Abstract. Purely dispersive partial differential equations as the Korteweg-de Vries equation, the nonlinear Schrödinger equation and higher dimensional generalizations thereof can have solutions which develop a zone of rapid modulated oscillations in the region where the corresponding dispersionless equations have shocks or blow-up. To numerically study such phenomena, fourth order time-stepping in combination with spectral methods is beneficial to resolve the steep gradients in the oscillatory region. We compare the performance of several fourth order methods for the KadomtsevPetviashvili and the Davey-Stewartson equations, two integrable equations in 2+1 dimensions: exponential time-differencing, integrating factors, time-splitting, implicit Runge-Kutta and Driscoll's composite Runge-Kutta method. The accuracy in the numerical conservation of integrals of motion is discussed.

Abstract. The formation of singularities in solutions to the dispersionless Kadomtsev-Petviashvili (dKP) equation is studied numerically for different classes of initial data. The asymptotic behavior of the Fourier coefficients is used to quantitatively identify the critical time and location and the type of the singularity. The approach is first tested in detail in 1 + 1 dimensions for the known case of the Hopf equation, where it is shown that the break-up of the solution can be identified with prescribed accuracy. For dissipative regularizations of this shock formation as the Burgers' equation and for dispersive regularizations as the Korteweg-de Vries equation, the Fourier coefficients indicate as expected global regularity of the solutions. The Kadomtsev-Petviashvili (KP) equation can be seen as a dispersive regularization of the dKP equation. The behavior of KP solutions for small dispersion parameter 1 near a break-up of corresponding dKP solutions is studied. It is found that the difference between KP and dKP solutions for the same initial data at the critical point scales roughly as 2/7 as for the Korteweg-de Vries equation.

Abstract. We present the first detailed numerical study of the semiclassical limit of the Davey-Stewartson II equations both for the focusing and the defocusing variant. We concentrate on rapidly decreasing initial data with a single hump. The formal limit of these equations for vanishing semiclassical parameter , the semiclassical equations, are numerically integrated up to the formation of a shock. The use of parallelized algorithms allows to determine the critical time tc and the critical solution for these 2 + 1-dimensional shocks. It is shown that the solutions generically break in isolated points similarly to the case of the 1 + 1-dimensional cubic nonlinear Schrödinger equation, i.e., cubic singularities in the defocusing case and square root singularities in the focusing case. For small values of , the full Davey-Stewartson II equations are integrated for the same initial data up to the critical time tc. The scaling in of the difference between these solutions is found to be the same as in the 1 + 1 dimensional case, proportional to 2/7 for the defocusing case and proportional to 2/5 in the focusing case.We document the Davey-Stewartson II solutions for small for times much larger than the critical time tc. It is shown that zones of rapid modulated oscillations are formed near the shocks of the solutions to the semiclassical equations. For smaller , the oscillatory zones become smaller and more sharply delimited to lens shaped regions. Rapid oscillations are also found in the focusing case for initial data where the singularities of the solution to the semiclassical equations do not coincide. If these singularities do coincide, which happens when the initial data are symmetric with respect to an interchange of the spatial coordinates, no such zone is observed. Instead the initial hump develops into a blow-up of the L∞ norm of the solution. We study the dependence of the blow-up time on the semiclassical parameter .

Nonlinear dispersive partial differential equations such as the nonlinear Schrödinger equations can have solutions that blow-up. We numerically study the long time behavior and potential blowup of solutions to the focusing Davey-Stewartson II equation by analyzing perturbations of the lump and the Ozawa exact solutions. It is shown in this way that the lump is unstable to both blowup and dispersion, and that blowup in the Ozawa solution is generic.

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