Propagation of chaos and the Hopf-Cole transformation.

*(English)*Zbl 0606.35041In a previous paper it has been shown that Burgers’ equation constitutes a limit of a contracted N-body problem when N tends to infinity. By proving that the L-operator may be transformed into a Laplacian \(\Delta\), by using an intertwining operator Q, an unsuccesful attempt was made to obtain a linearizing transformation for the Burgers equation in order to achieve a Hopf-Cole transformation.

In this paper the difficulties encountered previously were by-passed and a transformation closely related to the Hopf-Cole transformation was established. The transformation may be used to define the nonlinear evolution equations, which can be obtained by contraction from N-body linear problems, and distinguish the nonlinear equations which can be linearized by using the approximate approach established in this paper.

In this paper the difficulties encountered previously were by-passed and a transformation closely related to the Hopf-Cole transformation was established. The transformation may be used to define the nonlinear evolution equations, which can be obtained by contraction from N-body linear problems, and distinguish the nonlinear equations which can be linearized by using the approximate approach established in this paper.

Reviewer: P.Theocaris

##### MSC:

35K55 | Nonlinear parabolic equations |

35A30 | Geometric theory, characteristics, transformations in context of PDEs |

37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |

35Q99 | Partial differential equations of mathematical physics and other areas of application |

##### Keywords:

propagation of chaos; linearizing transformation; Hopf-Cole transformation; nonlinear evolution equations; contraction; N-body linear problems; approximate approach
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##### References:

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[3] | Hopf, E, The partial differential equation ut + uux = μuxx, Comm. pure appl. math., 3, 201-230, (1950) |

[4] | McKean, H.P, Propagation of chaos for a class of nonlinear parabolic equations, (), 177-194 |

[5] | \scE. Gutkin, Quantum nonlinear Schrödinger equation. I. Intertwining operators, Ann. Inst. Henri Poincaré, to appear. · Zbl 0614.35086 |

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