Existence of Global Solutions to the Nonlocal mKdV Equation on the Line  

作　　者：Anran LIU Engui FAN 

机构地区：[1]School of Mathematical Sciences,Fudan University,Shanghai 200433,China

出　　处：《Chinese Annals of Mathematics,Series B》2024年第4期497-528,共32页数学年刊（B辑英文版）

基　　金：supported by the National Natural Science Foundation of China(No.12271104)。

摘　　要：In this paper,the authors address the existence of global solutions to the Cauchy problem for the integrable nonlocal modified Korteweg-de Vries(nonlocal mKdV for short)equation under the initial data u0∈H^(3)(R)∩H^(1,1)(R)with the L^(1)(R)small-norm assumption.A Lipschitz L2-bijection map between potential and reflection coefficient is established by using inverse scattering method based on a Riemann-Hilbert problem associated with the Cauchy problem.The map from initial potential to reflection coefficient is obtained in direct scattering transform.The inverse scattering transform goes back to the map from scattering coefficient to potential by applying the reconstruction formula and Cauchy integral operator.The bijective relation naturally yields the existence of global solutions in a Sobolev space H^(3)(R)∩H^(1,1)(R)to the Cauchy problem.

关 键 词：Nonlocal mKdV equation Riemann-Hilbert problem Plemelj projection operator Lipschitz continuous Global solutions 

分 类 号：O175[理学—数学]