear ibvp. Replacing in this formula the forcing by the nonlinearity

We revisit the scattering problems for the 2D mass super-critical Schrödinger and Klein–Gordon equations with radial data below the ground state in the energy space. We give an alternative proof of energy scattering for both the defocusing and focusing cases using the ideas in the work of Dodson and Murphy [Proc. Am. Math. Soc. 145, 4859 (2017)]. Our results also include the exponential type nonlinearities

The Navier–Stokes–Voigt equations are a regularization of the Navier–Stokes equations. Sharing some asymptotic and statistical properties, they have been used in direct numerical simulations of the latter. In this article, we characterize the decay rate of the solutions to the Navier–Stokes–Voigt equations with damping. Applying the Fourier splitting method, we prove the H1 decay of weak solutions

In this paper, we first prove that the cubic, defocusing nonlinear Schrödinger equation on the two dimensional hyperbolic space with radial initial data in Hs(H2) is globally well-posed and scatters when s > 3/4. Then, we extend the result to nonlinearities of order p > 3. The result is proved by extending the high–low method of Bourgain in the hyperbolic setting and by using a Morawetz type estimate

The Heun–Racah and Heun–Bannai–Ito algebras are introduced. Specializations of these algebras are seen to be realized by the operators obtained by applying the algebraic Heun construct to the bispectral operators of the Racah and Bannai–Ito polynomials. The study supplements the results on the Heun–Askey–Wilson algebra and completes the description of the Heun algebras associated with the polynomial

A general theorem due to Howe of dual action of a classical group and a certain non-associative algebra on a space of symmetric or alternating tensors is reformulated in a setting of second quantization, and familiar examples in atomic and nuclear physics are discussed. The special case of orthogonal–orthogonal duality is treated in detail. It is shown that, like it was done by Helmers more than half

Given an arbitrary countably generated rigid C*-tensor category, we construct a fully faithful bi-involutive strong monoidal functor onto a subcategory of finitely generated projective bimodules over a simple, exact, separable, unital C*-algebra with a unique trace. The C*-algebras involved are built from the category using the Guionnet–Jones–Shlyakhtenko construction. Out of this category of Hilbert

We prove non-perturbative bounds on the time evolution of the probability distribution of operator size in the q-local Sachdev–Ye–Kitaev model with N fermions for any even integer q > 2 and any positive even integer N > 2q. If the couplings in the Hamiltonian are independent and identically distributed Rademacher random variables, the infinite temperature many-body Lyapunov exponent is almost surely

Newtonian and Schrödinger dynamics can be formulated in a physically meaningful way within the same Hilbert space framework. This fact was recently used to discover an unexpected relation between classical and quantum motions that goes beyond the results provided by the Ehrenfest theorem. The Newtonian dynamics was shown to be the Schrödinger dynamics of states constrained to a submanifold of the space

We develop generalized coherent states based on the Gazeau–Klauder formalism for a particle with position-dependent mass trapped in an infinite square well. We study the quantum statistical properties of these states by means of the Mandel parameter and the second-order correlation function. Our analysis reveals that the constructed coherent states exhibit sub-Poissonian statistics. Moreover, theoretical

We study a generalization of the Wigner function to arbitrary tuples of Hermitian operators. We show that for any collection of Hermitian operators A1, …, An and any quantum state, there is a unique joint distribution on Rn with the property that the marginals of all linear combinations of the Ak coincide with their quantum counterparts. In other words, we consider the inverse Radon transform of the

We study a magnetic Schrödinger Hamiltonian, with axisymmetric potential in any dimension. The associated magnetic field is unitary and non-constant. The problem reduces to a 1D family of singular Sturm–Liouville operators on the half-line indexed by a quantum number. We study the associated band functions. They have finite limits that are the Landau levels. These limits play the role of thresholds

We present the quantum and classical mechanics formalisms for a particle with a position-dependent mass in the context of a deformed algebraic structure (named κ-algebra), motivated by the Kappa-statistics. From this structure, we obtain deformed versions of the position and momentum operators, which allow us to define a point canonical transformation that maps a particle with a constant mass in a

The relevance in physics of non-Hermitian operators with real eigenvalues is being widely recognized not only in quantum mechanics but also in other areas, such as quantum optics, quantum fluid dynamics, and quantum field theory. In this note, a quantum system described by a non-Hermitian Hamiltonian, which is constituted by two types of interacting bosons, is investigated. The real eigenvalues of

We present an interpretation of the functions appearing in the Wei–Norman factorization of the evolution operator for a Hamiltonian belonging to the SU(1,1) algebra in terms of the classical solutions of the Generalized Caldirola–Kanai (GCK) oscillator (with time-dependent mass and frequency). Choosing P2, X2, and the dilation operator as a basis for the Lie algebra, we obtain that, out of the six

We study several versions of a quantum steganography problem in which two legitimate parties attempt to conceal a cypher in a quantum cover transmitted over a quantum channel without arising suspicion from a warden who intercepts the cover. In all our models, we assume that the warden has an inaccurate knowledge of the quantum channel and we formulate several variations of the steganography problem

A quantum walk describes the discrete unitary evolution of a quantum particle on a discrete graph. Some quantum walks, referred to as the Weyl and Dirac walks, provide a description of the free evolution of relativistic quantum fields in the small wave-vector regime. The clash between the intrinsic discreteness of quantum walks and the continuous symmetries of special relativity is resolved by giving

In quantum theory, it is known for a pair of noncommutative observables that there is no state on which they take simultaneously definite values and that there is no joint measurement of them. They are called preparation uncertainty and measurement uncertainty, respectively, and research has unveiled that they are not independent from but related with each other in a quantitative way. This study aims

It is shown that a sequence {Φn} of quantum channels strongly converges to a quantum channel Φ0 if and only if there exist a common environment for all the channels and a corresponding sequence {Vn} of Stinespring isometries strongly converging to a Stinespring isometry V0 of the channel Φ0. A quantitative description of the above characterization of the strong convergence in terms of appropriate metrics

Causal fermion systems incorporate local gauge symmetry in the sense that the Lagrangian and all inherent structures are invariant under local phase transformations of the physical wave functions. In the present paper, it is explained and worked out in detail that, despite this local gauge freedom, the structures of a causal fermion system give rise to distinguished gauges where the local gauge freedom

Recently, a detailed correspondence was established between, on one side, four- and five-dimensional large-N supersymmetric gauge theories with N = 2 supersymmetry and adjoint matter and, on the other side, integrable 1 + 1-dimensional quantum hydrodynamics. Under this correspondence, the phenomenon of dimensional transmutation, familiar in asymptotically free quantum field theories, gets mapped to

The equation of motion for cohomogeneity-one Nambu–Goto strings in flat space Rn,1 has been investigated. We first classify possible forms of the Killing vector fields in Rn,1 after appropriate action of the Poincaré group. Then, all possible forms of the Hamiltonian for the cohomogeneity-one Nambu–Goto strings are determined. It has been shown that the system always has the maximum number of functionally

The aim of this series of papers is to generalize the ambient approach of Duval et al. regarding the embedding of Galilean and Carrollian geometries inside gravitational waves with parallel rays. In this paper (Paper I), we propose a generalization of the embedding of torsionfree Galilean and Carrollian manifolds inside larger classes of gravitational waves. On the Galilean side, the quotient procedure

In this paper, the d-dimensional quantum harmonic oscillator with a pseudo-differential time quasi-periodic perturbation iψ̇=(−Δ+V(x)+ϵW(ωt,x,−i∇))ψ,x∈Rd is considered, where ω ∈ (0,2π)n, V(x)≔∑j=1dvj2xj2,vj≥v0>0, and W(θ, x, ξ) is a real polynomial in (x, ξ) of degree at most two, with coefficients belonging to Cℓ in θ∈Tn for the order ℓ satisfying ℓ ≥ 2n + β, 0 < β < 1. Using the techniques developed

This paper demonstrates sufficient conditions for the existence of a positive measure set of invariant KAM (Kolmogrov-Arnol'd-Moser) tori in a singly thermostated, one degree-of-freedom Hamiltonian vector field. This result is applied to four important single thermostats in the literature, and it is shown that in each case, if the Hamiltonian is real-analytic and well-behaved, then the thermostated

Linearization of a Hamiltonian system around an equilibrium point yields a set of Hamiltonian symmetric spectra: If λ is an eigenvalue of the linearized generator, −λ and λ¯ (hence, −λ¯) are also eigenvalues—the former implies a time-reversal symmetry, while the latter guarantees the reality of the solution. However, linearization around a singular equilibrium point (which commonly exists in noncanonical

We study steady bifurcation for the coupled system body-liquid consisting of a sphere freely falling in a Navier–Stokes liquid under the action of gravity. In particular, we show that, under the assumption that for the bifurcating solution, the translational velocity of the sphere is parallel to the gravity, bifurcation takes place, provided that 1 is a simple eigenvalue of a suitable linear operator

In this work, we derive a mathematical model for an open system that exchanges particles and momentum with a reservoir from their joint Hamiltonian dynamics. The complexity of this many-particle problem is addressed by introducing a countable set of n-particle phase space distribution functions just for the open subsystem, while accounting for the reservoir only in terms of statistical expectations

A quantum surface (QS) is an equivalence class of pairs (D, H) of simply connected domains D⊊C and random distributions H on D induced by the conformal equivalence for random metric spaces. This distribution-valued random field is extended to a QS with N + 1 marked boundary points (MBPs) with N∈Z≥0. We propose the conformal welding problem for it in the case of N∈Z≥1. If N = 1, it is reduced to the

We consider the quantum Sherrington–Kirkpatrick (SK) spin-glass model with a transverse field and provide a formula for its free energy in the thermodynamic limit, valid for all inverse temperatures β > 0. To characterize the free energy, we use the path integral representation of the partition function and approximate the model by a sequence of finite-dimensional vector-spin glasses with Rd-valued

Explicit examples of positive crystalline measures and Fourier quasicrystals are constructed using pairs of stable polynomials, answering several open questions in the area.

We construct Darboux transformations of arbitrary order for generalized Schrödinger-type equations, the potentials of which are second-degree polynomials in the energy. Our equations are allowed to contain first-derivative terms with arbitrary coefficients, such as they occur, for example, in position-dependent mass scenarios. Our Darboux transformations are shown to be applicable to equations from

We find the large number asymptotics of the Legendre functions Pn(cos θ) and Qn(cos θ) uniform in 0 ≤ θ ≤ π by reducing the procedure to regular perturbations. The corresponding results for the associated functions Pnm(cos⁡θ) and Qnm(cos⁡θ) are also discussed.

We consider the problem of variable separation for the classical integrable Hamiltonian systems possessing gl(n)-valued Lax matrices satisfying quadratic Poisson brackets of Freidel and Maillet [Phys. Lett. B 262(2-3), 278 (1991)] with spectral-parameter dependent a-b-c-d tensors. We formulate, in terms of the corresponding a-b-c-d tensors, a sufficient condition that guarantees that the separating

In this article, we prove a reducibility result for the linear Schrödinger equation on a Zoll manifold with quasi-periodic in time pseudo-differential perturbation of order less than or equal to 1/2. As far as we know, this is the first reducibility result for an unbounded perturbation on a compact manifold different from the torus.

In this paper, we consider the Cauchy problem for the N − abc family of the Camassa–Holm type equation with both dissipation and dispersion. First, we establish the global well-posedness of the strong solutions under certain conditions on the initial datum. Then, we investigate the propagation speed with compactly supported initial data. This result improves earlier ones reported in the literature

This paper deals with a high order viscoelastic wave equation with nonlinear boundary damping–source interactions. By the combination of Galerkin approximation and potential well methods, we obtain the global existence of weak solutions. Under some appropriate assumptions on the boundary damping–source terms and the relaxation function g, we establish general decay and blow-up results associated with

We consider an L2-critical cubic Dirac equation in one space-dimension known as the Thirring model. Global well-posedness in L2 for this equation was proved by Candy. Here, we show that Candy’s result is sharp in the sense that the equation is ill posed in Lp for 1 ≤ p < 2 and in the massless case also in Hs with s < 0.

In this paper, we consider a coupled system of two wave equations. One of these equations is conservative, and the other has damping and delay terms. If the damping acts with more force than the delay term, we show polynomial stability for strong solutions of the system. Explicit decay rates are found, and the optimality of those rates is discussed. On the other hand, if the damping acts with the same

We investigate the existence of nontrivial solutions for a class of Schrödinger–Poisson systems with steep potential well V and the nonlinearity a(x)|u|p−2u(2 < p < 4) in R3. Such a problem cannot be studied by applying variational methods in a standard way, even by restricting its corresponding energy functional on the Nehari manifold, because Palais–Smale sequences may not be bounded. In this paper

A new fractional function space Xk(⋅),α(⋅)(Ω) with variable exponents k, α and its relaxed properties are established in this paper. Under this configuration, the comparison principle of the fractional α(⋅)-Laplace operator and a priori estimate of weak solutions for a variable-order fractional α(⋅)-Laplacian equation are obtained. Then, the application of a fixed point theorem ensures the existence

This paper proves some uniform regularity estimates for a two-phase model with a magnetic field and a related system in Td.

In this paper, we mainly devote to investigate the periodic Dullin–Gottwald–Holm equation. By overcoming the difficulties caused by the complicated mixed nonlinear structure, a very useful priori estimate is derived in Lemma 2.7. Based on Hα1-conservation and L∞-estimate of solution, some new blow-up phenomena are derived for the periodic Dullin–Gottwald–Holm equation under different initial conditions

This article concerns the formation of finite-time singularities in solutions to quasilinear hyperbolic systems with small initial data. We propose a universal test function method that works for many nonlinear hyperbolic systems arising from physical applications. We first present a simpler proof of the main result in the work of Sideris [Commun. Math. Phys. 101(4), 475–485 (1985)]: the global classical

In this paper, we study the existence and instability of normalized standing waves for the fractional Schrödinger equation i∂tψ = (−Δ)sψ − f(ψ), where 0 < s < 1, f(ψ) = |ψ|pψ with 4sN

The aim of this paper is to improve the previous work on the relativistic Vlasov–Maxwell system, one of the most important equations in plasma physics. Recently, Bardos et al. [Q. Appl. Math. 78, 193–217 (2020)] presented a proof of an Onsager type conjecture on the renormalization property and the entropy conservation laws for the relativistic Vlasov–Maxwell system. Particularly, the authors proved

In this paper, we study the following fractional Schrödinger equation involving critical exponent ε2s(−Δ)su+V(x)u=|u|2s*−2u+λf(u), x∈RN, where 0 < s < 1, N > 2s, 2s*=2NN−2s, λ > 0, and f is a continuous superlinear but subcritical function. Under some suitable conditions, we obtain the concentration behavior of solutions for the above equation. This improves the results in the work of Shang and Zhang

In this paper, the well-posedness and optimal convergence rates of subsonic irrotational flows through a three-dimensional infinitely long nozzle with a smooth obstacle inside are established. More precisely, the global existence and uniqueness of the uniformly subsonic flow are obtained via variational formulation as long as the incoming mass flux is less than a critical value. Furthermore, with the

We study the ground state of a large number N of 2D extended anyons in an external magnetic field. We consider a scaling limit where the statistics parameter α is proportional to N−1 when N → ∞, which allows the statistics to be seen as a “perturbation around the bosonic end.” Our model is that of bosons in a magnetic field interacting through long-range magnetic potential generated by magnetic charges

Complex-valued functions defined on a finite interval [a, b] generalizing power functions of the type (x−x0)n for n ≥ 0 are studied. These functions called Φ-generalized powers, Φ being a given nonzero complex-valued function on the interval, were considered to construct a general solution representation of the Sturm–Liouville equation in terms of the spectral parameter [V. V. Kravchenko and R. M.

We develop a mathematically rigorous theory for the quantum transfer processes in degenerate donor–acceptor dimers in contact with a thermal environment. We explicitly calculate the transfer rates and the acceptor population efficiency. The latter depends critically on the initial donor state. We show that quantum coherence in the initial state enhances the transfer process. If the electron is initially

We present a study of the properties of Bargmann Invariants (BIs) and Null Phase Curves (NPCs) in the theory of the geometric phase for finite dimensional systems. A recent suggestion to exploit the Majorana theorem on symmetric SU(2) multispinors is combined with the Schwinger oscillator operator construction to develop efficient operator-based methods to handle these problems. The BI is described

Adiabatic passage techniques, used to drive a system from one quantum state into another, find widespread applications in physics and chemistry. We focus on techniques to spatially transport a quantum amplitude over a strongly coupled system, such as STImulated Raman Adiabatic Passage (STIRAP) and Coherent Tunneling by Adiabatic Passage (CTAP). Previous results were shown to work on certain graphs

We prove decomposition rules for quantum Rényi mutual information, generalizing the relation I(A : B) = H(A) − H(A|B) to inequalities between Rényi mutual information and Rényi entropy of different orders. The proof uses Beigi’s generalization of Reisz–Thorin interpolation to operator norms [J. Math. Phys. 54(12), 122202 (2013)] and a variation of the argument developed by Dupuis [J. Math. Phys. 56

This paper is another step in our pursuit of nonperturbative Minkowskian quantum field theory. Instead of studying the Schwinger functions arising from the Euclidean approach, we study the Wightman distributions directly. We use the non-rigorous Feynman integral for a given quantum field theory as a guide for the rigorous definition of the characteristic functional corresponding to the Lagrangian,