Search

In this two-paper series, we prove the invariance of the Gibbs measure for a threedimensional wave equation with a Hartree nonlinearity. The novelty lies in the singularity of the Gibbs measure with respect to the Gaussian free field. In this paper, we focus on the dynamical aspects of our main result. The local theory is based on a paracontrolled approach, which combines ingredients from dispersive equations, harmonic analysis, and random matrix theory. The main contribution, however, lies in the global theory. We develop a new globalization argument, which addresses the singularity of the Gibbs measure and its consequences. [ABSTRACT FROM AUTHOR]

In this paper, the Cauchy problem for linear and nonlinear wave equations is studied. The equation involves abstract operator A in a Banach space E and convolution terms. Here, assuming enough smoothness on the initial data and on coefficients, the existence, uniqueness and regularity properties of local and global solutions are established in terms of fractional powers of a given sectorial operator A. We obtain the regularity properties of a wide class of wave equations by choosing the space E and the operator A which appear in the field of physics. [ABSTRACT FROM AUTHOR]

Schrödinger equation was proposed in theoretical physics to provide information and the behavior of a system of particles. The Schrödinger equation was recently solved for a combination of different potentials using different traditional methodologies. The studies do not consider the effect of a barrier between the potentials. In reality, the barrier between the potentials has an effect on the energy eigenvalues. In the present work, a Kratzer- Mie-type potential, a combination of Kratzer and Mie-type-constant potentials, was proposed as the interacting potential. The solutions of the radial Schrödinger equation was obtained in the presence of the combined potentials by considering a barrier between the potentials. The energy equation and the corresponding wave functions were explicitly calculated. The calculations were performed by using the methodology of supersymmetric quantum mechanics. This method involves the proposition of superpotential function as a solution to its Riccati differential equation. Numerical results were generated for some diatomic molecules using the energy equation and the diatomic model parameters. Some expectation values for the combined potential were calculated using Hellmann Feynman Theorem, and the effect of the barrier on the expectation values were numerically studied. It was observed that when a particle moves from the lower end to a higher end of a barrier, it absorbs energy from the system for some time. This same behavior was also noted when the particle penetrates through a barrier. Thus, its vibration will only depend on the initial energy, which is absorbed. It was equally seen that as the width of the barrier becomes larger than the height, the energy of the system decreases drastically. [ABSTRACT FROM AUTHOR]

This paper is dedicated to the improvement of the efficiency of the leapfrog method for linear and semilinear second-order differential equations. In numerous situations the strict CFL condition of the leapfrog method is the main bottleneck that thwarts its performance. Based on Chebyshev polynomials new methods have been constructed for linear problems that exhibit a much weaker CFL condition than the leapfrog method (at a higher cost). However, these methods fail to produce the correct long-time behavior of the exact solution which can result in a bad approximation quality. In this paper we introduce a new class of leapfrog-Chebyshev methods for semilinear problems. For the linear part, we use Chebyshev polynomials while the nonlinearity is treated by the standard leapfrog method. The method can be viewed as a multirate scheme because the nonlinearity is evaluated only once in each time step whereas the number of evaluations of the linear part corresponds to the degree of the Chebyshev polynomial. In contrast to existing literature (which is restricted to linear problems), we suggest to stabilize the scheme and we introduce a new starting value required for the two-step method. A new representation formula for the approximations obtained by using generating functions allows us to fully understand the stability and the long-time behavior of the stabilized and the unstabilized scheme. In particular, for linear problems we prove that these new schemes approximately preserve a discrete energy norm over arbitrarily long times. The stability analysis shows that stabilization is essential to guarantee a favorable CFL condition for the multirate scheme, which is closely related to local time-stepping schemes. We also show error bounds of order two for semilinear problems and that a special choice of the stabilization yields order four for linear problems. Finally, we discuss the efficient implementation of the new schemes and give generalizations to fully nonlinear equations. [ABSTRACT FROM AUTHOR]

We study systems of two coupled wave equations in one space dimension, with one control, spatially supported on an arbitrarily small interval. We obtain the controllability of such systems under certain conditions on the coupling. To do this we apply the "fictitious control method" in two cases: general systems with a controllable linearized system, and a particular case where the linearized system is not controllable, namely, a cubic coupling. In the latter case, our proof requires finding nontrivial trajectories of the control system that go from 0 to 0 and having a controllable linearized system. We build these trajectories by adapting (in one space dimension) a construction developed by Jean-Michel Coron, Sergio Guerrero, and Lionel Rosier for the study of coupled parabolic systems. [ABSTRACT FROM AUTHOR]

This work focuses on a wave equation with almost-periodic forcing ψ ( ω t , x ) subject to Dirichlet boundary conditions. For the equation we obtain almost-periodic solutions corresponding to full-dimensional invariant tori of an associated infinite dimensional dynamical system. The proof is based on the reducibility via an improved KAM theory. [ABSTRACT FROM AUTHOR]

In the present paper we investigate the fractal boundary value problems for the Fredholm and Volterra integral equations, heat conduction and wave equations by using the local fractional decomposition method. The operator is described by the local fractional operators. The four illustrative examples are given to elaborate the accuracy and reliability of the obtained results. [ABSTRACT FROM AUTHOR]