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Tracing complex singularities with spectral methods. (English) Zbl 0519.76002
Summary: A numerical method for investigating the possibility of blow-up after a finite time is introduced for a large class of nonlinear evolution problems. With initial data analytic in the space variable(s), the solutions have for any $$t > 0$$ complex-space singularities at the edge of an analyticity strip of width $$\delta(t)$$. Loss of regularity corresponds to the vanishing of $$\delta(t)$$. Numerical integration by high resolution spectral methods reveals the large wavenumber behavior of the Fourier transform of the solutions, from which $$\delta(t)$$ is readily obtained. Its time evolution can be traced down to about one mesh length. By extrapolation of $$\delta(t)$$, such numerical experiments provide evidence suggesting finite-time blow-up or all-time regularity. The method is tested on the inviscid and viscous Burgers equations and is applied to the one-dimensional nonlinear Schrödinger equation with quartic potential and to the two-dimensional incompressible Euler equation, all with periodic boundary conditions. In the latter case evidence is found suggesting that existing all-time regularity results can be substantially sharpened.

##### MSC:
 76M99 Basic methods in fluid mechanics 35Q99 Partial differential equations of mathematical physics and other areas of application 76B10 Jets and cavities, cavitation, free-streamline theory, water-entry problems, airfoil and hydrofoil theory, sloshing 76D10 Boundary-layer theory, separation and reattachment, higher-order effects 76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
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