By K. Ueno

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37 Lecture 5 Non-linear WKB (Hydrodynamics of Weakly Deformed Solution Lattices). Consider an evolutional system of non-linear differential equations Ψt = K(Ψ, Ψx , Ψxx , . ) where K is a polynomial with constant coefficients in the variables (Ψ, Ψx , . ). Here Ψ may be a vector function. e. functions such that each derivative is much smaller then the previous one: Ψ(n) << Ψ(n−1) << Ψ(n−2) << . . << Ψ. It is convenient to introduce an algebraic “filtration” with the properties f (Ψ) ≥ 0, f (Ψ(k) ) ≥ k, f (AB) ≥ f (A) + f (B).

We are going to introduce now the notion of “Physical Coordinates” or “Liouville” coordinates, which are important in soliton theory. Definition. Let a HTPB be given by the tensor (g pq (u)) and the quantities (bpq k (u)), written in the following Liouville form: there exists a tensor field pq γ (u) such that g pq (u) = γ pq (u) + γ qp (u), bpq k (u) = ∂γ pq (u) ; ∂uk coordinates (u) enjoying this property will be called Liouville or Physical ones. Remarks. 33 1. Obviously for any Lie algebra the natural linear coordinates are Liouville ones because we may define: k γ pq = bpq k u .

The main result is: Lemma. For any HT system for which the coordinates (r1 , . . e. ) the following indentity is true: ∂i W k = Γkki , i = k. Wi − W k All these systems commute (in fact all of them are HT Hamiltonian systems generated by the same HTPB). The last equation might be considered as an equation for finding all the HT flows commuting with each other, provided the symbols Γkki are known. In particular we have for any diagonal metric: Γkki = ∂i log |gk (r)|1/2 , gk = 1/g k (r). 35 As a consequence we have ∂i ∂j W k Wj − W k ∂i W k , i = j = k.

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