MODIFIED SCATTERING FOR BIPOLAR NONLINEAR SCHRÖDINGER–POISSON EQUATIONS

2004 ◽  
Vol 14 (10) ◽  
pp. 1481-1494 ◽  
Author(s):  
CHENGCHUN HAO ◽  
LING HSIAO ◽  
HAILIANG LI
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Behavior ◽  
Small Data ◽  
Scattering Operator ◽  
Smooth Solutions ◽  
Nonlinear Schrödinger ◽  
Poisson Equations ◽  
The Cauchy Problem

In this paper, we study the asymptotic behavior in time and the existence of the modified scattering operator of the globally defined smooth solutions to the Cauchy problem for the bipolar nonlinear Schrödinger–Poisson equations with small data in the space ℝ3.

Singularities of solutions for first order quasilinear hyperbolic systems

1983 ◽  
Vol 94 (1-2) ◽  
pp. 137-147 ◽  
Author(s):  
Lee Da-tsin(Li Ta-tsien) ◽  
Shi Jia-hong
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Hyperbolic Systems ◽  
Smooth Solutions ◽  
Small Initial Data ◽  
Quasilinear Hyperbolic Systems ◽  
First Order ◽  
Global Smooth Solutions ◽  
The Cauchy Problem

SynopsisIn this paper, the existence of global smooth solutions and the formation of singularities of solutions for strictly hyperbolic systems with general eigenvalues are discussed for the Cauchy problem with essentially periodic small initial data or nonperiodic initial data. A result of Klainerman and Majda is thus extended to the general case.

scholarly journals Comparison between the Cauchy problem and the scattering problem for the Landau damping in the Vlasov-HMF equation

2021 ◽  
pp. 1-24
Author(s):  
Dario Benedetto ◽  
Emanuele Caglioti ◽  
Stefano Rossi
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Behavior ◽  
Scattering Problem ◽  
New Method ◽  
Landau Damping ◽  
The Cauchy Problem ◽  
The One ◽  
Perturbative Result

We analyze the analytic Landau damping problem for the Vlasov-HMF equation, by fixing the asymptotic behavior of the solution. We use a new method for this “scattering problem”, closer to the one used for the Cauchy problem. In this way we are able to compare the two results, emphasizing the different influence of the plasma echoes in the two approaches. In particular, we prove a non-perturbative result for the scattering problem.

scholarly journals Asymptotic Behavior of Solutions to the Damped Nonlinear Hyperbolic Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Yu-Zhu Wang
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Behavior ◽  
Hyperbolic Equation ◽  
Contraction Mapping ◽  
Dimensional Space ◽  
Contraction Mapping Principle ◽  
Nonlinear Hyperbolic Equation ◽  
Asymptotic Behavior Of Solutions ◽  
Initial Value ◽  
The Cauchy Problem

We consider the Cauchy problem for the damped nonlinear hyperbolic equation inn-dimensional space. Under small condition on the initial value, the global existence and asymptotic behavior of the solution in the corresponding Sobolev spaces are obtained by the contraction mapping principle.

scholarly journals Asymptotics for the Ostrovsky-Hunter Equation in the Critical Case

2017 ◽  
Vol 2017 ◽  
pp. 1-21
Author(s):  
Fernando Bernal-Vílchis ◽  
Nakao Hayashi ◽  
Pavel I. Naumkin
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Behavior ◽  
Pseudodifferential Operator ◽  
Critical Case ◽  
Large Time ◽  
Asymptotic Behavior Of Solutions ◽  
The Cauchy Problem ◽  
Time Existence ◽  
Time Asymptotic Behavior

We consider the Cauchy problem for the Ostrovsky-Hunter equation ∂x∂tu-b/3∂x3u-∂xKu3=au, t,x∈R2,  u0,x=u0x, x∈R, where ab>0. Define ξ0=27a/b1/4. Suppose that K is a pseudodifferential operator with a symbol K^ξ such that K^±ξ0=0, Im K^ξ=0, and K^ξ≤C. For example, we can take K^ξ=ξ2-ξ02/ξ2+1. We prove the global in time existence and the large time asymptotic behavior of solutions.

scholarly journals Asymptotic expansion of small analytic solutions to the quadratic nonlinear Schrödinger equations in two-dimensional spaces

2002 ◽  
Vol 29 (9) ◽  
pp. 501-516
Author(s):  
Nakao Hayashi ◽  
Pavel I. Naumkin
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Behavior ◽  
Asymptotic Expansion ◽  
Initial Data ◽  
Asymptotic Representation ◽  
Analytic Solutions ◽  
Two Dimensional ◽  
Nonlinear Schrödinger ◽  
Scattering States ◽  
Suitable Norm

We study asymptotic behavior in time of global small solutions to the quadratic nonlinear Schrödinger equation in two-dimensional spacesi∂tu+(1/2)Δu=𝒩(u),(t,x)∈ℝ×ℝ2;u(0,x)=φ(x),x∈ℝ2, where𝒩(u)=Σj,k=12(λjk(∂xju)(∂xku)+μjk(∂xju¯)(∂xku¯)), whereλjk,μjk∈ℂ. We prove that if the initial dataφsatisfy some analyticity and smallness conditions in a suitable norm, then the solution of the above Cauchy problem has the asymptotic representation in the neighborhood of the scattering states.

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