Global well-posedness for the derivative nonlinear Schrödinger equation

Finite gap integration of the derivative nonlinear Schrödinger equation: A Riemann–Hilbert method

We show that the derivative nonlinear Schrodinger (DNLS) equation is globally well-posed in the weighted Sobolev space H2,2(ℝ). Our result exploits the complete integrability of the DNLS equation and removes certain spectral conditions on the initial data required by our previous work, thanks to Zhou’s analysis (Comm. Pure Appl. Math. 42:7 (1989), 895–938) on spectral singularities in the context of inverse scattering.

https://msp.org/apde/2020/13-5/apde-v13-n5-p09-s.pdf

Global existence for the derivative nonlinear Schrödinger equation with arbitrary spectral singularities

We study the derivative nonlinear Schrodinger (DNLS) equation for general initial conditions in weighted Sobolev spaces that can support bright solitons (but exclude spectral singularities). We sho...

Global well-posedness for the derivative non-linear Schrödinger equation

In this paper, we study the long-time dynamics and stability properties of the sine-Gordon equation $$f_{tt}-f_{xx}+\sin f=0.$$ Firstly, we use the nonlinear steepest descent for Riemann-Hilbert problems to compute the long-time asymptotics of the solutions to the sine-Gordon equation whose initial condition belongs to some weighted Sobolev spaces. Secondly, we study the asymptotic stability of the sine-Gordon equation. It is known that the obstruction to the asymptotic stability of the sine-Gordon equation in the energy space is the existence of small breathers which is also closely related to the emergence of wobbling kinks. Combining the long-time asymptotics and a refined approximation argument, we analyze the asymptotic stability properties of the sine-Gordon equation in weighted energy spaces. Our stability analysis gives a criterion for the weight which is sharp up to the endpoint so that the asymptotic stability holds.

Long-time asymptotics and stability for the sine-Gordon equation

We extend the global existence result for the derivative NLS equation to the case when the initial datum includes a finite number of solitons. This is achieved by an application of the B\"{a}cklund transformation that removes a finite number of zeros of the scattering coefficient. By means of this transformation, the Riemann--Hilbert problem for meromorphic functions can be formulated as the one for analytic functions, the solvability of which was obtained recently.

https://www.intlpress.com/site/pub/files/_fulltext/journals/dpde/2017/0014/0003/DPDE-2017-0014-0003-a003.pdf

The derivative NLS equation: global existence with solitons

We consider the massive Thirring model in the laboratory coordinates and explain how the inverse scattering transform can be developed with the Riemann–Hilbert approach. The key ingredient of our technique is to transform the corresponding spectral problem to two equivalent forms: one is suitable for the spectral parameter at the origin and the other one is suitable for the spectral parameter at infinity. Global solutions to the massive Thirring model are recovered from the reconstruction formulae at the origin and at infinity.

Inverse Scattering for the Massive Thirring Model

The massive Thirring model (MTM) was introduced in 1958 by the Austrian physicist Walter Thirring in the context of relativistic quantum field theory. It describes the self-interaction of a Dirac field in one space dimension. From the analytical point of view, this system of non-linear partial differential equations is of special interest, because it has a representation in terms of a Lax pair, consisting of two linear operators $L$ and $A$. Thanks to the Lax pair, the MTM admits an exact solution by the inverse scattering transform (IST). Since the dependence of $L$ and $A$ on the spectral parameter $\lambda$ is singular at the origin and at infinity, the IST cannot be defined for initial data of low regularity as straightforward as it is done for other equations, the NLS equation for instance. One key ingredient of the present thesis is to transform the known Lax pair to two equivalent Lax pairs: one is suitable for the spectral parameter at the origin and the other one is suitable at infinity. Using the equivalent operators the direct scattering transform is developed for an optimal $L^2$-based Sobolev space. The inverse scattering map is then formulated in terms of two Riemann--Hilbert problems whose solvability is proven.
As it is also known from other nonlinear dispersive equations one can create \emph{solitons} for the MTM. These special solutions are waves that move at constant speed and do not change in shape. They can refuse to disperse only because of the presence of the nonlinearity in the equation. It is relatively simple to characterize solitons, based on their scattering data. Using suitable Riemann-Hilbert techniques it is possible to analyse the interaction of two (or more) solitons. Furthermore, it can be shown precisely that each soliton will eventually enter the light cone $\set{|t|>|x|}$. Using the so-called $\db$--method (nonlinear steepest descent) we show that outside the light cone any solution (not only solitons) converges to zero with a rate of $\sim|t|^{-3/4}$. Inside the light-cone there are basically two different possibilities. Assuming that the initial data is free of solitons we use the $\db$--method and some well-known model Riemann--Hilbert--problems to show that the solution of the MTM scatters to a linear solution modulo phase correction. This linear solution can be computed explicitly from the scattering data and its amplitude decays with a rate of $\sim|t|^{-1/2}$. The second possibility is that the initial data contains finitely many solitons. Then, as the main result of the thesis, we prove that any solution breaks up into finitely many single solitons that travel at different speeds and thus, diverge. The remainder term is $\mathcal{O}(|t|^{-1/2})$.\medskip\\
Summarizing, the present thesis provides an analytical proof of the soliton resolution conjecture for the MTM. This result also implies the asymptotic stability of solitons.

Long-time asymptotics for the massive Thirring model

We extend the global existence result for the derivative NLS equation to the case when the initial datum includes a finite number of solitons. This is achieved by an application of the Backlund transformation that removes a finite number of zeros of the scattering coefficient. By means of this transformation, the Riemann--Hilbert problem for meromorphic functions can be formulated as the one for analytic functions, the solvability of which was obtained recently.

We prove the existence of global solutions to the DNLS equation with initial data in a large subset of $H^2(\mathbb R)\cap H^{1,1}(\mathbb R)$ containing a neighborhood of all solitons. We use the inverse scattering transform method, which was recently developed by D. Pelinovsky and Y. Shimabukuro, and an auto-Backlund transform in order to include solitons.

Aaron Saalmann

Papers

The derivative NLS equation: global existence with solitons

Inverse Scattering for the Massive Thirring Model

Long-time asymptotics for the massive Thirring model

The derivative NLS equation: global existence with solitons

Global existence for the derivative NLS equation in the presence of solitons