Global existence for nonlinear Klein-Gordon equations in infinite homogeneous waveguides in two dimensions

In this paper we prove a global existence result for nonlinear Klein-Gordon equations with small data in infinite homogeneous waveguids, R²×M, where M=(M,g) is a Zoll manifold. The method is based on the normal forms, the eigenfunction expansion for M and the special distribution of eigenvalues of Laplace-Beltrami on Zoll manifold.     

Scattering operator for nonlinear Klein-Gordon equations in higher space dimensions

We prove the existence of the scattering operator in the neighborhood of the origin in the weighted Sobolev space Hβ,1 with for the nonlinear Klein-Gordon equation with a power nonlinearity

     

Wave operators to a quadratic nonlinear Klein-Gordon equation in two space dimensions

We study asymptotics around the final states of solutions to the nonlinear Klein-Gordon equations with quadratic nonlinearities in two space dimensions , where . We prove that if the final states

     

Global existence of small solutions for quadratic quasilinear Klein-Gordon systems in two space dimensions

Consider a quasi-linear system of two Klein-Gordon equations with masses m₁, m₂. We prove that when m₁≠2m₂ and m₂≠2m₁, such a system has global solutions for small, smooth, compactly supported Cauchy data. This extends a result proved by Sunagawa (J. Differential Equations 192 (2) (2003) 308) in the semi-linear case. Moreover, we show that global existence holds true also when m₁=2m₂ and a convenient null condition is satisfied by the nonlinearities.     

The defocusing energy-supercritical NLS in higher dimensions

We consider the defocusing nonlinear Schr?dinger equations iu_t +△u =|u|~(p_u) with p being an even integer in dimensions d≥ 5. We prove that an a priori bound of critical norm implies global well-posedness and scattering for the solution.     

Local Existence and WKB Approximation of Solutions to Schrödinger–Poisson System in the Two-Dimensional Whole Space

We consider the Schrödinger–Poisson system in the two-dimensional whole space. A new formula of solutions to the Poisson equation is used. Although the potential term solving the Poisson equation may grow at the spatial infinity, we show the unique existence of a time-local solution for data in the Sobolev spaces by an analysis of a quantum hydrodynamical system via a modified Madelung transform. This method has been used to justify the WKB approximation of solutions to several classes of nonlinear Schrödinger equation in the semiclassical limit.     

A new method to solve the damped nonlinear Klein-Gordon equation

This paper discusses a damped nonlinear Klein-Gordon equation in the reproducing kernel space and provides a new method for solving the damped nonlinear Klein-Gordon equation based on the reproducing kernel space.Two numerical examples are given for illustrating the feasibility and accuracy of the method.     

Interaction of modulated pulses in nonlinear oscillator chains

We formally derive and rigorously justify the modulation equations of lowest order for the interaction of two modulated pulses on a one-dimensional nonlinear oscillator chain. We show that solutions with the initial form of the assumed ansatz preserve this form over time intervals with positive macroscopic length, and we show a bound on the possible shift of the envelope caused by the interaction. Thus, we rigorously justify and quantify the statement that under the given conditions there is almost no interaction of the modulated pulses.     

Trilinear Lp estimates with applications to the Cauchy problem for the Hartree-type equation

In this paper, $L^p$ estimates for a trilinear operator associated with the Hartree type nonlinearity are proved. Moreover, as application of these estimates, it is proved that after a linear transformation, the Cauchy problem for the Hartree-type equation becomes locally well posed in the Bessel potential and homogeneous Besov spaces under certain regularity assumptions on the initial data. This notion of well-posedness and the functional framework to solve the equation were firstly proposed by Y. Zhou.     

Solitons in coupled nonlinear Schrödinger equations with variable coefficients

We study the coupled nonlinear Schrodinger equation with variable coefficients (VCNLS), which can be used to describe the interaction among the modes in nonlinear optics and Bose–Einstein condensation. By constructing an explicit transformation, which maps VCNLS to the classical coupled nonlinear Schrödinger equations (CNLS), we obtain Bright–Dark and Bright–Bright solitons for VCNLS. Furthermore, the optical super-lattice potentials (or periodic potentials) and hyperbolic cosine potentials with parameters are designed, which are two kinds of important potentials in physics. This method can be used to design a large variety of external potentials in VCNLS, which could be meaningful for manipulating solitons experimentally.     

Stable approximation of solutions to irregular nonlinear operator equations in a Hilbert space under large noise

The class of regularized Gauss-Newton methods for solving inexactly specified irregular nonlinear equations is examined under the condition that additive perturbations of the operator in the problem are close to zero only in the weak topology. By analogy with the well-understood conventional situation where the perturbed and exact operators are close in norm, a stopping criterion is constructed ensuring that the approximate solution is adequate to the errors in the operator.     

Asymptotic stability of solitons to 1D nonlinear Schrödinger equations in subcritical case

We prove the asymptotic stability of solitary waves to 1D nonlinear Schrödinger equations in the subcritical case with symmetry and spectrum assumptions. One of the main ideas is to use the vector fields method developed by S. Cuccagna, V. Georgiev, and N. Visciglia [Comm. Pure Appl. Math., 2013, 6: 957–980] to overcome the weak decay with respect to t of the linearized equation caused by the one dimension setting and the weak nonlinearity caused by the subcritical growth of the nonlinearity term. Meanwhile, we apply the polynomial growth of the high Sobolev norms of solutions to 1D Schrödinger equations obtained by G. Staffilani [Duke Math. J., 1997, 86(1): 109–142] to control the high moments of the solutions emerging from the vector fields method.     

On dispersive properties of discrete 2D Schrödinger and Klein-Gordon equations

We derive the long-time asymptotics for solutions of the discrete 2D Schrödinger and Klein-Gordon equations.     

Existence of breathers for discrete nonlinear Schrödinger equations

In this paper, we obtain a new sufficient condition on the existence of breathers for the discrete nonlinear Schrödinger equations by using critical point theory in combination with periodic approximations. The classical Ambrosetti–Rabinowitz superlinear condition is improved.     

Sharp condition of global existence for second-order derivative nonlinear Schrödinger equations in two space dimensions

This paper discusses a class of second-order derivative nonlinear Schrödinger equations which are used to describe the upper-hybrid oscillation propagation. By establishing a variational problem, applying the potential well argument and the concavity method, we prove that there exists a sharp condition for global existence and blow-up of the solutions to the nonlinear Schrödinger equation. In addition, we also answer the question: how small are the initial data, the global solutions exist?