Asymptotics of solutions to the fourth order Schrodinger type equation with a dissipative nonlinearity

In this paper, the asymptotic behavior in time of solutions to the one-dimensional fourth order nonlinear Schrodinger type equation with a cubic dissipative nonlinearity λ|u| 2 u ,w hereλ is a complex constant satisfying Im λ< 0, is studied. This nonlinearity is a long-range inter- action. The local Cauchy problem at infinite initial time (the final value problem) to this equation is solved for a given final state with no size restriction on it. This implies the existence of a unique solution for the equation approaching some modified free dynamics as t → +∞ in a suit- able function space. Our modified free dynamics decays like (tlog t) −1/2 as t →∞ .

[1]  Akihiro Shimomura,et al.  Asymptotic Behavior of Solutions for Schrödinger Equations with Dissipative Nonlinearities , 2006 .

[2]  N. Hayashi,et al.  Domain and Range of the Modified Wave Operator for Schrödinger Equations with a Critical Nonlinearity , 2006 .

[3]  J. Segata.  Remark on well-posedness for the fourth order nonlinear Schrödinger type equation , 2004 .

[4]  瀬片 純市.  Well-posedness for the Fourth Order Nonlinear Schrodinger Type Equation Related to the Vortex Filament (流体と気体の数学解析 研究集会報告集) , 2003 .

[5]  R. Carles.  Geometric Optics and Long Range Scattering¶for One-Dimensional Nonlinear Schrödinger Equations , 2001 .

[6]  Jean-Claude Saut,et al.  Dispersion estimates for fourth order Schr?odinger equations , 2000 .

[7]  J. Ginibre,et al.  Long Range Scattering and Modified Wave Operators for some Hartree Type Equations II , 1998, math/9903073.

[8]  Nakao Hayashi,et al.  Asymptotics for large time of solutions to the nonlinear Schrödinger and Hartree equations , 1998 .

[9]  J. Ginibre,et al.  Long range scattering for non-linear Schrödinger and Hartree equations in space dimensionn≥2 , 1993 .

[10]  Tohru Ozawa,et al.  Long range scattering for nonlinear Schrödinger equations in one space dimension , 1991 .

[11]  Jacqueline E. Barab,et al.  Nonexistence of asymptotically free solutions for a nonlinear Schrödinger equation , 1984 .

[12]  A. Shimomura,et al.  GLOBAL EXISTENCE AND ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO THE FOURTH ORDER ̈ NONLINEAR SCHR ODINGER TYPE EQUATION , 2007 .

[13]  Luis Vega,et al.  Oscillatory integrals and regularity of dispersive equations , 1991 .

[14]  A. Shimomura,et al.  GLOBAL EXISTENCE AND ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO THE FOURTH ORDER NONLINEAR SCHRÖDINGER TYPE EQUATION , 2022 .