Article
Mathematics, Applied
Jie Chen, Baoxiang Wang
Summary: In this paper, we investigate the almost sure scattering for the Klein-Gordon equations with Sobolev critical power and obtain the almost sure scattering with random initial data under certain conditions. We employ the method of induction on scales and bushes argument, and utilize the mass term of the Klein-Gordon equation to control the increment of energy.
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS
(2022)
Article
Mathematics
Miguel Ballesteros, Diego Iniesta, Ivan Naumkin, Clemente Pena
Summary: This paper discusses the initial-boundary value problem for the nonlinear Klein-Gordon equation in a quarter-plane with Dirichlet non zero boundary conditions. By constructing the wave and scattering operators, the influence of the boundary data on the asymptotic behavior of solutions is studied. It is demonstrated that non zero boundary conditions require a modification of the critical value of the nonlinear term.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2022)
Article
Mathematics, Applied
Raphael Cote, Xu Yuan
Summary: This paper considers the behavior of solutions to the nonlinear damped Klein-Gordon equation. It is shown that when only one nonlinear object appears asymptotically for large times, the nonlinear object necessarily takes the form of a bound state. Convergence of solutions is established for non-degenerate states or degenerate excited states satisfying a simplicity condition, with exponential or algebraic rates respectively. An example is provided where the solution converges exactly at a rate of t(-1) to the excited state.
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS
(2022)
Article
Automation & Control Systems
Jun Liu, Weiping Yan, Can Zhang
Summary: This paper investigates the stabilizability of a quasilinear Klein-Gordon-wave system with variable coefficients in Rn. The stabilizability of linear wave-type equations with Kelvin-Voigt damping has been considered by Liu-Zhang and Yu-Han for different systems. In this paper, it is shown that there exists a linear feedback control law that can exponentially stabilize the quasilinear Klein-Gordon-wave system under certain smallness conditions, with the feedback control being a strong Kelvin-Voigt damping.
SIAM JOURNAL ON CONTROL AND OPTIMIZATION
(2023)
Article
Mathematics, Applied
Jianyi Chen, Zhitao Zhang, Guijuan Chang, Jing Zhao
Summary: This paper studies the nonlinear Klein-Gordon systems arising from relativistic physics and quantum field theories, focusing on the existence, regularity, and asymptotic behavior of time-periodic solutions to the linearly coupled problem as e goes to 0. By constructing critical points of an indefinite functional via variational methods, solutions are obtained and their asymptotic behavior is characterized, showing convergence to solutions of wave equations without coupling terms as epsilon -> 0. Through careful analysis, interesting results regarding the higher regularity of periodic solutions are obtained, different from elliptic regularity theory.
ADVANCED NONLINEAR STUDIES
(2021)
Article
Mathematics
C. Buriol, L. G. Delatorre, V. H. Gonzalez Martinez, D. C. Soares, E. H. G. Tavares
Summary: The study deals with a nonlinear Klein-Gordon system in an inhomogeneous medium, with local damping distributed around the boundary according to the Geometric Control Condition. It is shown that the energy of the system exponentially goes to zero for initial data within bounded sets of finite energy phase-space.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics
Yue Ma
Summary: This article develops techniques for strong couplings in two-dimensional wave Klein-Gordon systems, distinguishing the roles of different decay factors and developing a method to exchange one type of decay for another. The global existence result of a model problem is established, as well as the global existence result for the Klein-Gordon-Zakharov model system.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics
Nicolas Besset
Summary: The study demonstrates asymptotic completeness for the charged Klein-Gordon equation under certain conditions, with scattering interpreted as asymptotic transports along principal null geodesics in an extended spacetime.
JOURNAL OF GEOMETRIC ANALYSIS
(2021)
Article
Physics, Particles & Fields
Marco Matone
Summary: In this study, Friedmann's equations are formulated as second-order linear differential equations using the Schwarzian derivative technique. This representation is equivalent to eigenvalue problems and suggests the existence of a measurement problem in the equations. The study also explores the relationship between Klein-Gordon operators and the Klein-Gordon Hamilton-Jacobi equations, revealing a new symmetry of Friedmann's equations in flat space when presented in linear form.
EUROPEAN PHYSICAL JOURNAL C
(2021)
Article
Multidisciplinary Sciences
U. H. M. Zaman, Mohammad Asif Arefin, M. Ali Akbar, M. Hafiz Uddin
Summary: Nonlinear fractional partial differential equations have wide applications in various engineering and research fields, such as shallow-water, oceanography, fluid dynamics, acoustics, plasma physics, optical fiber system, turbulence, nonlinear biological systems, and control theory. This research focuses on constructing new closed form solutions for traveling waves of fractional order nonlinear coupled type Boussinesq-Burger (BB) and coupled type Boussinesq equations. By using the subsidiary extended tanh-function technique and conformable derivatives, we obtained new results and simplified the solution process. Various wave forms of solitons were achieved, and the physical sketch was visualized using mathematical software. The suggested technique is reliable and provides more general exact solutions of close form traveling waves.
Article
Mathematics, Applied
Yan Guo, Chunyan Huang, Benoit Pausader, Klaus Widmayer
Summary: In this study, we investigated the stability of long-time nonlinear dynamics in the inviscid setting under constant rotation. Through the development of an anisotropic framework that utilizes available symmetries, we demonstrated that axisymmetric initial data of sufficiently small size epsilon lead to solutions that exist for a long time of at least epsilon-M and disperse, regardless of the speed of rotation.
COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS
(2023)
Article
Mathematics, Applied
Sangdon Jin, Jinmyoung Seok
Summary: In this study, it is shown that there is a correspondence between positive solitary waves of Nonlinear Maxwell-Klein-Gordon equations and Nonlinear Schrodinger-Poisson equations under the nonrelativistic limit. The existence or multiplicity of positive solutions depends on the choices of parameters in the equations. Additionally, a new result of existence of positive solutions with lower order nonlinearity is presented.
CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics, Applied
Senhao Duan, Yue Ma
Summary: In this paper, we establish the global stability of a type of totally geodesic wave maps and the Klein-Gordon-Zakharov system in R2+1. By constructing auxiliary systems and revealing hidden structures, we are able to establish the global stability.
SIAM JOURNAL ON MATHEMATICAL ANALYSIS
(2022)
Article
Mathematics, Applied
Qihong Shi, Yaqian Jia, Jianxiong Cao
Summary: Motivated by the potential singularity in the stationary state, we revisit the initial boundary value problem of the classical Klein-Gordon-Schrodinger (KGS) system in one space dimension. Our result demonstrates the existence of singular solutions, some of which may exhibit exponential growth with respect to the spatial variable.
APPLIED MATHEMATICS LETTERS
(2022)
Article
Mathematics, Applied
Hans Lindblad, Jonas Luhrmann, Avy Soffer
Summary: This study investigates the asymptotic behavior of small solutions to one-dimensional Klein-Gordon equations with variable coefficient quadratic nonlinearities. It discovers a striking resonant interaction between specific spatial frequencies of the variable coefficient and the temporal oscillations of the solutions, leading to a novel type of modified scattering behavior. Sharp decay estimates and asymptotics are established in both resonant and non-resonant cases.
ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
(2021)