Special Session 53: Mathematical Theory on the Klein-Gordon Equation and Related Models

Asymptotic stability of traveling waves for one-dimensional nonlinear Schrodinger equations

Charles Collot CY Cergy Paris Universite France Co-Author(s):    Pierre Germain

Abstract: We consider one dimensional nonlinear Schrodinger equations around a traveling wave. We prove its asymptotic stability for general nonlinearities, under the hypotheses that the orbital stability condition of Grillakis-Shatah-Strauss is satisfied and that the linearized operator does not have a resonance and only has 0 as an eigenvalue. As a by-product of our approach, we show modified scattering for the radiation remainder. Our proof combines for the first time modulation techniques and the study of space-time resonances. We rely on the use of the distorted Fourier transform, akin to the work of Buslaev and Perelman and, and of Krieger and Schlag, and on precise renormalizations, computations and estimates of space-time resonances to handle its interaction with the soliton. This is joint work with Pierre Germain.

Modified scattering for a non-local derivative NLS

Nobu Kishimoto Kyoto University Japan Co-Author(s):    Kiyeon Lee

Abstract: We consider asymptotic behavior of small solutions to a one-dimensional nonlinear Schr\{o}dinger equation with a non-local cubic derivative nonlinear term, which has dissipative effect. In the periodic setting, dissipation becomes prominent and the initial value problem is known to be ill-posed backward in time even for small data. In contrast, on the real line we show global existence of solutions and modified scattering behavior in both time directions for small data in weighted Sobolev space.

Numerical approximation of discontinuous solutions of the semilinear wave equation

Buyang Li The Hong Kong Polytechnic University Hong Kong Co-Author(s):    Jiachuan Cao, Buyang Li, Yanping Lin, Fangyan Yao

Abstract: A high-frequency recovered fully discrete low-regularity integrator is constructed to approximate rough and possibly discontinuous solutions of the semilinear wave equation. The proposed method, with high-frequency recovery techniques, can capture the discontinuities of the solutions correctly without spurious oscillations and approximate rough and discontinuous solutions with a higher convergence rate than pre-existing methods. Rigorous analysis is presented for the convergence rates of the proposed method in approximating discontinuous solutions of bounded variation in one dimension (which allow jump discontinuities). The proposed method is proved to have almost first-order convergence under the stepsize condition $\tau =O(1/N)$, where $\tau$ and $N$ denote the time stepsize and the number of Fourier terms in the space discretization, respectively. Numerical examples are presented in both one and two dimensions to illustrate the advantages of the proposed method in improving the accuracy in approximating rough and discontinuous solutions of the semilinear wave equation.

Scattering for defocusing energy sub-critical wave equation with inverse square potential

Baoping Liu Peking University Peoples Rep of China Co-Author(s):    Haiming Du

Abstract: We consider the defocusing energy sub-critical nonlinear wave equation with inverse square potential, and prove global wellposedness and scattering for radial data lying in critical Sobolev space. The main ingredients for our proof include the Fourier truncation method, the hyperbolic coordinate transformation, and the radial endpoint Strichartz estimate for wave equations with potential of critical decay.

Instability of standing waves for cubic-quintic NLS with delta potential

Masahito Ohta Tokyo University of Science Japan Co-Author(s):

Abstract: We consider a nonlinear Schr\{o}dinger equation with the cubic-quintic combination of repulsive and attractive nonlinearities, and an attractive delta potential in one space dimension. The stability and instability of standing wave solutions are studied.

Numerical study of the logarithmic Schrodinger equation with repulsive harmonic potential

Chunmei Su Tsinghua University Peoples Rep of China Co-Author(s):

Abstract: We consider the nonlinear Schrodinger equation with a logarithmic nonlinearity and a repulsive harmonic potential. Depending on the parameters of the equation, the solution may or may not be dispersive. When dispersion occurs, it does with an exponential rate in time. To control this, we change the unknown function through a generalized lens transform. This approach neutralizes the possible boundary effects, and could be used in the case of the nonlinear Schrodinger equation without potential. We then employ standard splitting methods on the new equation via a nonuniform grid, after the logarithmic nonlinearity has been regularized. We also discuss the case of a power nonlinearity and give some results concerning the error estimates of the first-order Lie-Trotter splitting method for both cases of nonlinearities. Finally extensive numerical experiments are reported to investigate the dynamics of the equations.

The modulational approximation to the water waves

Qingtang Su Academy of Mathematics and System Sciences Peoples Rep of China Co-Author(s):

Abstract: In this talk I will discuss the modulational approximation to the irrotational and incompressible water waves. In particular, I will show how to use this to study the instability of the water waves.

Late-time asymptotics for the Klein-Gordon equation on a Schwarzschild black hole

Maxime Van de Moortel Rutgers University USA Co-Author(s):    Federico Pasqualotto, Yakov Shlapentokh-Rothman, Maxime Van de Moortel

Abstract: The late-time behavior of the linear Klein-Gordon equation on a Schwarzschild geometry, which models the simplest black hole in General Relativity, has long posed a significant challenge due to the presence of stable (timelike) trapping. We present our recent resolution of this problem, uncovering an unexpected contrast between solutions with exponentially decaying initial data and those with polynomial decay. These results lay the groundwork for future exploration of the nonlinear dynamics of the Schwarzschild black hole for the Einstein-Klein-Gordon equations.

Global well-posedness of the defocusing, cubic nonlinear wave equation outside of the ball with radial data

Guixiang Xu Beijing Normal University Peoples Rep of China Co-Author(s):    Guixiang Xu and Pengxuan Yang

Abstract: We consider the defocusing, cubic nonlinear wave equation with zero Dirichlet boundary value in the exterior $\Omega = \mathcal{R}^3\backslash \bar{B}(0,1)$. We make use of the distorted Fourier transformto establish the dispersive estimate and the global-in-time (endpoint) Strichartz estimate of the linear wave equation outside of the ball with radial data. As an application, we combine the Fourier truncation method with the energy method to show global well-posedness of radial solution to the defoucusing, cubic nonlinear wave equation outside of a ball in the Sobolev space $\left(\dot H^{s}_{D}(\Omega) \cap L^4(\Omega) \right)\times \dot H^{s-1}_{D}(\Omega)$ for $s>3/4$.

Energy Transfer and Radiation in Hamiltonian Nonlinear Klein-Gordon Equations

Zhaojie Yang Fudan University Peoples Rep of China Co-Author(s):    Zhen Lei, Jie Liu, Zhaojie Yang

Abstract: We consider Klein-Gordon equations with cubic nonlinearity in three spatial dimensions, which are Hamiltonian perturbations of the linear one with potential. It is assumed that the corresponding Klein-Gordon operator admits an arbitrary number of possibly degenerate eigenvalues in $(0, m)$, and hence the unperturbed linear equation has multiple time-periodic solutions known as bound states. In 1999, Soffer and Weinstein discovered a mechanism called Fermi`s Golden Rule for this nonlinear system in the case of one simple but relatively large eigenvalue $\Omega \in (m/3 , m)$, by which energy is transferred from discrete to continuum modes and the solution still decays in time. In our first work, we solved the general one simple eigenvalue case. In our second work, we solved this problem in full generality: multiple and simple or degenerate eigenvalues in $(0, m)$. Indeed, we obtained the sharp rate of energy transfer from one discrete state to continuum modes in the general case. The proof is based on a kind of pseudo-one-dimensional cancellation structure in each eigenspace, a renormalized damping mechanism, and an enhanced damping effect. It also relies on a refined Birkhoff normal form transformation and an accurate generalized Fermi`s Golden Rule building upon the results of Bambusi and Cuccagna. This is a joint work with Prof. Zhen Lei and Dr. Jie Liu.

Long time behaviors for damped Klein-Gordon and wave equations

Lifeng Zhao University of Science and Technology of China Peoples Rep of China Co-Author(s):

Abstract: Soliton resolution conjecture is a long standing problem for nonlinear dispersive equations. The soliton resolution results for damped Klein-Gordon and damped energy critical wave equation will be stated. In addition, some specific examples of multi-bubble solutions with precise dynamics are constructed.

Some recent results on vortex patch problems

Maolin Zhou Nankai University Peoples Rep of China Co-Author(s):

Abstract: In this talk, we will discuss some problems about vortex patch problems. It mainly consists of two parts: (1) boundary regularity, especially why the singular angle must be 90 degree; (2) degenerate bifurcation from annulus for particular parameters.