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| Existence of Global Solutions to the Nonlocal mKdV Equation on the Line |
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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： 427
Net amount： 213 |
Authors： |
Anran LIU; Engui FAN |
Foundation： |
the National Natural Science Foundation of China (No.12271104) |
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| Abstract： |
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 L1(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. |
Keywords： |
Nonlocal mKdV equation, Riemann-Hilbert problem, Plemelj projection
operator, Lipschitz continuous, Global solutions |
Classification： |
35P25, 35Q51, 35Q15, 35A01, 35G25 |
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