TY - JOUR
T1 - THE DERIVATIVE NONLINEAR SCHR̈ODINGER EQUATION
T2 - GLOBAL WELL-POSEDNESS AND SOLITON RESOLUTION
AU - JENKINS, ROBERT
AU - LIU, JIAQI
AU - PERRY, PETER
AU - SULEM, CATHERINE
N1 - Publisher Copyright:
© (2019) Brown University
PY - 2020
Y1 - 2020
N2 - We review recent results on global well-posedness and long-time behavior of smooth solutions to the derivative nonlinear Schr̈odinger (DNLS) equation. Using the integrable character of DNLS, we show how the inverse scattering tools and the method of Zhou [SIAM J. Math. Anal. 20 (1989), pp. 966–986] for treating spectral singularities lead to global well-posedness for general initial conditions in the weighted Sobolev space H2,2pRq. For generic initial data that can support bright solitons but exclude spectral singularities, we prove the soliton resolution conjecture: the solution is asymptotic, at large times, to a sum of localized solitons and a dispersive component, Our results also show that soliton solutions of DNLS are asymptotically stable.
AB - We review recent results on global well-posedness and long-time behavior of smooth solutions to the derivative nonlinear Schr̈odinger (DNLS) equation. Using the integrable character of DNLS, we show how the inverse scattering tools and the method of Zhou [SIAM J. Math. Anal. 20 (1989), pp. 966–986] for treating spectral singularities lead to global well-posedness for general initial conditions in the weighted Sobolev space H2,2pRq. For generic initial data that can support bright solitons but exclude spectral singularities, we prove the soliton resolution conjecture: the solution is asymptotic, at large times, to a sum of localized solitons and a dispersive component, Our results also show that soliton solutions of DNLS are asymptotically stable.
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U2 - 10.1090/qam/1553
DO - 10.1090/qam/1553
M3 - Article
AN - SCOPUS:85085750369
SN - 0033-569X
VL - 78
SP - 33
EP - 73
JO - Quarterly of Applied Mathematics
JF - Quarterly of Applied Mathematics
IS - 1
ER -