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Global Dynamics of small data solutions to the Derivative Nonlinear Schr\"odinger equation
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Abstract
In this paper, we consider the derivative nonlinear Schr\"odinger (DNLS) equation. While the existence theory has been intensely studied, properties like dispersive estimates for the solutions have not yet been investigated. Here we address this question for the problem with small and localized data, and show that a dispersive estimate for the solution holds globally in time. For the proof of our result we use vector field methods combined with the \emph{testing by wave packets method}, whose implementation in this problem is novel.
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Modified scattering phenomena are encountered in the study of global properties for nonlinear dispersive partial differential equations in situations where the decay of solutions at infinity is borderline and scattering fails just barely. An interesting example is that of problems with cubic nonlinearities in one space dimension.
The method of testing by wave packets was introduced by the authors as a tool to efficiently capture the asymptotic equations associated to such flows, and thus establish the modified scattering mechanism in a simpler, more efficient fashion, and at lower regularity. In these expository notes we describe how this method can be applied to problems with general dispersion relations.
This paper is dedicated to the study of the derivative nonlinear Schrödinger equation on the real line. The local well-posedness of this equation in the Sobolev spaces \(H^s({\mathop {{\mathbb {R}}}\nolimits })\) is well understood since a couple of decades, while the global well-posedness is not completely settled. For the latter issue, the best known results up-to-date concern either Cauchy data in \(H^{\frac{1}{2}}({\mathop {{\mathbb {R}}}\nolimits })\) with mass strictly less than \(4\pi \) or general initial conditions in the weighted Sobolev space \(H^{2, 2}({\mathop {{\mathbb {R}}}\nolimits })\). In this article, we prove that the derivative nonlinear Schrödinger equation is globally well-posed for general Cauchy data in \(H^{\frac{1}{2}}({\mathop {{\mathbb {R}}}\nolimits })\) and that furthermore the \(H^{\frac{1}{2}}\) norm of the solutions remains globally bounded in time. The proof is achieved by combining the profile decomposition techniques with the integrability structure of the equation.
We study the Derivative Nonlinear Schr\"odinger (DNLS). equation for general initial conditions in weighted Sobolev spaces that can support bright solitons (but exclude spectral singularities corresponding to algebraic solitons). We show that the set of such initial data is open and dense in a weighted Sobolev space, and includes data of arbitrarily large $L^2$-norm. We prove global well-posedness on this open and dense set. In a subsequent paper, we will use these results and a steepest descent analysis to prove the soliton resolution conjecture for the DNLS equation with the initial data considered here and asymptotic stability of $N-$soliton solutions.
This article represents a first step toward understanding the long time dynamics of solutions for the Benjamin-Ono equation. While this problem is known to be both completely integrable and globally well-posed in $L^2$, much less seems to be known concerning its long time dynamics. Here, we prove that for small localized data the solutions have (nearly) dispersive dynamics almost globally in time. An additional objective is to revisit the $L^2$ theory for the Benjamin-Ono equation and provide a simpler, self-contained approach.
This article is concerned with the small data problem for the cubic nonlinear
Schr\"odinger equation (NLS) in one space dimension, and short range
modifications of it. We provide a new, simpler approach in order to prove that
global solutions exist for data which is small in $H^{0,1}$. In the same
setting we also discuss the related problems of obtaining a modified scattering
expansion for the solution, as well as asymptotic completeness.
Ill-posedness is established for the initial value problem (IVP) associated to the derivative nonlinear Schrödinger equation for data in $H^s(\mathbb{R}), s < 1/2$ . This result implies that best result concerning local well-posedness for the IVP is in Hs(R), s ≥ 1/2. It is also shown that the (IVP) associated to the generalized Benjamin-Ono equation for data below the scaling is in fact ill-posed.
A nonlinear evolution equation is derived for Alfvén waves propagating along the magnetic field in a cold plasma. The equation provides new types of solitary waves. The phase of a solitary wave is coupled nonlinearly with its amplitude, and the propagation velocity is restricted within the range determined by the asymptotic amplitude and the wave number.
The stability of circularly polarized waves of finite amplitude propagating parallel to the magnetic field is studied. A set of equations for slowly varying waves of arbitrary amplitude is obtained. A discussion of the stability of the waces is based on this set of equations. Earlier results are confirmed; in addition we find that finite amplitude always promotes stability. An amplitude dependent stability condition for long waves, previously obtained by the author, is confirmed.
The envelope formalism for the description of a small-amplitude parallel-propagating Alfvén wave train is tested against direct numerical simulations of the Hall-MHD equations in one space dimension where kinetic effects are neglected. It turns out that the magnetosonic-wave dynamics departs from the adiabatic approximation not only near the resonance between the speed of sound and the Alfvén wave group velocity, but also when the speed of sound lies between the group and phase velocities of the Alfvén wave. The modulational instability then does not anymore affect asymptotically large scales and strong nonlinear effects can develop even in the absence of the decay instability. When the Hall-MHD equations are considered in the long-wavelength limit, the weakly nonlinear dynamics is accurately reproduced by the derivative nonlinear Schrödinger equation on the expected time scale, provided no decay instabilities are present. The stronger nonlinear regime which develops at later time is captured by including the coupling to the nonlinear dynamics of the magnetosonic waves.
A class of self-similar solutions to the derivative nonlinear Schrödinger equations is studied. Especially, the asymptotics of profile functions are shown to posses a logarithmic phase correction. This logarithmic phase correction is obtained from the nonlinear interaction of profile functions. This is a remarkable difference from the pseudo-conformally invariant case, where the logarithmic correction comes from the linear part of the equations of the profile functions.
We show the well-posedness in $H^{\frac 12 }$ of
the Cauchy problem for a certain class of one dimensional nonlinear
Schrödinger equations with the derivative nonlinearity. This is an
improvement of results in $H^1$ by N. Hayashi and T. Ozawa [2,3,4,20].
Our results can cover the derivative nonlinear Schrödinger equation.
Our proof is based on the Fourier restriction norm method and the gauge
transformation.
A pair of coupled, nonlinear, partial differential equations which describe the evolution of low‐frequency, large‐scale‐length perturbations propagating parallel, or nearly parallel, to the equilibrium magnetic field in high‐β plasma have been obtained. The equations account for irreversible resonant particle effects. In the regime of small but finite propagation angles, the pair of equations collapses into a single Korteweg‐de Vries equation (neglecting irreversible terms) which agrees with known results.
A 2×2 matrix linear ordinary differential equation with large parameter τ and irregular singular point of fourth order at infinity is considered. The leading (in τ) order of the monodromy data of this equation is calculated in terms of its coefficients. Isomonodromic deformations of the equation are self-similar solutions of the modified nonlinear Schrödinger equation, and therefore inversion of the expressions obtained for the monodromy data gives the leading term in the time asymptotic behavior of the self-similar solution. The application of these results to the type IV Painlevé equation is considered in detail.
The lifespan of small data solutions for Intermediate Long Wave equation (ILW)
- M Ifrim
- J-C Saut
M. Ifrim and J-C. Saut. The lifespan of small data solutions for Intermediate Long Wave equation
(ILW). Preprint available at: https://arxiv.org/abs/2305.05102, 2023.
Application of the reductive perturbation method to long hydromagnetic waves parallel to the magnetic field in a cold plasma
- E Mjoelhus
E. Mjoelhus. Application of the reductive perturbation method to long hydromagnetic waves parallel
to the magnetic field in a cold plasma. Technical report, Bergen Univ.(Norway). Dept. of Applied
Mathematics, 1974.