
 
Existence of Global Solutions to the Nonlocal mKdV Equation on the Line 
 
Citation： 
Anran LIU,Engui FAN.Existence of Global Solutions to the Nonlocal mKdV Equation on the Line[J].Chinese Annals of Mathematics B,2024,45(4):497~528 
Page view： 523
Net amount： 249 
Authors： 
Anran LIU; Engui FAN 
Foundation： 
the National Natural Science Foundation of China (No.12271104) 


Abstract： 
In this paper, the authors address the existence of global solutions to the
Cauchy problem for the integrable nonlocal modified Kortewegde Vries (nonlocal mKdV
for short) equation under the initial data u0 ∈ H 3(R)∩H 1,1(R) with the L1(R) smallnorm
assumption. A Lipschitz L2bijection map between potential and reflection coefficient
is established by using inverse scattering method based on a RiemannHilbert 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. 
Keywords： 
Nonlocal mKdV equation, RiemannHilbert problem, Plemelj projection
operator, Lipschitz continuous, Global solutions 
Classification： 
35P25, 35Q51, 35Q15, 35A01, 35G25 

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