Tzitzeica solitons vs. relativistic Calogero-Moser 3-body clusters

(1.1) has a curious history. It first arose a century ago in the work of the Rumanian mathematician Tzitzeica [1,2]. He arrived at it from the viewpoint of the geometry of surfaces, obtaining an associated linear representation and a Bäcklund transformation. For many decades after Tzitzeica’s work, the equation (1.1) was not studied, two papers by Jonas [3,4] being a notable exception. Thirty years ago, it was reintroduced within the area of soliton theory, independently by Dodd and Bullough [5] and Zhiber and Shabat [6], cf. also Mikhailov’s paper [7]. In this setting, (1.1) is viewed as an integrable relativistic theory for a field Ψ(t, y) in two space-time dimensions, written in terms of light cone (characteristic) coordinates, t = u -v, y = u + v.

(1.2)

Accordingly, the PDE (1.1) is known under various names, and has been studied from several perspectives, including geometry [1][2][3][4], classical soliton theory [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20], and quantum soliton theory [21][22][23][24][25][26]. Moreover, it has shown up within the context of gas dynamics [27,28].

The principal aim of this paper is to tie in a class of soliton solutions to the Tzitzeica equation (1.1) with integrable particle dynamics of relativistic Calogero-Moser type. (A survey covering both relativistic and nonrelativistic Calogero-Moser systems can be found in [29].) The intimate relation of the latter integrable particle systems to soliton solutions of various evolution equations (including the sine-Gordon, Toda lattice, KdV and modified KdV equations) was already revealed in the paper in which they were introduced [30], and was elaborated on in [31]. Later on, the list of equations whose soliton solutions are connected to the relativistic Calogero-Moser systems was considerably enlarged [32][33][34]. In all of these cases, the N solitons correspond to N point particles.

The novelty of the present soliton-particle correspondence is that the Tzitzeica Nsoliton solutions at issue correspond to an integrable reduction of the 3N -body relativistic Calogero-Moser dynamics. Physically speaking, a Tzitzeica soliton may be viewed as a lowest energy bound state of three Calogero-Moser ‘quarks’, one of which has negative charge, whereas the other two have positive charge.

A crucial ingredient for establishing the correspondence is the relation between an extensive class of 2D Toda solitons and the relativistic Calogero-Moser systems, already studied in [33]. Indeed, the relation can be combined with the link between the Tzitzeica equation and the 2D Toda equation. The latter link has been known for quite a while, and we proceed to sketch it in a form that suits our later requirements.

Assume that φ n is a solution to the 2D Toda equation in the form [35] φ n,uv = exp(φ n -φ n-1 ) -exp(φ n+1 -φ n ), n ∈ Z, (1.3) which has the symmetry property

and which is moreover 3-periodic, i.e., φ n+3 = φ n .

(1.5)

Then one has in particular φ 0 = 0, φ 2 = -φ 1 , (1.6) so that Ψ = φ 1 (1.7)

satisfies (1.1). Conversely, a solution Ψ to (1.1) yields a solution φ n to (1.3) satisfying (1.4) and (1.5) when one sets

The point is now that there exist soliton solutions to (1.3) that can be made to satisfy the extra requirements (1.4)- (1.5), hence yielding soliton solutions to (1.1). The relevant 2D Toda solitons are those found by the Kyoto school [36,37]. These solitons also formed the starting point for [33]. They are most easily expressed in tau-function form, the relation of τ n to φ n being given by φ n = ln(τ n+1 /τ n ), n ∈ Z.

(1.9)

In terms of τ n , the evolution equation becomes

and the extra features (1.4) and (1.5) amount to

and τ n+3 = τ n .

(1.12)

As we show in Section 2, one can make special parameter choices in the 2D Toda 2N -soliton solutions τ n (u, v) so that they satisfy (1.11)- (1.12). The function Ψ = ln(τ 2 /τ 1 )

(1.13) then satisfies (1.1), and can be viewed as a Tzitzeica N -soliton solution. (In Lie algebraic terms, the successive requirements (1.11) and (1.12) amount to reductions

2 .) More specifically, the 2D Toda 2N -solitons of Section 2 are of the form

where the dependence of the 2N × 2N (Cauchy type) matrix C and diagonal matrix D on the parameters a, b, ξ 0 ∈ C 2N is suppressed. To satisfy the B ∞ restriction (1.11) and to prepare for the 3-periodicity restriction (1.12), these 6N parameters are expressed in terms of 2N parameters φ, θ ∈ C N and a coupling parameter c. We then show that the tau-functions have period l for c equal to π/l, so that the Tzitzeica restrictions are satisfied for c = π/3.

In Section 3 we make a further parameter change, trading φ 1 , . . . , φ N for ‘positions’ q 1 , . . . , q N . This reparametrization ensures in particular that the summand involving all exponentials has coefficient 1. Restricting attention to the parameter set

the tau-functions take their simplest and most natural form. In particular, for parameter

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