Analysis and Mathematical Physics最新文献

In this paper, we give a short summary of fractal calculus. We introduce the concept of fractal variation of calculus and derive the general form of the fractal Euler equation, along with an alternate form. We explore applications of the fractal Euler equation, including the optical fractal path near the event horizon of a black hole and determining the shortest distance in fractal space. Examples and illustrative plots are provided to demonstrate the detailed behavior of these equations and their practical implications.

Let (s,bin mathbb {R}). This article is devoted to establishing the Fourier-analytical characterization of the pointwise multiplier space (M(B^{s,b}_{p,q}(mathbb {R}^n))) for the logarithmic Besov space (B^{s,b}_{p,q}(mathbb {R}^n)) in the endpoint cases, that is, (p,qin {1,infty }). The authors first obtain such a characterization for the cases where (p=1) and (q=infty ) and where (p=infty ) and (q=1). Applying this, the authors then establish the duality formula (M(B^{s,b}_{p,q}(mathbb {R}^n))=M(B^{-s,-b}_{p',q'}(mathbb {R}^n)),) where (s,bin mathbb {R}), (p,qin [1,infty ]), and (p') and (q') are respectively the conjugate indices of p and q. This duality principle is further applied to establish the Fourier-analytical characterization of (M(B^{s,b}_{p,q}(mathbb {R}^n))) in the cases where (p=infty =q) and where (p=1=q).

This work derives novel exact solutions of the Biswas–Milovic nonlinear Schrödinger equation by employing the innovative Extended Modified Auxiliary Equation Mapping Technique, augmented with enhanced sensitivity analysis. The resulting bright, kink, anti-kink, and periodic soliton solutions provide deep insights into the complex dynamics of nonlinear wave propagation. To unravel the intricate behaviors of these solitons, we analyze phase trajectories, density distributions, and streamlines, with a particular focus on their sensitivity to initial conditions. Stability is rigorously evaluated through a Hamiltonian formalism, ensuring both analytical rigor and structural robustness. Collectively, these findings enrich the theoretical understanding of soliton dynamics and open new pathways for practical applications in advanced physical systems.

Fukaya and Oh studied the correspondence between pseudoholomorphic disks in (T^{*}M) which are bounded by Lagrangian sections ({L_{i}^{epsilon }}) and gradient trees in M which consist of gradient curves of ({f_{i}-f_{j}}). Here, (L_{i}^{epsilon }) is defined by (L_{i}^{epsilon }=) graph((epsilon df_{i})). They constructed approximate pseudoholomorphic disks in the case (epsilon >0) is sufficiently small. When (M=mathbb {R}) and Lagrangian sections are affine, pseudoholomorphic disks (w_{epsilon }) can be constructed explicitly. In this paper, we show that pseudoholomorphic disks (w_{epsilon }) converges to the gradient tree in the limit (epsilon rightarrow +0) when the number of Lagrangian sections is three and four.

In this paper, we define generalized monotonicity concepts related to equilibrium problems generated by trifunctions. We then study the existence of solutions to mixed equilibrium problems described as the sum of a maximal monotone trifunction and a pseudomonotone trifunction in Brézis sense. The main tools for this study are a Thikonov regularization procedure with respect to the generalized duality mapping and recession analysis adapted to trifunctions. An application consists in an existence result for a noncoercive hemivariational inequality.

The aim of this paper is to investigate the right-sided quaternionic free metaplectic transformation (QFMT) and its associated uncertainty principles (UPs) for (mathbb {R}^{2d})-dimensional quaternionic-valued signals. First, we establish the fundamental mathematical properties of the QFMT, including partial derivatives, the inversion formula, Parseval’s theorem, and the Hausdorff–Young inequality. Next, we establish various UPs within this framework, such as the Rènyi and Shannon entropy UPs and Donoho–Stark’s UP in terms of concentration. Finally, we derive (L^a)-bandlimited variant of the Donoho–Stark UP in the QFMT domain.

Conformal geodesics form an invariantly defined family of unparametrized curves in a conformal manifold generalizing unparametrized geodesics/paths of projective connections. The equation describing them is of third order, and it was an open problem whether they are given by an Euler–Lagrange equation. In dimension 3 (the simplest, but most important from the viewpoint of physical applications) we demonstrate that the equation for unparametrized conformal geodesics is variational.

We consider the Schrödinger operator on the quantum graph whose edges connect the points of ({{mathbb {Z}}}). The numbers of the edges connecting two consecutive points n and (n+1) are read along the orbits of a shift of finite type. We prove that the Lyapunov exponent is potitive for energies E that do not belong to a discrete subset of ([0,infty )). The number of points E of this subset in ([(pi (j-1))^2, (pi j)^2]) is the same for all (jin {{mathbb {N}}}).

In this paper, we investigate the existence of multiple weak solutions for a Schrödinger-Kirchhoff type elliptic system involving nonlocal ((alpha _1(cdot ), ldots , alpha _N(cdot )))-Laplacian operator. The system is modeled as follows:

We apply the three critical points theorem to establish sufficient conditions for the existence of at least three weak solutions under appropriate assumptions on the system’s parameters and nonlinearity terms. This work extends the analysis of elliptic systems involving variable exponent spaces and nonlocal operators, offering novel insights into their mathematical structure and solution properties.