Well-posedness for One-dimensional Derivative Nonlinear Schrödinger Equations
نویسندگان
چکیده
where u = u(t, x) : R → C is a complex-valued wave function, both λ 6= 0 and k > 5 are real numbers. A great deal of interesting research has been devoted to the mathematical analysis for the derivative nonlinear Schrödinger equations [3, 4, 6, 7, 8, 9, 10, 11, 13, 18, 21]. In [13], C. E. Kenig, G. Ponce and L. Vega studied the local existence theory for the Cauchy problem of the derivative nonlinear Schrödinger equations
منابع مشابه
Almost Critical Well-posedness for Nonlinear Wave Equations with Qμν Null Forms in 2d
In this paper we prove an optimal local well-posedness result for the 1+2 dimensional system of nonlinear wave equations (NLW) with quadratic null-form derivative nonlinearities Qμν . The Cauchy problem for these equations is known to be ill-posed for data in the Sobolev space H with s ≤ 5/4 for all the basic null-forms, except Q0, thus leaving a gap to the critical regularity of sc = 1. Follow...
متن کاملWellposedness of Cauchy problem for the Fourth Order Nonlinear Schrödinger Equations in Multi-dimensional Spaces
We study the well-posedness of Cauchy problem for the fourth order nonlinear Schrödinger equations i∂t u=−ε u+ 2u+ P (( ∂ x u ) |α| 2, ( ∂ x ū ) |α| 2 ) , t ∈R, x ∈Rn, where ε ∈ {−1,0,1}, n 2 denotes the spatial dimension and P(·) is a polynomial excluding constant and linear terms. © 2006 Elsevier Inc. All rights reserved.
متن کاملGlobal Well-posedness and Scattering for Derivative Schrödinger Equation
In this paper we mainly study the Cauchy problem for the derivative nonlinear Schrödinger equation in d-dimension (d ≥ 2). We obtain some global well-posedness results with small initial data. The crucial ingredients are L e , L ∞,2 e type estimates, and inhomogeneous local smoothing estimate (L e estimate). As a by-product, the scattering results with small initial data are also obtained.
متن کاملGlobal well-posedness and scattering for the fourth order nonlinear Schrödinger equations with small data
Abstract: For n > 3, we study the Cauchy problem for the fourth order nonlinear Schrödinger equations, for which the existence of the scattering operators and the global well-posedness of solutions with small data in Besov spaces Bs 2,1(R n) are obtained. In one spatial dimension, we get the global well-posedness result with small data in the critical homogeneous Besov spaces Ḃs 2,1. As a by-pr...
متن کاملGlobal Well-posedness for Nonlinear Schrödinger Equations with Energy-critical Damping
We consider the Cauchy problem for the nonlinear Schrödinger equations with energy-critical damping. We prove the existence of global intime solutions for general initial data in the energy space. Our results extend some results from [1, 2].
متن کامل