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The formation of singularities in solutions to the dispersionlessKadomtsev-Petviashvili (dKP) equation is studied numerically for differentclasses of initial data. The asymptotic behavior of the Fourier coefficients isused to quantitatively identify the critical time and location and the type ofthe singularity. The approach is first tested in detail in 1+1 dimensions forthe known case of the Hopf equation, where it is shown that the break-up of thesolution can be identified with prescribed accuracy. For dissipativeregularizations of this shock formation as the Burgers' equation and fordispersive regularizations as the Korteweg-de Vries equation, the Fouriercoefficients indicate as expected global regularity of the solutions. TheKadomtsev-Petviashvili (KP) equation can be seen as a dispersive regularizationof the dKP equation. The behavior of KP solutions for small dispersionparameter $\epsilon\ll 1$ near a break-up of corresponding dKP solutions isstudied. It is found that the difference between KP and dKP solutions for thesame initial data at the critical point scales roughly as $\epsilon^{2/7}$ asfor the Korteweg-de Vries equation.

Numerical study of shock formation in the dispersionless Kadomtsev-Petviashvili equation and dispersive regularizations