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In this paper, we consider the following elliptic system of Hamiltonian type on a bounded domain: (0.1) { − Δ u = K 1 (| y |) | v | p − 1 v , in B 1 (0) , − Δ v = K 2 (| y |) | u | q − 1 u , in B 1 (0) , u = v = 0 on ∂ B 1 (0) , where K 1 (r) and K 2 (r) are positive bounded functions, B 1 (0) is the unit ball in R N , (p , q) is a pair of positive numbers lying on the critical hyperbola: 1 p + 1 + 1 q + 1 = N − 2 N. We investigate the local uniqueness and the nondegeneracy of the bubbling solution for problem (0.1). Our proof is based on the local Pohozaev identities, blow-up analysis, and the properties of Greens function. Our results are quite different from single equation, which is mainly caused by the non-cooperative nonlinear terms and (p , q) lying on the critical hyperbola. We believe that the various new ideas and technique computations that we used in this paper would be very useful to deal with other related problems of Hamiltonian type with interact critical exponents. [ABSTRACT FROM AUTHOR]

In this paper, we study the following (p , q) -Laplacian equation with L p -constraint: { − Δ p u − Δ q u + λ | u | p − 2 u = f (u) , in R N , ∫ R N | u | p d x = c p , u ∈ W 1 , p (R N) ∩ W 1 , q (R N) , where 1 < p < q < N , Δ i = div (| ∇ u | i − 2 ∇ u) , with i ∈ { p , q } , is the i -Laplacian operator, λ is a Lagrange multiplier and c > 0 is a constant. The nonlinearity f is assumed to be continuous and satisfying weak mass supercritical conditions. The purpose of this paper is twofold: to establish the existence of ground states, and to reveal the basic behavior of the ground state energy E c as c > 0 varies. Moreover, we introduce a new approach based on the direct minimization of the energy functional on the linear combination of Nehari and Pohozaev constraints intersected with the closed ball of radius c p in L p (R N). The analysis developed in this paper allows to provide the general growth assumptions imposed to the reaction f. [ABSTRACT FROM AUTHOR]

In this paper we consider a class of p -evolution equations of arbitrary order with variable coefficients depending on time and space variables (t , x). We prove necessary conditions on the decay rates of the coefficients for the well-posedness of the related Cauchy problem in Gevrey spaces. [ABSTRACT FROM AUTHOR]

Multiparametric quantum semigroups M q ˆ , p ˆ (n) are generalization of the one-parameter general linear semigroups M q (n) , where q ˆ = (q i j) and p ˆ = (p i j) are 2 n 2 parameters satisfying certain conditions. In this paper, we study the algebra of multiparametric Manin matrices using the R-matrix method. The systematic approach enables us to obtain several classical identities such as Muir's identities, Newton's identities, Capelli-type identities, Cauchy-Binet's identity both for determinant and permanent as well as a rigorous proof of the MacMahon master equation for the quantum algebra of multiparametric Manin matrices. Some of the generalized identities are also lifted to multiparameter q -Yangians. [ABSTRACT FROM AUTHOR]

We consider the Hamilton-Jacobi equation H (x , D u) + λ (x) u = c , x ∈ M , where M is a connected, closed and smooth Riemannian manifold. The functions H (x , p) and λ (x) are continuous. H (x , p) is convex, coercive with respect to p , and λ (x) changes the signs. The first breakthrough to this model was achieved by Jin-Yan-Zhao [11] under the Tonelli conditions. In this paper, we consider more detailed structure of the viscosity solution set and large time behavior of the viscosity solution on the Cauchy problem. To the best of our knowledge, it is the first detailed description of the large time behavior of the HJ equations with non-monotone dependence on the unknown function. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Cauchy problem for the three dimensional parabolic-elliptic Keller-Segel equation with the large plane Couette-Poiseuille flow. Without advection, there exist solution with arbitrarily mass which blow up in finite time. Firstly, we introduce three dimensional plane Couette-Poiseuille flow and study the enhanced dissipation effect of such flows by resolvent estimate method. Next, we show that the enhanced dissipation effect of such flows can suppress blow-up of solution to three dimensional parabolic-elliptic Keller-Segel equation and establish global classical solution with large initial data. [ABSTRACT FROM AUTHOR]

In this paper, we establish a new regularity criterion in terms of the magnetic field for the local-in-time smooth solution of the two-dimensional non-resistive MHD equations. In particular, our result settles an open problem remarked by Lei et al. (2010), [11]. [ABSTRACT FROM AUTHOR]

In this paper, we use Aleksandrov's reflection principle to prove symmetry of solutions to F (∇ S n 2 u + u I) = f (u , u 2 + | ∇ S n u | 2 ) , where u is the support function of a convex body, and F is a function of principal radii. As a corollary, alongside [Ivaki, arXiv:2307.06252 ], we provide an alternative proof of the uniqueness of solution to the isotropic Gaussian-Minkowski problem. [ABSTRACT FROM AUTHOR]

In this paper we prove the existence of highest, cusped, traveling wave solutions for the fractional KdV equations f t + f f x = | D | α f x for all α ∈ (− 1 , 0) and give their exact leading asymptotic behavior at zero. The proof combines careful asymptotic analysis and a computer-assisted approach. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of ground state solutions for the following (p , q) -Choquard equation: − Δ p u − Δ q u + | u | p − 2 u + | u | q − 2 u = (I α ⁎ F (u)) f (u) in R N , where 2 ≤ p < q < N , Δ s is the s -Laplacian operator, with s ∈ { p , q } , I α is the Riesz potential of order α ∈ ((N − 2 q) + , N) , F ∈ C 1 (R , R) is a general nonlinearity of Berestycki-Lions type and F ′ = f. Furthermore, we analyze the regularity, symmetry and decay properties of these solutions. In particular, we extend the results in [33] to the (p , q) -Laplacian setting. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with traveling wave phenomena of the inhomogeneous half-wave equation, which models the energy of a spin zero particle in the Coulomb field. We study the Gagliardo-Nirenberg and critical Hardy-Sobolev inequalities with velocity 0 < | v | < 1 and obtain the estimates for the best constants and optimizers of inequalities. Moreover, we establish the non-scattering results with small traveling wave for energy subcritical and critical cases. [ABSTRACT FROM AUTHOR]

This paper is concerned with the stability of planar viscous shock wave for the 3-dimensional compressible Navier-Stokes-Poisson (NSP) system in the domain Ω : = R × T 2 with T 2 = (R / Z) 2. The stability problem of viscous shock under small 1-dimensional perturbations was solved in Duan-Liu-Zhang [7]. In this paper, we prove the viscous shock is still stable under small 3-d perturbations. Firstly, we decompose the perturbation into the zero mode and non-zero mode. Then we can show that both the perturbation and zero-mode time-asymptotically tend to zero by the anti-derivative technique and crucial estimates on the zero-mode. Moreover, we can further prove that the non-zero mode tends to zero with exponential decay rate. The key point is to estimate the non-zero mode of nonlinear terms involving electronic potential, see Lemma 6.1 below. [ABSTRACT FROM AUTHOR]

In this paper we study the asymptotic dynamics for a class of semilinear parabolic problems with Dirichlet boundary conditions in domains with a small hole of size proportional to a positive parameter ε. In other words, we prove that the family of attractors behaves continuously as ε → 0. We will also provide the convergence rates in terms of the parameter. [ABSTRACT FROM AUTHOR]

This paper concerns the structural stability of subsonic steady-states to the bipolar Euler-Poisson equations under small perturbation of doping profiles. Here, the electron density is imposed with degenerate sonic boundary and considered in interiorly subsonic case, while the hole density is considered in fully subsonic case. Unlike the unipolar model, we show that the structural stability in bipolar model holds regardless of the type of doping profile and propose a new version of comparison principle, which captures the intrinsic negative correlation between electrons and holes. To overcome the difficulty caused by the degenerate effect at the sonic boundary, we introduce two different weight functions to handle the singularity near both sides of endpoints separately. The approaches adopted to prove the structural stability include the local singularity analysis, the monotonicity method, the continuation argument and the squeezing skill. Several numerical simulations are performed which support our theoretical results. [ABSTRACT FROM AUTHOR]

This paper aims to study a semilinear transfer equation with a delay term defined over a non-cylindrical domain. We prove the existence and regularity of weak solutions as well as the existence, regularity and finiteness of the fractal dimension of pullback attractors on tempered universes that depend on a non-increasing function. We address the problem of estimating the fractal dimension of pullback attractors, under current techniques, which consists of readjusting the proof presented in [9, Lemma 1.3] and extending it to families of normed spaces parameterized in time, see Theorem 4.11. [ABSTRACT FROM AUTHOR]

In this paper we classify all monic, quartic, polynomials d (x) ∈ Z [ x ] for which the Pell equation f (x) 2 − d (x) g (x) 2 = 1 has a non-trivial solution with f (x) , g (x) ∈ Z [ x ]. [ABSTRACT FROM AUTHOR]

This present paper is dedicated to the study of the Cauchy problem of the two-dimensional Euler-Boussinesq-Bénard equations which couple the incompressible Euler equations for the velocity and a transport equation with critical dissipation for the temperature. We show that there is a global unique solution to this model with Yudovich's type data. This settles the global regularity problem which was remarked by Wu and Xue (2012) [44]. [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic behavior of the solutions of the p -Laplacian equation with mixed homogeneous Neumann-Dirichlet boundary conditions. It is posed in a two-dimensional rough thin domain with two different composites periodically distributed. Each composite has its own periodicity and roughness order. Here, we obtain distinct homogenized limit equations which will depend on the relationship among the roughness and thickness orders of each one. [ABSTRACT FROM AUTHOR]

In the paper, we revisit the uniform continuity properties of the data-to-solution map of the Camassa–Holm equation on the real-line case. We show that the data-to-solution map of the Camassa–Holm equation is not uniformly continuous on the initial data in Besov spaces B p , r s (R) with s > 1 2 and 1 ≤ p , r < ∞ , which improves the previous works Himonas et al. (2007) [23] , Li et al. (2020) [32] and Li et al. (2021) [31]. Furthermore, we present a strengthening of our previous work in Li et al. (2020) [32] and prove that the data-to-solution map for the Camassa–Holm equation is nowhere uniformly continuous in B p , r s (R) with s > max ⁡ { 1 + 1 / p , 3 / 2 } and (p , r) ∈ [ 1 , ∞ ] × [ 1 , ∞). The method applies also to the b-family of equations which contain the Camassa–Holm and Degasperis–Procesi equations. [ABSTRACT FROM AUTHOR]

In this paper we begin the study of set-theoretic type solution of the braid equation. Our theory includes set-theoretical solutions as basic examples. More precisely, the linear solution associated to a set-theoretic solution on a set X can be regarded as coming from the coalgebra kX , where k is a field and the elements of X are grouplike. We introduce and study a broader class of linear solutions associated in a similar way to more general coalgebras. We show that the relationships between set-theoretical solutions, q -cycle sets, q -braces, skew-braces, matched pairs of groups and invertible 1-cocycles remain valid in our setting. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the noncompact prescribed Chern scalar curvature problem which reduces to solve a Kazdan-Warner type equation on noncompact non-Kähler manifolds. By introducing an analytic condition on noncompact manifolds, we establish related existence results. As its another application, we further give a new geometric proof of a classical multiplicity theorem of Ni (1982) [28]. [ABSTRACT FROM AUTHOR]

The extended cubic Camassa-Holm equation can be derived as asymptotic model from shallow-water approximation to the 2D incompressible Euler equations. This model encompasses both quadratic and cubic nonlinearities and the solution of the extended cubic Camassa-Holm equation possible development of singularities in finite time. This paper is devoted to the continuation of solutions to the extended cubic Camassa-Holm equation beyond wave breaking, and the global existence and uniqueness of the Hölder continuous energy conservative solutions for the Cauchy problem of the extended cubic Camassa-Holm type equation are investigated. An equivalent semilinear system was first introduced by a new set of independent and dependent variables, which can resolve all singularities due to possible wave breaking. Returning to the original variables, depending continuously on the initial data, the existence of the global conservative weak solutions can be obtained. Moreover, by analyzing the evolution of the quantities u and v = 2 arctan ⁡ u x along each characteristic, the uniqueness of the global conservative solutions for the Cauchy problem with general initial data u 0 ∈ H 1 (R) was proved. [ABSTRACT FROM AUTHOR]

In this paper, we use a natural iteration technique to prove the existence of solutions to nonlinear Dirichlet problems. Among the examples included is the prescribed mean curvature equation. The nature of the technique allows applications to unbounded domains, as we show with domains of the form R n − 1 × (− d / 2 , d / 2). [ABSTRACT FROM AUTHOR]

In this paper, we investigate several properties of the positive solution to the nonlocal periodic-parabolic problem { u t = d [ ∫ Ω J (x − y) u (y , t) d y − u (x , t) ] + λ u − b (x) q (t) u p in Ω × [ 0 , T ] , u (x , 0) = u (x , T) in Ω , where Ω is a bounded smooth domain in R N (N ≥ 1), λ > 0 is a parameter, d > 0 and p > 1 are constants, q (t) is a positive and T -periodic continuous function, b (x) is a nonnegative continuous function, and is strictly positive only outside a domain Ω ¯ 0 ⊂ Ω. So the equation has a degeneracy over Ω 0. In such a case, it is known that there is a unique positive solution u λ if and only if λ is between the principle eigenvalues λ p (Ω) and λ p (Ω 0) of the associated nonlocal diffusion operator (over Ω and Ω 0 , respectively). We show that u λ is stable under perturbations of λ and b (x) in the equation, and when λ increases to λ p (Ω 0) , u λ converges to ∞ uniformly over Ω ¯ × [ 0 , T ] but a clear pattern is exhibited by u λ / ‖ u λ ‖ ∞ : u λ (x , t) ‖ u λ ‖ ∞ → { 0 for (x , t) ∈ (Ω ¯ ∖ Ω ¯ 0) × [ 0 , T ] , ϕ (x) for (x , t) ∈ Ω ¯ 0 × [ 0 , T ] , where ϕ (x) is the normalised positive eigenfunction associated to λ p (Ω 0). The proof relies on a general regularity result for the solutions of these types of nonlocal equations, which is of independent interest. [ABSTRACT FROM AUTHOR]

In this paper, we study the propagation dynamics for a class of integro-difference equations with a shifting habitat. We first use the moving coordinates to transform such an equation to an integro-difference equation with a new kernel function containing the shifting speed c. In two directions of the spatial variable, the resulting equation has two limiting equations with spatial translation invariance. Under the hypothesis that each of these two limiting equations has both leftward and rightward spreading speeds, we establish the spreading properties of solutions and the existence of nontrivial forced waves for the original equation by appealing to the abstract theory of nonmonotone semiflows with asymptotic translation invariance. Further, we prove the stability and uniqueness of forced waves under appropriate conditions. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the finite time blow-up for the high-order Camassa-Holm-Fokas-Olver-Rosenau-Qiao equations, which is a generalization of the Camassa-Holm equation and the Fokas-Olver-Rosenau-Qiao equation. We explore how high-order nonlinearities affect the dispersive dynamics and breakdown mechanism of solutions. Firstly, we established the local well-posedness for the Cauchy problem in the framework of Besov spaces. Then, we derive the precise blow-up mechanism for strong solutions by means of the transport equation theory. Finally, a sufficient condition on initial data that leads to the finite time blow-up of the second-order derivative of the solutions is described in detail. [ABSTRACT FROM AUTHOR]

This paper is concerned with the stationary solutions to the one-dimensional full (non-isentropic) compressible Navier-Stokes-Korteweg equations with far field states and boundary data in the half line. When the fluids enter into and blow out the region through the boundary, respectively, the unique existence of the stationary solutions to the one-dimensional full compressible Navier-Stokes-Korteweg equations in the half line is shown provided that the boundary strength is small enough. Moreover, we also give the spatial decay rates of the stationary solutions. The main ingredient of the proof is the manifold theory and the center manifold theorem that take the accurate analysis of the cubic characteristic equations. [ABSTRACT FROM AUTHOR]

In this paper we consider determining a minimal surface embedded in a Riemannian manifold Σ × R. We show that if Σ is a two dimensional Riemannian manifold with boundary, then the knowledge of the associated Dirichlet-to-Neumann map for the minimal surface equation determine Σ up to an isometry. [ABSTRACT FROM AUTHOR]

This paper introduces a new notion of dispersion for kinetic equations solely based on the conservation laws and independent of the specific type of interactions. We present new a-priori estimates for kinetic PDEs and improve the Bony-type functional approach. [ABSTRACT FROM AUTHOR]

In this paper, we consider a kind of singular integrals T λ f (x) = p.v. ∫ R n Ω (y) | y | n − λ f (x − y) d y for any f ∈ L q (R n) , 1 < q < ∞ and 0 < λ < n , which appear in the generalized 2D dissipative quasi-geostrophic (QG) equation (0.1) ∂ t θ + u ⋅ ∇ θ + κ Λ 2 β θ = 0 , (x , t) ∈ R 2 × R + , κ > 0 , where u = − ∇ ⊥ Λ − 2 + 2 α θ , α ∈ [ 0 , 1 2 ] and β ∈ (0 , 1 ]. Firstly, we give a uniform sparse domination for this kind of singular integral operators. Secondly, we obtain the uniform quantitative weighted bounds for the operator T λ with rough kernel. As an application, we obtain the uniform quantitative weighted bounds for the commutator [ b , T λ ] with rough kernel and study solutions to the generalized 2D dissipative quasi-geostrophic (QG) equation. [ABSTRACT FROM AUTHOR]

This paper proposes an optimal control problem for a parabolic equation with a nonlocal nonlinearity. The system is described by a parabolic equation involving a nonlinear term that depends on the solution and its integral over the domain. We prove the existence and uniqueness of the solution to the system and the boundedness of the solution. Regularity results for the control-to-state operator, the cost functional and the adjoint state are also proved. We show the existence of optimal solutions and derive first-order necessary optimality conditions. In addition, second-order necessary and sufficient conditions for optimality are established. [ABSTRACT FROM AUTHOR]

In this paper, we prove that if a solution to the Muskat problem with different densities and the same viscosity is sufficiently smooth, the solution is analytic in a region that degenerates at the turnover points, provided some additional conditions are satisfied. This paper studies the analyticity of the solution near turnover points, complementing the result in [38]. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following indefinite dual fractional parabolic equation involving the Marchaud fractional time derivative ∂ t α u (x , t) + (− Δ) s u (x , t) = a (x) f (u (x , t)) in R n × R , where α , s ∈ (0 , 1) , and the functions a and f are nondecreasing. We prove that there is no positive bounded solutions. To this end, we first show that all positive bounded solutions u (⋅ , t) must be strictly monotone increasing along the direction determined by a (x). Then by mollifying the first eigenfunction for fractional Laplacian (− Δ) s and constructing an appropriate subsolution for the Marchaud fractional operator ∂ t α − 1 , we derive a contradiction and thus obtain the non-existence of solutions. To overcome the challenges caused by the dual non-locality of the operator ∂ t α + (− Δ) s , we introduce several new ideas and novel techniques. These novel approaches are not only applicable to the specific problem at hand but can also be extended to address various other fractional problems, be they elliptic or parabolic, including those featuring dual nonlocalities associated with the Marchaud time derivatives. [ABSTRACT FROM AUTHOR]

In this paper, we systematically study the existence, asymptotic behaviors, uniqueness, and nonlinear orbital stability of traveling-wave solutions with small propagation speeds for the generalized surface quasi-geostrophic (gSQG) equation. Firstly we obtain the existence of a new family of global solutions via the variational method. Secondly we show the uniqueness of maximizers under our variational setting. Thirdly by using the variational framework, the uniqueness of maximizers and a concentration-compactness principle we establish some stability theorems. Moreover, after a suitable transformation, these solutions constitute the desingularization of traveling point vortex pairs. [ABSTRACT FROM AUTHOR]

In this paper we present a new identity to associate the conservation laws of stress–energy tensor with the field equations in Yang–Mills theory. The Lorentz force is included with a consistent formulation as in Maxwell theory. [ABSTRACT FROM AUTHOR]

Exponential/extended Runge–Kutta–Nyström (ERKN) methods have been validated by numerous theoretical and numerical results to be more efficient and suitable than classical Runge–Kutta–Nyström (RKN) methods in dealing with highly oscillatory Hamiltonian systems. Once numerical solutions are required to achieve a prescribed local error tolerance for practical problems, the embedded ERKN pairs with adaptive stepsize are needed. Given the higher efficiency of high-order methods than low-order methods, this paper focuses on high-order embedded ERKN pairs. Using the particular mapping from RKN methods into ERKN methods, we establish an approach to constructing high-order embedded ERKN pairs. By diagonalizing the frequency matrix, an implementation algorithm with less calculation amount than the direct calculation procedure is proposed for high-dimensional problems. In addition, the dispersion and dissipation of the ERKN pairs ERKN4(3) and ERKN8(6) are analyzed. Furthermore, the application of ERKN pairs to an important highly oscillatory problem, i.e., the wave equation prescribed with different boundary conditions is discussed. For the periodic boundary condition, a special implementation algorithm incorporated with FFT techniques is presented, which decreases the calculation amount in one time step from O (N 2) to O (N log ⁡ N). Finally, numerical results with the FPU problem, the Klein–Gordon equation in the nonrelativistic limit regime, and a two-dimensional wave equation demonstrate the superiority of ERKN pairs over classical RKN pairs. • An effective approach is presented to establish high-order embedded ERKN pairs. • A special implementation algorithm by diagonalizing the frequency matrix is proposed to reduce the calculation amount. • The application of the embedded ERKN pairs to wave equations prescribed with different boundary conditions is discussed. [ABSTRACT FROM AUTHOR]

An explicit formula for the number of solutions of the equation in the title is given when a certain condition, depending only on the exponent and the characteristic of the field, holds. This formula improves the one given by the authors in an earlier paper. [ABSTRACT FROM AUTHOR]

A (projective, geometrically irreducible, non-singular) curve X defined over a finite field F q 2 is maximal if the number N q 2 of its F q 2 -rational points attains the Hasse-Weil upper bound, that is N q 2 = q 2 + 2 g q + 1 where g is the genus of X. An important question, also motivated by applications to algebraic-geometry codes, is to find explicit equations for maximal curves. For a few curves which are Galois covered of the Hermitian curve, this has been done so far ad hoc, in particular in the cases where the Galois group has prime order. In this paper we obtain explicit equations of all Galois covers of the Hermitian curve with Galois group of order p 2 where p is the characteristic of F q 2 . Doing so we also determine the F q 2 -isomorphism classes of such curves and describe their full F q 2 -automorphism groups. [ABSTRACT FROM AUTHOR]

In this paper, we study the prescribed Q -curvature flow equation on a arbitrary even dimensional closed Riemannian manifold (M , g) , which was introduced by S. Brendle in [3] , where he proved the flow exists for long time and converges at infinity if the GJMS operator is weakly positive with trivial kernel and ∫ M Q d μ < (n − 1) ! Vol (S n). In this paper we study the critical case that ∫ M Q d μ = (n − 1) ! Vol (S n) , we will prove the convergence of the flow under some geometric hypothesis. In particular, this gives a new proof of Li-Li-Liu's existence result in [21] in dimension 4 and extend the work of Li-Zhu [22] in dimension 2 to general even dimensions. In the proof, we give a explicit expression of the limit of the corresponding energy functional when the blow up occurs. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following superlinear elliptic problem (P) { − Δ u = λ | u | p − 2 u + f (x , u) , in Ω , u = 0 , on ∂ Ω , where λ > 0 and 2 < p < 2 + δ for some δ > 0 small. The nonlinearity f satisfies the Ambrosetti-Rabinowitz condition and other appropriate hypotheses such that u = 0 is a local minimizer of the associated energy functional of equation (P). Our main novelties are threefold. Firstly, using the properties of Gromoll-Meyer pairs in Morse theory, we prove that equation (P) has at least one nontrivial solution close to 0. Moreover, four nontrivial solutions are obtained with assumptions on f at infinity, and none of these solutions depends on the gaps of consecutive eigenvalues of operator −Δ. Therefore, our results differ significantly from those of the paper by Li and Li (2016) [16]. Secondly, under the assumptions of the paper above, we can obtain the existence of a fifth nontrivial solution of equation (P) for λ = 1. Finally, by using minimax methods and Morse theory, we also obtain the existence of five nontrivial solutions of equation (P) based on the relationship between parameter λ and eigenvalues of operator −Δ. [ABSTRACT FROM AUTHOR]

This paper studies the existence and uniqueness problem for the generalized Benjamin-Bona-Mahony (gBBM) equation with quasi-periodic initial data on the real line. We obtain an existence and uniqueness result in the classical sense with arbitrary time horizon under the assumption of polynomially decaying initial Fourier data using the combinatorial analysis method developed in earlier papers by Christ [6] , Damanik-Goldstein [11] , and the present authors [12]. Our result is valid for exponentially decaying initial Fourier data and hence can be viewed as a Cauchy-Kovalevskaya theorem in the space variable for the gBBM equation with quasi-periodic initial data. [ABSTRACT FROM AUTHOR]

In this paper, we propose a unified framework for studying the Cahn–Hilliard equation with two distinct types of dynamic boundary conditions, namely, the Allen–Cahn and Cahn–Hilliard types. Using this unified framework, we develop a linear, second-order, and energy-stable scheme based on the multiple scalar auxiliary variables (MSAV) approach. We design efficient and decoupling algorithms for solving the corresponding linear system in which the unknown variables are intricately coupled both in the bulk and at the boundary. Several numerical experiments are shown to validate the proposed scheme, and to investigate the effect of different dynamical boundary conditions on the dynamics of phase evolution under different scenarios. [ABSTRACT FROM AUTHOR]

This paper describes in terms of Artin-Schreier equations field extensions whose Galois group is isomorphic to any of the four non-cyclic groups of order p 3 or the ten non-Abelian groups of order p 4 , p an odd prime, over a field of characteristic p. [ABSTRACT FROM AUTHOR]

In this paper we prove the orthonormal Strichartz estimates for the higher order and fractional Schrödinger, wave, Klein-Gordon and Dirac equations with potentials. As in the case of the Schrödinger operator, the proofs are based on the smooth perturbation theory by T. Kato. However, for the Klein-Gordon and Dirac equations, we also use a method of the microlocal analysis in order to prove the estimates for wider range of admissible pairs. As applications we prove the global existence of a solution to the higher order or fractional Hartree equation with potentials which describes the dynamics of infinitely many particles. We also give a local existence result for the semi-relativistic Hartree equation with electromagnetic potentials. As another application, the refined Strichartz estimates are proved for higher order and fractional Schrödinger, wave and Klein-Gordon equations. [ABSTRACT FROM AUTHOR]

The extended Fisher–Kolmogorov (EFK) equation has been used to describe some phenomena in physical, material and biological systems. In this paper, we propose a full-rank splitting scheme and a rank-adaptive splitting approach for this equation. We first use a finite difference method to approximate the space derivatives. Then, the resulting semi-discrete system is split into two stiff linear parts and a nonstiff nonlinear part. This leads to our full-rank splitting scheme. The convergence of the proposed scheme is proved rigorously. Based on the frame of the full-rank splitting scheme, we design a rank-adaptive splitting approach for obtaining a low-rank solution of the EFK equation. Numerical examples show that our methods are robust and accurate. They can also preserve the energy dissipation. • The EFK equation is split into three subproblems, then a full-rank splitting scheme is established. The convergence of this scheme is analyzed. • A rank-adaptive low-rank approach is proposed for the EFK equation. To the best of our knowledge, this is new in the literature for the equation. • Numerical examples show that our methods are robust and accurate. They can also preserve energy dissipation. [ABSTRACT FROM AUTHOR]

Systems exhibiting Saddle-Node (SN) bifurcations are often characterized by drastic amplitude and phase jumps, representing a crucial state in engineering scenarios. The accurate and efficient prediction of SN points is fundamental for the comprehensive understanding and control of dynamical systems. This paper derives an equation for locating SN points based on the dimensionless governing equation for the in-plane primary resonance of a suspended cable. It reveals that the SN points for the cable are influenced by three key parameters: the cable's effective nonlinearity Γ e m , the excitation parameter F 2 , and damping ratio. Importantly, when the cable is subjected solely to horizontal end excitation, the product of Γ e m and F 2 emerges as a new parameter, Λ e m. The effects of parameter Λ e m (Γ e m), damping, axial, and vertical excitation amplitudes on SN points are investigated. Findings indicate that these key parameters more significantly affect the SN1 (the one near peck point) than SN2, and slight variations in Λ e m (Γ e m) or vertical excitation amplitude can lead to substantial alterations in SN1. The effect of Λ e m (Γ e m) on SN1 is asymmetric, with the values of σ and a being significantly higher when Λ e m (Γ e m) is positive than when negative. The computational results of the SN position equation closely align with experimental observations and the literature, demonstrating good computational efficiency. [ABSTRACT FROM AUTHOR]

Latitude on the choice of initialisation is a shared feature between one-step extended state-space and multi-step methods. The paper focuses on lattice Boltzmann schemes, which can be interpreted as examples of both previous categories of numerical schemes. We propose a modified equation analysis of the initialisation schemes for lattice Boltzmann methods, determined by the choice of initial data. These modified equations provide guidelines to devise and analyze the initialisation in terms of order of consistency with respect to the target Cauchy problem and time smoothness of the numerical solution. In detail, the larger the number of matched terms between modified equations for initialisation and bulk methods, the smoother the obtained numerical solution. This is particularly manifest for numerical dissipation. Starting from the constraints to achieve time smoothness, which can quickly become prohibitive for they have to take the parasitic modes into consideration, we explain how the distinct lack of observability for certain lattice Boltzmann schemes—seen as dynamical systems on a commutative ring—can yield rather simple conditions and be easily studied as far as their initialisation is concerned. This comes from the reduced number of initialisation schemes at the fully discrete level. These theoretical results are successfully assessed on several lattice Boltzmann methods. • We study the initialization of general lattice Boltzmann methods introducing an ad hoc modified equation analysis. • We find the constraints to obtain consistent initialization schemes, preserving second-order for the overall method. • We finely describe initial boundary layers due to dissipation mismatches between bulk and initialization schemes. • We introduce the observability of a lattice Boltzmann scheme, characterizing those with easily-mastered initializations. • We test the introduced analytical tools and their effectiveness through several—very conclusive—numerical experiments. [ABSTRACT FROM AUTHOR]

Efficient numerical schemes for solving the Poisson equation with jump conditions are of great interest for a variety of problems, including the modeling of electroporation phenomena and filamentary discharges. In this paper, we propose a modification to a finite volume scheme, namely the discrete dual finite volume method, in order to account for jump conditions with surface charges, i.e. with a source term. Our numerical tests demonstrate second-order convergence even with highly distorted meshes. We then apply the proposed method to model electroporation phenomena in biological cells by proposing a model that considers the thickness of the cell membrane as a separate domain, which differs from the literature. We show the advantages of the proposed method in this context through numerical experiments. • The discrete dual finite volume scheme is extended to solve the Poisson equation with jump conditions and surface charges. • The method is shown to exhibit a second-order convergence through canonical numerical tests. • The method is applied to the electroporation phenomena, where accurate modeling of the potential at the membrane is obtained. • Numerical experiments on the stationary and non-stationary case are provided. [ABSTRACT FROM AUTHOR]

Parareal is a well-known parallel-in-time algorithm that combines a coarse and fine propagator within a parallel iteration. It allows for large-scale parallelism that leads to significantly reduced computational time compared to serial time-stepping methods. However, like many parallel-in-time methods it can fail to converge when applied to non-diffusive equations such as hyperbolic systems or dispersive nonlinear wave equations. This paper explores the use of exponential integrators within the Parareal iteration. Exponential integrators are particularly interesting candidates for Parareal because of their ability to resolve fast-moving waves, even at the large stepsizes used by coarse propagators. This work begins with an introduction to exponential Parareal integrators followed by several motivating numerical experiments involving the nonlinear Schrödinger equation. These experiments are then analyzed using linear analysis that approximates the stability and convergence properties of the exponential Parareal iteration on nonlinear problems. The paper concludes with two additional numerical experiments involving the dispersive Kadomtsev-Petviashvili equation and the hyperbolic Vlasov-Poisson equation. These experiments demonstrate that exponential Parareal methods offer improved time-to-solution compared to serial exponential integrators when solving certain non-diffusive equations. • Exponential Parareal notably reduces time-to-solution for non-diffusive equations. • Linear analysis accurately predicts Parareal performance on nonlinear problems. • Repartitioning is essential for stabilizing exponential integrators within Parareal. [ABSTRACT FROM AUTHOR]

The paper aims to study the geometry and physics of the Raychaudhuri equation (RE) in the background of a homogeneous and anisotropic space–time described by Kantowski–Sachs (KS) metric. Role of anisotropy/shear in the context of convergence and possible avoidance of singularity has been analyzed subject to a physically motivated constraint. Moreover, using a suitable transformation the first order RE has been converted to a second order differential equation analogous to a Harmonic Oscillator and criterion for convergence has been shown to be associated with the time varying frequency of the Oscillator. Finally, the paper points out a geometric and physical notion of anisotropy along with their corresponding behavior towards convergence. [ABSTRACT FROM AUTHOR]