Instability of Standing Wave for the Klein–Gordon–Hartree Equation

The instability property of the standing wave uω(t, x) = eiωtφ(x) for the Klein–Gordon– Hartree equation     

Scattering for focusing combined power-type NLS

We consider the scattering of Cauchy problem for the focusing combined power-type Schr¨odinger equation. In the spirit of concentration-compactness method, we will show that, H1 solution will scatter under some condition on its energy and mass. We adapt some variance argument, following the idea of Ibrahim–Masmoudi–Nakanishi.     

变系数Sine-Gordon方程的反散射变换

In this paper, the variable coefficient Sine-Gordon type equation uxt=a(t)sinu β(t)uxx k(t)(xux)x is discussed. It is related to the eigenvalue problem Vx=QV.The structure equation and the evolution laws of scattering data for the second equation are derived and the inverse scattering solution of the first equation is obtained.     

THE CAUCHY PROBLEM OF THE HARTREE EQUATION

In this paper, we systematically study the wellposedness, illposedness of the Hartree equation, and obtain the sharp local wellposedness, the global existence in Hs, s≥1 and the small scattering result in Hs for 2 〈γ〈 n and s ≥ 2/γ - 1. In addition, we study the nonexistence of nontrivial asymptotically free solutions of the Hartree equation.     

A NEW APPROACH AND ERROR ANALYSIS FOR RECONSTRUCTING THE SCATTERED WAVE BY THE POINT SOURCE METHOD

Consider an inverse scattering problem by an obstacle D belong to R^2 with impedance boundary. We investigate the reconstruction of the scattered field u^s from its far field pattern u^∞ using the point source method. First, by applying the boundary integral equation method, we provide a new approach to the point-source method of Potthast by classical potential theory. This extends the range of the point source method from plane waves to scattering of arbitrary waves. Second, by analyzing the behavior of the Hankel function, we obtain an improved strategy for the choice of the regularizing parameter from which an improved convergence rate （compared to the result of [15]） is achieved for the reconstruc- tion of the scattered wave. Third, numerical implementations are given to test the validity and stability of the inversion method for the impedance obstacle.     

Notes on Global Existence for the Nonlinear Schrdinger Equation Involves Derivative

In this paper, we consider the scattering for the nonlinear Schr¨odinger equation with small,smooth, and localized data. In particular, we prove that the solution of the quadratic nonlinear Schr¨odinger equation with nonlinear term |u|2involving some derivatives in two dimension exists globally and scatters. It is worth to note that there exist blow-up solutions of these equations without derivatives. Moreover, for radial data, we prove that for the equation with p-order nonlinearity with derivatives, the similar results hold for p ≥2d+32d-1and d ≥ 2, which is lower than the Strauss exponents.     

On initial data of the monopole equation

The space-time monopole equation is the reduction of anti-self-dual Yang-Mills equations in R2,2 to R2,1. This equation is a non-linear wave equation, and can be encoded in a Lax pair. An equivalent Lax pair is used by Dai and Terng to construct monopoles with continuous scattering data, and then the equation can be linearized by the scattering data, allowing one to use the inverse scattering method to solve the Cauchy problem with rapidly decaying small initial data. In this paper, we use the terminology of holomorphic bundle and transversality of certain maps, parametrized by initial data, to give more initial data, with which we can use scattering method to solve the Cauchy problem of the monopole equation up to gauge transformation.     

Scattering Theory for the Defocusing Fourth Order NLS with Potentials

Based on the endpoint Strichartz estimates for the fourth order Schr?dinger equation with potentials for n ≥ 5 by [Feng, H., Soffer, A., Yao, X.: Decay estimates and Strichartz estimates of the fourth-order Schr?dinger operator. J. Funct. Anal., 274, 605–658(2018)], in this paper, the authors further derive Strichartz type estimates with gain of derivatives similar to the one in [Pausader,B.: The cubic fourth-order Schr?dinger equation. J. Funct. Anal., 256, 2473–2517(2009)]. As their applications, we combine the classical Morawetz estimate and the interaction Morawetz estimate to establish scattering theory in the energy space for the defocusing fourth order NLS with potentials and pure power nonlinearity 1 +8/n p 1 +8/(n-4) in dimensions n ≥ 7.     

THE SASA-SATSUMA EQUATION ON A NON-ZERO BACKGROUND: THE INVERSE SCATTERING TRANSFORM AND MULTI-SOLITON SOLUTIONS

We concentrate on the inverse scattering transformation for the Sasa-Satsuma equation with 3×3 matrix spectrum problem and a nonzero boundary condition. To circumvent the multi-value of eigenvalues, we introduce a suitable two-sheet Riemann surface to map the original spectral parameter k into a single-valued parameter z. The analyticity of the Jost eigenfunctions and scattering coefficients of the Lax pair for the Sasa-Satsuma equation are analyzed in detail. According to the analyticity of the...     

Uniqueness on identification of cubic convolution nonlinearity

We shall consider the inverse scattering problem for time dependent version of Hartree equation and nonlinear Klein-Gordon equation. The uniqueness theorem on identifying the cubic convolution nonlinearity from the knowledge of the scattering operator will be shown.     

Nonrelativistic Limit of Scattering Theory for Nonlinear Klein-Gordon Equations

We study the nonrelativistic limit of the nonlinear Klein-Gordon equation and prove that the wave operators, the inverses of them, and the scattering operator for the naturally modulated equation converge to those for the singular limit, the nonlinear Schrödinger equation.     

Wick型随机KG方程的精确解

利用白噪声分析、Hermite变换和双曲正切法来研究随机偏微分KleinGordon方程,并在Kondratiev分布空间(S)-1-上分别获得了变系数Klein-Gordon方程和Wick型随机Klein-Gordon方程的精确解和白噪声泛函解.     

Global existence and uniform decay of a damped Klein-Gordon equation in a noncylindrical domain

In this paper, we consider a damped Klein-Gordon equation in a noncylindrical domain. This work is devoted to proving the existence of global solutions and decay for the energy of solutions for a damped Klein-Gordon equation in a noncylindrical domain.     

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.     

浅水波方程和Klein-Gordon方程新的精确行波解

采用了一种新的方法来求解浅水波方程和Klein-Gordon的行波解.在该方法下,Klein-Gordon方程和浅水波方程都得到了其精确的周期孤立波解,从而该方法的有效性得到了验证.     

Solution of the Cauchy problem for the Boiti-Leon-Pempinelli equation

The Cauchy problem for the (2+1)-dimensional nonlinear Boiti-Leon-Pempinelli (BLP) equation is studied within the framework of the inverse problem method. Evolution equations generated by the system of BLP equations under study are derived for the resolvent, Jost solutions, and scattering data for the two-dimensional Klein-Gordon differential operator with variable coefficients. Additional conditions on the scattering data that ensure the stability of the solutions to the Cauchy problem are revealed. A recurrence procedure is suggested for constructing the polynomial integrals of motion and the generating function for these integrals in terms of the spectral data.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 109, No. 2, pp. 163–174, November, 1996.     

一类非线性Klein-Gordon方程的数值解

本文主要研究了一类非线性Klein-Gordon方程.利用Fourier谱方法对一类非线性Klein-Gordon方程的求解,给出了求解的离散过程,并通过了数值模拟与文献结果进行了对比.结果表明这种方法对于求解此类非线性Klein-Gordon方程具有很好的效果.     

Scattering theory for the Klein-Gordon equation

We develop the scattering theory for the Klein-Gordon equation. We follow the usual procedure of considering an equivalent equation, which is first order in time, in the Hilbert space of vector valued functions which have a finite energy norm. We prove existence and completeness of the wave operators, the intertwining relations, and the invariance principle as well. This is done for a large class of potentials. In particular, the magnetic potential may even be divergent at infinity. Electric and scalar potentials that behave at infinity as $\text{¦x¦}{}^{\mathrm{??\; ?\; l}}\text{, ? > 0}$ are contained in our class.     

Spacetime Causality in the Study of the Hankel Transform

We study Hilbert space aspects of the Klein-Gordon equation in two-dimensional spacetime. We associate to its restriction to a spacelike wedge a scattering from the past light cone to the future light cone, which is then shown to be (essentially) the Hankel transform of order zero. We apply this to give a novel proof, solely based on the causality of this spatio-temporal wave propagation, of the theorem of de Branges and V. Rovnyak concerning Hankel pairs with a support property. We recover their isometric expansion as an application of Riemann’s general method for solving Cauchy-Goursat problems of hyperbolic type. Communicated by Vincent Rivasseau Submitted: October 28, 2005; Accepted: February 17, 2006