On complex algebraic singularities of some genuinely nonlinear PDEs

Abstract

In this manuscript, we highlight a new phenomenon of complex algebraic singularity 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 around the smooth locus 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. Notice, moreover, that the monodromy of the Cauchy datum is Abelian, whereas one of the solutions is non-Abelian. Moreover, the singular locus 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. Later, we state a general conjecture that describes the expected phenomenon. We view this Conjecture as a working programme allowing us to develop interesting new Mathematics. We also state another Conjecture 2, which is a particular case of the general Conjecture 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 2. Then, we perform convincing, elaborate numerical tests that suggest that a Banach norm should exist for which the mapping F should be a contraction so that the solution (with the above specific algebraic structure) should be unique. This work is a continuation of Leichtnam (1993).

Keywords

Bibliographic record

BibTeX Citation

@article{Dutykh2025complexalgebraic,
  author = {Dutykh, D. and Leichtnam, E.},
  title = {On complex algebraic singularities of some genuinely nonlinear PDEs},
  journal = {Rend. Mat. Appl.},
  year = {2025},
  volume = {46},
  number = {1-2},
  pages = {63--150},
  abstract = {In this manuscript, we highlight a new phenomenon of complex algebraic singularity 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 around the smooth locus 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. Notice, moreover, that the monodromy of the Cauchy datum is Abelian, whereas one of the solutions is non-Abelian. Moreover, the singular locus 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. Later, we state a general conjecture that describes the expected phenomenon. We view this Conjecture as a working programme allowing us to develop interesting new Mathematics. We also state another Conjecture 2, which is a particular case of the general Conjecture 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 2. Then, we perform convincing, elaborate numerical tests that suggest that a Banach norm should exist for which the mapping F should be a contraction so that the solution (with the above specific algebraic structure) should be unique. This work is a continuation of Leichtnam (1993).},
  keywords = {inviscid Burgers equation, Initial value problems, singular locus, non-Abelian solutions, Nonlinear PDEs, Banach norm, genuinely nonlinear PDEs, fixed point algorithm, working programme, Abelian monodromy, Ramification, contraction mapping, Complex singularities, Cauchy datum, holomorphic ramification, complex algebraic singularities}
}