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).

翻译：本文中，我们揭示了一类广泛真正非线性偏微分方程（PDEs）解中复代数奇点形成的新现象。我们从唯一的柯西数据出发，该数据在光滑轨迹周围呈分枝全纯态且具有充分奇异性。随后，我们预期存在一个解，该解在由代数方程判别式消失所定义的奇异轨迹S周围呈分枝全纯态。此外，注意到柯西数据的单值性是阿贝尔的，而解的单值性是非阿贝尔的。同时，奇异轨迹S依赖于柯西数据，这与（仅针对线性问题陈述的）勒雷原理形成对比。这一现象源于偏微分方程是真正非线性的，且柯西数据具有充分奇异性。首先，我们研究无粘伯格斯方程的情形。随后，我们提出一个描述预期现象的通用猜想。我们将该猜想视为一个工作纲领，用以发展有趣的新数学。我们还提出了另一个猜想2，它是通用猜想的一个特例，但保留了该主题的所有特征和难度。接着，我们提出一个带映射F的新算法，使得F的不动点将给出与猜想2相关问题的解。然后，我们进行了令人信服且精细的数值测试，结果表明应存在一个巴拿赫范数，使得映射F在该范数下是压缩的，从而解（具有上述特定代数结构）应当是唯一的。这项工作是Leichtnam (1993) 的延续。