Article
Mathematics
Yongming Luo
Summary: We consider the problem of large data scattering for the 2D and 3D cubic-quintic nonlinear Schrodinger equation in the focusing-focusing regime. In the 2D case where the cubic nonlinearity is L2-critical, we establish a new type of scattering criterion uniquely determined by the mass of the initial data, which differs from the classical setting based on the Lyapunov functional. Finally, we formulate a solely mass-determining scattering threshold for the 3D cubic-quintic nonlinear Schrodinger equation in the focusing-focusing regime.
JOURNAL OF FUNCTIONAL ANALYSIS
(2022)
Article
Mathematics, Applied
Simon Becker, Jonathan Sewell, Euan Tebbutt
Summary: This study examines the computability of global solutions to linear Schrodinger equations with magnetic fields and the Hartree equation. It demonstrates that the solution can be globally computed with error control on the entire space if there exist a priori decay estimates in generalized Sobolev norms on the initial state. By employing weighted Sobolev norm estimates, the solution can be computed with uniform computational runtime in relation to initial states and potentials.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics
Noriyoshi Fukaya, Vladimir Georgiev, Masahiro Ikeda
Summary: We study the existence and stability properties of ground-state standing waves for a two-dimensional nonlinear Schrodinger equation with a point interaction and a focusing power nonlinearity. We prove that the standing wave is stable if its frequency is close to a negative eigenvalue and discuss the stability and instability cases for sufficiently large frequencies.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2022)
Article
Mathematics
Yongming Luo
Summary: In this study, the focusing intercritical NLS on a semiperiodic waveguide manifold is investigated. The author constructs a sharp threshold using the semivirial vanishing theory, which characterizes the bifurcation of scattering and blow-up solutions. However, due to the non-Lipschitz property of the derivative of the nonlinear potential in higher dimensions and the anisotropic nature of the domain, the previous proof cannot be extended. By applying an adapted version of the IMDM estimates, the large data scattering result continues to hold for d >= 5.
MATHEMATISCHE ANNALEN
(2023)
Article
Engineering, Electrical & Electronic
Kalim U. Tariq, Mustafa Inc, S. M. Raza Kazmi, Reem K. Alhefthi
Summary: In the theory of optical fibres, the nonlinear Schrodinger equation is a significant physical model for understanding optical soliton dynamics. The propagation of optical solitons in nonlinear optical fibres is of contemporary interest due to its applications in ultrafast signal routing systems and short light pulses in communications. The objective of this investigation is to develop soliton solutions for the resonance model using modern analytical techniques. Various periodic and singular bell shaped soliton solutions are obtained, and their stability and modulation instabilities are analyzed. The findings show impressive and competent avenue for obtaining advanced moving pulse shapes in recent years.
OPTICAL AND QUANTUM ELECTRONICS
(2023)
Article
Mathematics
Binhua Feng, Shihui Zhu
Summary: In this paper, a comprehensive study is made for the orbital stability of standing waves for the fractional Schrödinger equation with combined power-type nonlinearities. Different conditions are provided for the existence and stability of the standing waves, with a focus on their strong instability under certain parameters and criteria. This research contributes to understanding the behavior of standing waves in the fractional Schrödinger equation and provides insights into their stability properties.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematical & Computational Biology
Min Gong, Hui Jian, Meixia Cai
Summary: In this article, the global existence and stability issues of the nonlinear Schrödinger equation with partial confinement are considered. By establishing new cross-invariant manifolds and variational problems, a new sharp criterion for global existence is derived in different cases. The existence of orbitally stable standing waves is then obtained using the profile decomposition technique. This work extends and complements previous results.
MATHEMATICAL BIOSCIENCES AND ENGINEERING
(2023)
Article
Mathematics, Interdisciplinary Applications
Xiuye Liu, Jianhua Zeng
Summary: Dark solitons, localized nonlinear waves, have attracted significant attention due to their rich formation and dynamics in various fields. In this study, a purely nonlinear strategy is used to stabilize dark soliton stripes by introducing a quasi-one-dimensional Gaussian-like trap and combining it with an external linear harmonic trap. The results demonstrate complete stabilization of dark soliton stripes and a significant reduction in modulational instability.
CHAOS SOLITONS & FRACTALS
(2022)
Article
Astronomy & Astrophysics
Filip Ficek
Summary: The paper investigates solutions of the Schrodinger-Newton-Hooke (SNH) system in energy supercritical spatial dimensions, finding the existence of nonlinear ground and excited states. By studying spherically symmetric stationary solutions, it is shown that the frequency of the ground state exhibits different behaviors in various spatial dimensions.
Article
Physics, Fluids & Plasmas
Liwen Zou, XinHang Luo, Delu Zeng, Liming Ling, Li-Chen Zhao
Summary: This study addresses the problem of weak Gaussian perturbations triggering rogue waves from the perspective of computer vision using deep neural networks. It is found that these rogue waves have similar computer vision patterns, but no previous results have proven the similarity of these patterns for different perturbations due to the difficulty of automatic measurement. To solve this problem, the researchers propose a rogue wave detection network model and design a corresponding dataset. In detection experiments, the model achieves a high average precision on the test set. Additionally, a metric called the density of rogue wave units is derived to characterize the evolution of Gaussian perturbations and obtain statistical results.
Article
Astronomy & Astrophysics
Jonas P. Pereira, Michal Bejger, J. Leszek Zdunik, Pawel Haensel
Summary: The internal composition of neutron stars is still uncertain as only indirect inferences can be made about their interiors. This study assumes a hypothetical future scenario to estimate the observational accuracy in differentiating phase transitions in neutron stars. The results suggest that future gravitational wave detectors and electromagnetic missions may help assess some aspects of phase transitions in neutron stars.
Article
Engineering, Electrical & Electronic
Fazal Badshah, Kalim U. Tariq, Muhammad Aslam, Wen-Xiu Ma, S. Mohsan Raza Kazmi
Summary: In this article, the application of the nonlinear Schrodinger equation (NLSE) in optical fiber theory is investigated, exploring the evolution of perturbations in both stable and unstable media. Various analytical methods are utilized to obtain solutions for bright, dark, singular, optical, bell-shaped, and periodic solitons, and the stability of the results is validated. The modulation instability of the governing model is also studied to explain the behavior patterns caused by irregular refractive indices in an optical fiber.
OPTICAL AND QUANTUM ELECTRONICS
(2023)
Article
Mathematics, Applied
O. Y. Imanuvilov, M. Yamamoto
Summary: This article focuses on the uniqueness and stability of an inverse source problem for determining a spatially varying factor f(x) of a source term in the Schrödinger equation with time independent coefficients. By proving the logarithmic conditional stability estimate for the Cauchy problem with zero Dirichlet boundary conditions and the unique continuation from a small part of the lateral boundary, the results for the uniqueness and stability of the varying factor f(x) are established.
Article
Mathematics, Applied
Fei Justina Liu, Tai-Peng Tsai, Ian Zwiers
Summary: This note examines the existence and stability of standing waves for one dimensional nonlinear Schrodinger equations with nonlinearities that are the sum of three powers, through analytical and numerical methods. Special attention is given to the curves of non-existence and stability change on the parameter planes.
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS
(2021)
Article
Mathematics, Applied
Jeongho Kim, Bora Moon
Summary: We establish rigorous and quantified hydrodynamic limits of the Schrodinger-type equations, including the Schrodinger equation, Hartree equation, and Chern-Simons-Schrodinger equations, when the Planck constant is negligible. We focus on the case without Gross-Pitaevskii type self-interacting potential, and combine the modulated energy with the bounded Lipschitz distance between densities to overcome difficulties in deriving hydrodynamic limits.
JOURNAL OF EVOLUTION EQUATIONS
(2023)
Article
Mathematics, Applied
Torsten Lindstrom
Summary: This paper aims to analyze the mechanism for the interplay of deterministic and stochastic models in contagious diseases. Deterministic models usually predict global stability, while stochastic models exhibit oscillatory patterns. The study found that evolution maximizes the infectiousness of diseases and discussed the relationship between herd immunity concept and vaccination programs.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2024)
Article
Mathematics, Applied
Dong Deng, Hongxun Wei
Summary: This paper investigates the existence and nonexistence of time-periodic traveling waves for a diffusive influenza model with treatment and seasonality. By utilizing the next generation operator theory and Schauder's fixed point theorem, the conditions for the existence of time-periodic traveling wave solutions are obtained, along with the proof of nonexistence in certain cases and exponential decay for waves with critical speed.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2024)
Article
Mathematics, Applied
Xuan Ma, Yating Wang
Summary: In this paper, the dynamics of a rarefied gas in a finite channel is studied, specifically focusing on the phenomenon of Couette flow. The authors demonstrate that the unsteady Couette flow for the Boltzmann equation converges to a 1D steady state and derive the exponential time decay rate. The analysis holds for all hard potentials.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2024)
Article
Mathematics, Applied
Meng Zhao
Summary: In this paper, a reaction-diffusion waterborne pathogen model with free boundary is studied. The existence of a unique global solution is proved, and the longtime behavior is analyzed through a spreading-vanishing dichotomy. Sharp criteria for spreading and vanishing are obtained, which differs from the previous results by Zhou et al. (2018) stating that the epidemic will spread when the basic reproduction number is larger than 1.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2024)
Article
Mathematics, Applied
Gulsemay Yigit, Wakil Sarfaraz, Raquel Barreira, Anotida Madzvamuse
Summary: This study presents theoretical considerations and analysis of the effects of circular geometry on the stability of reaction-diffusion systems with linear cross-diffusion on circular domains. The highlights include deriving necessary and sufficient conditions for cross-diffusion driven instability and computing parameter spaces for pattern formation. Finite element simulations are also conducted to support the theoretical findings. The study suggests that linear cross-diffusion coupled with reaction-diffusion theory is a promising mechanism for pattern formation.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2024)
Article
Mathematics, Applied
Miaoqing Tian, Lili Han, Xiao He, Sining Zheng
Summary: This paper studies the attraction-repulsion chemotaxis system of two-species with two chemical substances. The behavior of solutions is determined by the interactions among diffusion, attraction, repulsion, logistic sources, and nonlinear productions in the system. The paper provides conditions for the global boundedness of solutions.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2024)
Article
Mathematics, Applied
Michal Borowski, Iwona Chlebicka, Blazej Miasojedow
Summary: This article provides a short proof of a sharp rearrangement estimate for a generalized version of a potential of Wolff-Havin-Maz'ya type. It characterizes the potentials that are bounded between rearrangement invariant spaces via a one-dimensional inequality of Hardy-type. By controlling very weak solutions to a broad class of quasilinear elliptic PDEs of non-standard growth, the special case of the mentioned potential infers the local regularity properties of solutions in rearrangement invariant spaces for prescribed classes of data.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2024)
Article
Mathematics, Applied
Young-Pil Choi, Jinwook Jung
Summary: This study investigates the global-in-time well-posedness of the pressureless Euler-alignment system with singular communication weights. A global-in-time bounded solution is constructed using the method of characteristics, and uniqueness is obtained via optimal transport techniques.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2024)
Article
Mathematics, Applied
Chuangxia Huang, Xiaodan Ding
Summary: In this paper, a diffusive Mackey-Glass model with distinct diapause and developmental delays is proposed based on the diapause effect. Some sufficient conditions for the existence of traveling wave fronts are obtained by constructing appropriate upper and lower solutions and employing inequality techniques. Two numerical examples are provided to demonstrate the reliability and feasibility of the proposed model.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2024)
Article
Mathematics, Applied
Hongxing Zhao
Summary: This paper investigates the flow of fluid through a thin corrugated domain saturated with porous medium, governed by the Navier-Stokes model. Asymptotic models are derived by comparing the relation between a and the size of the periodic cylinders. The homogenization technique based on the generalized Poincare inequality is used to prove the main results.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2024)
Article
Mathematics, Applied
Evgenii S. Baranovskii, Roman V. Brizitskii, Zhanna Yu. Saritskaia
Summary: This paper proves the solvability of optimal control problems for both weak and strong solutions of a boundary value problem associated with the nonlinear reaction-diffusion-convection equation with variable coefficients. In the case of strong solutions, the requirements for smoothness of the multiplicative control are reduced. The study of extremal problems is based on the proof of solvability of the corresponding boundary value problems and the qualitative analysis of their solution properties. The paper establishes existence results for weak solutions with large data, the maximum principle, and local existence and uniqueness of a strong solution. Furthermore, an optimal feedback control problem is considered, and sufficient conditions for its solvability in the class of weak solutions are obtained using methods of the theory of topological degree for set-valued perturbations.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2024)
Article
Mathematics, Applied
Antonia Chinni, Beatrice Di Bella, Petru Jebelean, Calin Serban
Summary: This article focuses on the multiplicity of solutions for differential inclusions involving the p-biharmonic operator, applying a variational approach and relying on non-smooth critical point theory.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2024)
Article
Mathematics, Applied
Zhong Tan, Saiguo Xu
Summary: This paper investigates the Rayleigh-Taylor instability of three-dimensional inhomogeneous incompressible Euler equations with damping in a horizontal slab. It is shown that the Euler system with damping is nonlinearly unstable around the given steady state if the steady density profile is non-monotonous along the height. A new variational structure is developed to construct the growing mode solution, and the difficulty in proving the sharp exponential growth rate is overcome by exploiting the structures in linearized Euler equations. Combined with error estimates and a standard bootstrapping argument, the nonlinear instability is established.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2024)
Article
Mathematics, Applied
Samuele Ricco, Andrea Torricelli
Summary: This paper presents a solution method for the autonomous obstacle problem, finding a necessary condition for the extremality of the unique solution using a primal-dual formulation. The proof is based on classical arguments of Convex Analysis and Calculus of Variations' techniques.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2024)
Article
Mathematics, Applied
Shuxin Ge, Rong Yuan, Xiaofeng Zhang
Summary: This paper studies an initial boundary value problem for a nonlocal parabolic equation with a diffusion term and convex-concave nonlinearities. By establishing the Lq-estimate and analyzing its energy, the existence of global solutions is proven and some blow-up conditions are obtained. Using the variational structure of the problem, the Mountain-pass theorem is utilized to demonstrate the existence of nontrivial steady-state solutions. The dynamical behavior of global solutions with relatively compact trajectories in H01 (Ω) is also established, showing uniform convergence to a non-zero steady state after a long time due to the energy functional satisfying the P.S. condition. Finally, an unstable steady states sequence is derived using another minimax theorem.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2024)
