Statistical Foundation of the Fluid Analogue of the Soliton Formalism

In view of the many applications to turbulent phenomena in fluids, plasmas, optics, astrophysics and nerve systems, a large effort has been given to investigate the solitons analytically and numerically. The analyses by means of the nonlinear Zakharov equations and the nonlinear Schrödinger equation have not been successful in obtaining a spectrum of strong turbulence and have concluded that these soliton equations are not suitable for fully developed turbulence. An examination of the mathematical foundation of these nonlinear equations reveals that their derivation had retained the nonlinearity from the modulation and had omitted the non-linearity from the emission of soliton waves by the finite velocity fluctuations. By a fully nonlinear analysis we develop a general soliton formalism for the description of the nonlinear evolution of soliton fluctuations in both plasmas and classical fluids. From the Navier-Stokes equations for plasmas and compressible fluids of two scales (fast and slow waves), we derive two equations of propagation of density waves. A fast soliton field is spontaneously created by rarefaction, and a slow density wave modulates the field intensity as a ponderomotive force. These are the two constitutive properties of solitons. They were axioms in plasmas and are not evident in fluids. The methods of statistical mechanics can be used in two contexts. Firstly, we demonstrate the constitutive properties by means of a Lagrangian-kinetic formalism of the fluctuation-dissipation theory, in self-consistency with the two-scale dynamics of the Navier-Stokes equations. The first of the two properties transforms the hyperbolic equation of propagation of the fast density wave into a parabolic equation of evolution for the field-envelope as driven by the soliton emission from the finite velocity fluctuations. The second property gives the modulation. This gives a statistical foundation to our soliton system in the generalized form of the Zakharov-Schrödinger equations. It serves to describe the microdynamical state of strong turbulence. Secondly in a separate paper, we shall develop a statistical theory of strong soliton turbulence by applying the group-kinetic method to our generalized soliton system. It is evident that the strong turbulence is governed by the reverse and direct cascades, as arising from the nonlinear emission of solitons and the nonlinear modulation, and is unrelated to the division into two scales to which the soliton model belongs.