Volume 46_1-2_(2025)_63-150

On complex algebraic singularities of some genuinely nonlinear PDEs
Denys Dutykh, Éric Leichtnam

Abstract. In this manuscript, we highlight a new phenomenon of complex algebraic singularities formation for solutions of a large class of genuinely nonlinear Partial Differential Equations (PDEs). We start from a unique Cauchy datum which is holomorphic ramified like x11/(k + 1)<\sup> (and its powers) around the smooth locus x1 = 0 and is sufficiently singular. Then, we expect the existence of a solution which should be holomorphic ramified around the singular locus S defined by the vanishing of the discriminant of an algebraic equation of degree k + 1. Notice, moreover, that the monodromy of the Cauchy datum is Abelian, whereas one of the solutions is non-Abelian and that S depends on the Cauchy datum in contrast to the Leray principle (stated for linear problems only). This phenomenon is due to the fact that the PDE is genuinely nonlinear and that the Cauchy datum is sufficiently singular. First, we investigate the case of the inviscid Burgers Equation (iBE). Later, we state a general Conjecture 9.2, which describes the expected phenomenon. We view this Conjecture 9.2 as a working programme allowing us to develop interesting new Mathematics. We also state Conjecture 7.1, which is a particular case of the general Conjecture 9.2 but keeps all the flavour and difficulty of the subject. Then, we propose a new algorithm with a map F such that a fixed point of F would give a solution to the problem associated with Conjecture 7.1. Then, we perform convincing, elaborate numerical tests which suggest that a Banach norm shouldexist for which the mapping F should be a contraction so that the solution (with the above specific algebraic structure) should be unique

Rend. Mat. Appl. (7) 46 (2025) 63-150;  pdf