Asymptotics for the higher-order derivative nonlinear Schrödinger equation
This paper is concerned with global existence and calculate the large time asymptotics of solutions for the derivative higher-order nonlinear Schrödinger equation with initial data \[\begin{cases} i\partial_tv+\frac{a}{2}\partial_x^2v-\frac{b}{4}\partial_x^4v=(\overline{\partial_xv})^2,~~~t>1, x\in\mathbb R,\\ v(1,x)=v_0(x),~~x\in\mathbb R, \end{cases} \tag{e1.1}\] where \(a, b>0\). Some difficulties are overcome by the authors via the method of paper [\textit{N. Hayashi} and the first author, Electron. J. Differ. Equ. 2008, Paper No. 15, 38 p. (2008; Zbl 1136.35324); ``A quadratic nonlinear Schrödinger equation in one space dimension'', J. Differ. Equ. 186, 165--185 (2002)], and the method similar to the normal forms of \textit{J. Shatah} [Commun. Pure Appl. Math. 38, 685--696 (1985; Zbl 0597.35101)] to transform the quadratic nonlinearity to critical cubic nonlinearities involving bilinear operators, and by developing the factorization technique originated in papers [\textit{N. Hayashi} and \textit{T. Ozawa}, Ann. Inst. Henri Poincaré, Phys. Théor. 48, No. 1, 17--37 (1988; Zbl 0659.35078); \textit{N. Hayashi} and the first author, J. Math. Phys. 56, No. 9, Article ID 093502, 25 p. (2015; Zbl 1328.35212); Z. Angew. Math. Phys. 59, No. 6, 1002--1028 (2008; Zbl 1190.35199)]. In such a way, the authors can show below that the nonlinearity in equation (1.1) due to it's oscillating structure acts as a critical one for large time, and some related results are generalized.
derivative nonlinear Schrödinger equation
scientific article; zbMATH DE number 7359955
scientific article