Abstract: When a PDE that generates an analytic semiflow blows up, its solutions may be continued in complex time around the singularity potentially producing a branched Riemann surface. The work Cho et al. [Jpn. J. Ind. Appl. Math. 33 (2016): 145-166] investigated this phenomena for the quadratic heat equation $u_t=u_{xx}+u^2$. When solutions are continued for purely imaginary time, a nonlinear Schr\"odinger equation (NLS) $i{u}_t=u_{xx}+u^2$ for $x\in \mathbb{T}$ is obtained, and the authors conjectured that this NLS is globally well-posed for real initial data. In this talk I will discuss how, by using a mix of analytical and computer-assisted techniques, we have shown that this equation exhibits rich dynamical structure punctuated by (presumably unstable)  blowup solutions. It is also of note that the nonlinearity here, a complex quadratic,  is essentially the same nonlinearity as in the Constantin-Lax-Majda equation, a 1D model for incompressible fluids.

 

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Organizers: 

Chueh-Hsin Chang (CCU), Jia-Yuan Dai (NTHU), Bo-Chih Huang (CCU), Chih-Chiang Huang (CCU), Chang-Hong Wu (NYCU) & Yuya Tokuta (Kyoto U.)