A vertex operator representation of solutions to the Gurevich-Zybin hydrodynamical equation

Nonlinear hydrodynamic equations are of constant interest still from classical works by B. Riemann, who had extensively studied them in general three-dimensional case, having paid special attention to their one-dimensional spatial reduction, for which he devised the generalized method of characteristics and Riemann invariants. These methods appeared to be very effective [1] in investigating many types of nonlinear spatially one-dimensional systems of hydrodynamical type and, in particular, the characteristics method in the form of a "reciprocal" transformation of variables has been used recently in studying a so called Gurevich-Zybin system [2,3] in [8] and a Whitham type system in [6,5]. Moreover, this method was further effectively applied to studying solutions to a generalized [5] (owing to D. Holm and M. Pavlov) Riemann type hydrodynamical system (1.1) D N t u = 0, D t := ∂/∂t + u∂/∂x, where N ∈ Z + and u ∈ C ∞ (R 2 ; R) is a smooth function. Making use of novel methods, devised in [24,7] and based both on the spectral theory [9,16,19,18] and the differential algebra techniques, the Lax type representations for the cases N = 1, 4 were constructed in explicit form.

In this work we are interested in constructing a so called vertex operator representation [13, 14, 21, ?] for solutions to the Gurevich-Zybin hydrodynamical hierarchy (1.1) at N = 2 :

making use an approach recently devised in [22,23] for the case of the classical AKNS hierarchy of integrable flows, and which can be easily generalized for treating the problem for arbitrary integers N ∈ Z + .

We begin with a Lax type linear spectral problem [8,5,4] for the equation (1.1) at N = 2 :

(2.1)

where, by definition, v := D t u, f ∈ L ∞ (R/2πZ; C 2 ) and λ ∈ C is a spectral parameter. Assume that a vector function (u, v) ⊤ ∈ M depends parametrically on the infinite set t := {t 1 , t 2 , t 3 , . . .} ∈ R Z+ in such a way that the generalized Floquet spectrum [9,15,18] σ(ℓ) := {λ ∈ C : sup x∈R ||f (x; λ)|| ∞ < ∞} of the linear problem (2.2) persists in being parametrically iso-spectral, that is dσ(ℓ)/dt j = 0 for all t j ∈ R. The iso-spectrality condition gives rise to a hierarchy of commuting to each other nonlinear bi-Hamiltonian dynamical systems on the functional manifold M in the general form

where K j : M → T (M ) and H j ∈ D(M ), j ∈ Z + , are, respectively, vector fields and conservation laws on the manifold M, which were before described in [5,4,7],

(2.4)

It is well known [15,18,9,16] that the Casimir invariants, determining conservation laws for dynamical systems (2.3), are generated by the suitably normalized monodromy matrix S(x; λ) ∈ End C 2 of the linear problem (2.2)

where F (y, x; λ) ∈ End C 2 is the matrix solution to the Cauchy problems Here G-⊂ G, where G := G+ ⊕ Gis the natural splitting into two affine subalgebras of positive and negative λ-expansions of the centrally extended [15,25] affine current sl(2)-algebra Ĝ := G ⊕ C :

The latter is endowed with the Lie commutator

where the scalar product is defined as

with the vector-functions α ± ∈ C ∞ (R/2πZ; R) satisfying the following determining functional relationships:

as ξ := 1/λ → 0 and existing when the condition ϕ(x, t) := u 2

x -2v x = 0 on the manifold M at t = 0 ∈ R N .

The fundamental matrix F (y, x; λ) ∈ End C 2 can be represented for all x, y ∈ R in the form

Consequently, if one sets y = x + 2π in this formula and defines the expression

satisfies the necessary condition (2.8) as λ → ∞.

Remark 2.1. The invariance of the expression (2.20) with respect to the generating vector field (2.3) on the manifold M derives from the representation (2.19), the equations (2.13) and

which follows naturally from the determining matrix flows (2.12) upon applying the translation y → y + 2π.

The matrix expression (2.21) gives rise to the following important functional relationships:

which allow to introduce in a natural way the vertex operator vector fields (2.24)

acting on an arbitrary smooth function η ∈ C ∞ (R Z+ ; R) by means of the shifting mappings:

(2.25) X ± λ η(x, t 1 , t 2 , …, t j , …) := η ± (x, t; λ) = = η(x, t 1 ± 1/λ, t 2 ± /(2λ 2 ), t 3 ± 1/(3λ 3 )…, t j ± 1/(jλ j ), …) as λ → ∞. Namely, we following proposition holds.

Proposition 2.2. The functional vertex operator expressions α(x, t; λ) = X - λ α(x, t) = α -(x, t; λ), (2.26) β(x, t; λ) = X + λ β(x, t)) = β + (x, t; λ) solve the functional equations (2.18), that is

where t ∈ R Z+ and ξ = 1/λ → 0.

Proof. To state this proposition it is enough to show that the following relationships hold:

for any parameter ξ → 0, where by definition

is a generating evolution vector field. Before doing this we find the evolution equation

on the matrix S(x; µ) as µ, λ → ∞, which entails the following differential relationships:

(2.31) dλ ). Using these relationships (2.31), one can easily obtain by means of simple, but rather cumbersome calculations, the needed relationships (2.28). As their direct consequences

This content is AI-processed based on open access ArXiv data.