Speaker：Yong Lv（Nanjing University）
Time：14:00-15:00 June 11 ，2020
Title：Space-time resonances and high-frequency Raman instabilities in the two-fluid Euler-Maxwell system
Abstract：We apply the symbolic flow method to the two-fluid Euler-Maxwell system, and show that space-time resonances induce high-frequency Raman and Langmuir instabilities. A consequence is that the Zakharov WKB approximation to Euler-Maxwell is unstable for non-zero group velocities. A key step in the proof is the reformulation of the set of resonant frequencies as the locus of weak hyperbolicity for linearized equations around the WKB solution. Due to large transverse variations in the WKB profile, the equation satisfied by the symbolic flow around resonant frequencies is a linear partial differential equation. At space-time resonances corresponding to the Raman or Langmuir instability, we observe a fast growth of the symbolic flow, which translates into an instability result for the original system.
Speaker：Liming Ling（South China University of Technology）
Time：15:15- 16:15 June 11 ，2020
Title： Infinite order rogue waves of the focusing nonlinear Schrödinger equations
Abstract：In this talk, we would like to introduce the fundamental rogue wave solutions of the focusing nonlinear Schrödinger equation in the limit of large order. Using a recently proposed Riemann–Hilbert representation
of the rogue wave solution of arbitrary order k, we establish the existence of a limiting profile of the rogue wave in the large-k limit when the solution is viewed in appropriate rescaled variables capturing the near-field region where the solution has the largest amplitude. The limiting profile is a new particular solution of the focusing nonlinear Schrödinger equation in the rescaled variables—the rogue wave
of infinite order—which also satisfies ordinary differential equations with respect to space and time. The spatial differential equations are identified with certain members of the Painlevé-III hierarchy. We compute the far-field asymptotic behavior of the near-field limit solution and compare the asymptotic formulas with the exact solution
using numerical methods for solving Riemann–Hilbert problems. In a certain transitional region for the asymptotics, the near-field limit function is described by a specific globally defined tritronquée solution of the Painlevé-II equation. These properties lead us to regard the rogue wave of infinite order as a new special function.（joint with Deniz Bilman and Peter D. Miller）
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