Speaker
Description
This lecture is cancelled, the talks will resume as usual at 10:20.
I will consider a solution $q(x, t)$ for the focusing nonlinear Schrödinger equation $iq_t + q_{xx} + 2|q|^2 q = 0 $ with initial values $q(x, 0) \approx A_1 e^{i \phi_1} e^{−2iB_1 x}$ as $x → −\infty$ and $q(x, 0) \approx A_2 e^{i\phi_{2}} e^{−2iB_2 x}$ as $x → +\infty$.
I’m interested in its long-time asymptotics. It is qualitatively different in sectors $\xi_{i+1} < \xi := \frac{x}{t} < \xi_{i}$ of the $(x, t)$ half-plane and the goal is to determine these sectors and the asymptotics of $q$ in each of them.
I will concentrate on the shock case ($B_1 < B_2$ ). The case $B_1 = B_2$ has already been studied by Biondini and Mantzavinos (CPAM 2017) and the rarefaction case ($B_2 < B_1$ ) is close to the case $A_1 = 0$ studied in a paper with Kotlyarov and Shepelsky (IMRN 2011).
The shock case has already been considered by Buckingham and Venakides (CPAM 2007). I will show it is actually rich in asymptotic scenarios. I will present these different scenarios. They depend on the relative values of the parameters $A_j , B_j$ . (This is joint work with Jonatan Lenells and Dmitry Shepelsky.)
