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Dissipation mechanics and exact solutions for nonlinear equations of dissipative type—Principle and application of dissipation mechanics (I)

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How (ceaselessly) Heaven revolves! How (constantly) Earth abides at rest! Do the Sun and the Moon contend about their respective places? Is there someone presiding over and directing those things? Who causes and maintains them without trouble and exertion? Or is there perhaps some secret mechanism in consequence of which they cannot but be as they are? Chuang Tsu, Heaven revolves.

Abstract

This work is the continuation and the distillation of the discussion of Refs. [1-4].

  1. (A)

    From complementarity principle we build up dissipation mechanics in this paper. It is a dissipative theory of correspondence with the quantum mechanics. From this theory we can unitedly handle problems of macroscopic non-equilibrium thermodynamics and viscous hydrodynamics, and handle each dissipative and irreversible problems in quantum mechanics. We prove the basic equations of dissipation mechanics to eigenvalues equations of correspondence with the Schrödinger equation or Dirac equation in this paper.

  2. (B)

    We unitedly merge the basic nonlinear equations of dissipative type, especially the Navier-Stokes equation as a basic equation for macroscopic non-equilibrium thermodynamics and viscous hydrodynamics into integrability condition of basic equation of dissipation mechanics. And we can obtain their exact solutions by the inverse scattering method in this paper.

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Abbreviations

A :

amplitude of wave function

A o :

scalar potential of electromagnetic field

A k (k=1,2,3):

vector potential of electromagnetic field

A(x,t,ζ):

elements of matrix Hamiltonian function (operator)

B(x,t,ζ):

elements of matrix Hamiltonian function (operator)

C(x,t,ζ):

elements of matrix Hamiltonian function (operator)

c :

velocity of light

e :

electric charge of particle

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Authors and Affiliations

  1. Centre of Fundamental Physics, University of Science and Technology of China, Hefei

    Shen Hui-chuan

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  1. Shen Hui-chuan
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Communicated by Chien Wei-zang

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Hui-chuan, S. Dissipation mechanics and exact solutions for nonlinear equations of dissipative type—Principle and application of dissipation mechanics (I). Appl Math Mech 7, 1125–1142 (1986). https://doi.org/10.1007/BF01896976

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