Dispersionless BKP Hierarchy and Quadrant L\"owner Equation

We show that $N$-variable reduction of the dispersionless BKP hierarchy is described by a L\"owner type equation for the quadrant.


Introduction
Since an unexpected connection of the dispersionless KP hierarchy (the Benney equations) to the chordal Löwner equation was found in the seminal paper [9], similar examples have been found by [14,15,25,26] and others. The dispersionless integrable hierarchies are obtained as quasi-classical limits of "dispersionful" integrable hierarchies and thus the Lax operators of the latter are replaced by Lax "functions" (see, for example, [23]).
On the other hand, Löwner type equations are differential equations characterising oneparameter families of conformal mappings between families of domains with growing slits and fixed reference domains (the upper half plane for the chordal Löwner equation and the unit disk or its exterior for the radial Löwner equation). For details of the Löwner equations in the context of complex analysis we refer to [1,7,19].
The key point of the "unexpected" relation is that the Lax functions satisfying the onevariable reduction condition are solutions of the Löwner type equations. In order to explain more precisely, let us recall the first and most typical example found in [9]: the relation between the dispersionless KP hierarchy and the chordal Löwner equation. We follow formulation in [26]. Suppose that L KP (w; t) is a solution of the dispersionless KP hierarchy, which depends on t = (t 1 , t 2 , t 3 , . . . ) only through a single variable λ. Namely, there exists a function f KP (w; λ) = ∞ n=0 u n (λ)w 1−n (u 0 = 1, u 1 = 0) of two variables (w, λ) and a function λ(t) of t such that L KP (w; t) = f KP (w; λ(t)) = ∞ n=0 u n (λ(t))w 1−n =w + u 2 (λ(t))w −1 + u 3 (λ(t))w −2 + · · · .
Then the inverse function g KP (z; λ) of f KP (w; λ) inw satisfies the chordal Löwner equation 1 (5.2) (g = g KP ,ũ = u 2 ) with respect to λ. There is also a way back and we can construct a solution of the dispersionless KP hierarchy from a solution of the chordal Löwner equation. Similar connections of the dispersionless mKP/Toda hierarchies, the Dym hierarchy, the universal Whitham hierarchy to the various Löwner type equations were found in [14,15,24,25,26]. This paper is a contribution to the Special Issue in honor of Anatol Kirillov and Tetsuji Miwa. The full collection is available at http://www.emis.de/journals/SIGMA/InfiniteAnalysis2013.html 1 The chordal Löwner equation was first introduced in [12] and rediscovered independently by Gibbons and Tsarev in [9] and by Schramm in [20].
The goal of the present paper is to add another example: we show that, in a similar sense, the one-variable reduction of the dispersionless BKP hierarchy, a variant of the dispersionless KP hierarchy, is characterised by the quadrant Löwner equation (3.2), which is satisfied by a family of conformal mappings from a quadrant slit domain to the quadrant. It is desirable to explain fundamental reason of such relation between dispersionless integrable hierarchies and Löwner type equations in complex analysis.
The other direction, namely, construction of a solution of the dispersionless BKP hierarchy from a solution of the quadrant Löwner type equation, can be generalised to the N -variable case. The celebrated Gibbons-Tsarev system (see, for example, [8,9,16,17]) arises as the compatibility condition of the parameter functions in the quadrant Löwner equation. Our Gibbons-Tsarev system is a particular case in [18] and analysed there in more general context.
Let us comment several differences from previously known results, which are due to our restricting ourselves to the dispersionless BKP hierarchy specifically and not studying general hydrodynamic type integrable systems comprehensively. (Some of the comments also apply to the reduction of similar systems, which we studied earlier in [24,25,26].) • We use the relation of the Grunsky coefficients and the dispersionless tau function (2.7) systematically as in [24,25,26]. We believe that this makes proofs transparent.
• The quadrant Löwner equation (3.2) can be obtained if we put q = g 2 , B 0 = 2u and Instead of studying more general equations in [18], we identify our particular equation (3.2) as the Löwner type equation for the conformal mappings of quadrant domains, which makes it possible to construct an explicit example of solutions by means of complex analysis in Section 6.
• Although Tsarev's generalised hodograph method is mostly applied to the (1+1)-or (2+1)dimensional systems (for example, in [16,17,28]), there is no need to restrict the number of the independent variables of the dispersionless hierarchy. We show in this paper that the compatibility condition for Tsarev's generalised hodograph method is satisfied for any independent variables of the dispersionless BKP hierarchy simultaneously. See Lemma 4.1 and its proof for details. This also follows from the method by Kodama and Gibbons [11].
This paper is organised as follows. In Section 2 we review the results on the dispersionless BKP hierarchy. Using its dispersionless Hirota equation, we prove that the one-variable reduction of the dispersionless BKP hierarchy is described by the quadrant Löwner equation in Section 3. Conversely we prove in Section 4 that a solution of the dispersionless BKP hierarchy can be constructed from a function depending on N parameters satisfying the quadrant Löwner equations. In Section 5 we explain that the quadrant Löwner equation is obtained by "folding" the chordal Löwner equation by the square root, w = w − 2u(λ), where 2u(λ) is the suitably chosen centre of the folding. This shows that the quadrant Löwner equation is satisfied by slit mappings between quadrant domains. The last Section 6 is devoted to the simplest non-trivial example.

Dispersionless BKP hierarchy
The dispersionless BKP hierarchy (the dBKP hierarchy for short) was first introduced probably by Kupershmidt in [13] as a special case of the Kupershmidt hydrodynamic chain. The formulation of the dBKP hierarchy as a dispersionless limit of the BKP hierarchy by Date, Kashiwara and Miwa [5], which is relevant to us, was due to Takasaki [21,22] and subsequently studied by Bogdanov and Konopelchenko [3], Chen and Tu [4]. The theory can be developed almost in the same way as that of the dispersionless KP hierarchy in [23]. We refer further developments from the viewpoint of the Kupershmidt hydrodynamic chain to [2,13] and [18, especially § 10] (and references therein).
Here we briefly review necessary facts on the dBKP hierarchy, following mainly [21,22]. Let L(w; t) be a (formal) Laurent series in w of the form the coefficients of which depend on t = (t 1 , t 3 , t 5 , . . . ). We identify another independent variable x with t 1 . The truncation operation (·) ≥0 of the Laurent series is defined by n∈Z a n w n ≥0 = n≥0 a n w n .
The Poisson bracket {a(w; x), b(w; x)} is defined in the same way as the dispersionless KP hierarchy The dispersionless BKP hierarchy is the system of Lax equations where B n := (L n ) ≥0 . We call L the Lax function of the dBKP hierarchy. What we need later is the dispersionless Hirota equation of the tau function, which is equivalent to the above Lax representation. In order to define the tau function and also for later use, we introduce several terms in complex analysis. (See [7,19] for details in the context of complex analysis.) The Faber polynomials Φ n (z) and the Grunsky coefficients b mn are defined for a function (or a Laurent series 3 ) of the form The Faber polynomial Φ n (w) is a monic n-th order polynomial in w, coefficients of which are polynomials in b k 's. The Grunsky coefficient b mn is a polynomial in b k 's and symmetric in indices m and n. They are connected by the relation When g(z) is an odd function, i.e., b even = 0, we have Hence the Faber polynomial Φ n (w) is an odd (resp. even) polynomial in w for odd (resp. even) n, and consequently Φ n (g(z)) is an odd (resp. even) function in z. Therefore the relation (2.4) shows that b nm = 0 if n and m are of different parity. We can pick up the Faber polynomials and the Grunsky coefficients with odd indices by the following generating functions In these terms the dispersionless BKP hierarchy is stated in the following way.
for any odd m and n.
The function F (t) is called the free energy or the logarithm of the tau function τ dBKP (t): F (t) = log τ dBKP (t). The proof of the proposition is the same as the corresponding statements for the dispersionless KP hierarchy in § 3.3 of [27].
Since the first Faber polynomial Φ 1 (w; t) of k(z; t) is w, the relation (2.4) means Substituting this expression into (2.6), the above condition (2.7) is equivalent to the following equation where the operator D(z) is defined by Equation (2.9) is the dispersionless Hirota equation for the dBKP hierarchy first obtained in [3] (see also [22]).
3 From the dBKP hierarchy to the quadrant Löwner equation One of the main theorems of the present paper is the following.
Theorem 3.1. Suppose that L(w; t) is a solution of the dBKP hierarchy whose dependence on t = (t 1 , t 3 , . . . ) is only through a single variable λ. Namely, there exists a function f (w; λ) of w and λ of the form and a function λ(t) of t such that L(w; t) = f (w; λ(t)).
We call the equation (3.2) the quadrant Löwner equation by the reason which we shall explain in Section 5. As in the case of the chordal Löwner equation, we call the function V (λ) the driving function.

From the quadrant Löwner equation to the dBKP hierarchy
In this section we prove the converse statement of Theorem 3.1 in a generalised form (N -variable diagonal reduction). Let f (w; λ) be a function of the following form where λ = (λ 1 , . . . , λ N ) is a set of N variables. We denote its inverse function in w by g(z; λ): f (g(z; λ); λ) = z, g(f (w; λ); λ) = w. Let us consider a system of differential equations for g(z; λ) of the type (3.2) in each variable λ i , namely, the quadrant Löwner equations with a driving function V i for i = 1, . . . , N , where u = u 1 . Or equivalently we consider a system of linear partial differential equations for f When N 2, the compatibility conditions [∂ λ i − A i , ∂ λ j − A j ] = 0 of this linear system boil down to the following equations for V i (or V 2 i ) and u for i, j = 1, . . . , N , i = j. This is the Gibbons-Tsarev system in our case, which is well-known in the literature. (See the remark below.) We assume that these conditions hold and that g and f satisfy (4.1) and (4.2) respectively.
The converse statement to Theorem 3.1 is the following.
This is the so-called diagonal N -variable reduction. We shall discuss the solvability of (4.5) later.

Remark 4.2.
We can consider more general N -variable reduction, changing the equation (4.1) to ∂g ∂λ i (z; λ) = R i (g(z; λ); λ) and correspondingly the equation (4.2) to where the coefficient R i (w; λ) is an odd rational function with 2N simple poles at w = ±V j (λ) (j = 1, . . . , N ) which vanishes at ∞: R i (w; λ) → 0 (w → ∞). In other words, Because of the equation Proof of Theorem 4.1. This is proved in the same way as Proposition 5.5 of [26]. From (2.6) it follows that −2 1≤m,n, odd under the assumption (4.1). The right-hand side is rewritten as by virtue of (3.6) and (4.5). Therefore (4.7) implies that mn ∂b mn /∂t k is symmetric in (m, n, k), from which follows the existence of a function F (t) satisfying (2.7). According to Proposition 2.1, L(w; t) = f (w; λ(t)) is a solution of the dBKP hierarchy.
The system of the first order partial differential equations (4.5) can be solved by Tsarev's generalised hodograph method [28].
This is essentially Theorem 10 of [28], which is well-known to experts, but since the number of the independent variables are infinite in our case, we briefly review the proof and check that it really works. To handle the infinite number of equations simultaneously, we make use of the generating function of the Faber polynomials. We also note that this lemma also follows directly from the generalised hodograph method in the slightly improved version made by Kodama and Gibbons [11] for finitely many independent variables.
Proof . This proof is parallel to the proof of Lemma 4.1 of [24]. First we prove that χ i,n (n is odd) satisfies the same equation (4.8) as R i . Since the generating function for ∂χ i,n /∂λ j is obtained by differentiating (3.6). The result is Substituting the compatibility condition (4.3) and the quadrant Löwner equations (4.1), we obtain 1≤n, odd On the other hand, it follows from (3.6) and the definition of χ i,n that 1≤n, odd Comparing (4.10) and (4.11), we have for odd n and i, j = 1, . . . , N , i = j. An important point is that this equation holds for all n simultaneously. Namely, the coefficient of the equation (4.12) does not depend on n.
(i) Lengthy computation with the help of (4.3) and (4.4) shows that The right-hand side is symmetric in (j, k). Hence we have This together with the expression (4.12) is the compatibility condition of (4.8), as shown in § 3 of [28].
(ii) By differentiating the relation (4.9) by t 1 and t k (k is odd) we obtain where Because of (4.8), (4.12) and the hodograph relation (4.9) the above expression becomes (The fact that the coefficient Γ ij does not depend on n is essential here.) Therefore (4.13) reduces to This proves (4.5).

Quadrant Löwner equation
In this section we show that the quadrant Löwner equation (3.2) is satisfied by slit mappings between quadrant domains. In fact this equation is connected to the chordal Löwner equation as follows.
Here the driving function (ii) Conversely, letg(z; λ) =z +ũ(λ)z −1 + · · · be a solution of the chordal Löwner equation (5.2) with the driving function U (λ). Let V (λ) be a solution of the following ordinary differential equation If we define u(λ) by As is well known (cf., for example, [1]), a one-parameter family of conformal mappings from a slit domain to the upper half plane satisfies the chordal Löwner equation 4 . Hence the equation (3.2) is essentially an equation for conformal mappings from a slit domain to the quadrant (Fig. 1). By this reason we call (3.2) the quadrant Löwner equation.
Actually, we can "fold" the functiong(z 2 ; λ) at any point c(λ) on the real axis instead of 2u(λ) to obtain the conformal mapping to the quadrant: g c (z; λ) = g(z 2 ; λ) − c(λ). As is shown in the proof, the shift by 2u(λ) is necessary to normaliseg by the hydrodynamic normalisation condition.
Proof . Both statements are consequences of straightforward computation.
Hence summarising, we have (5.6) By comparing the coefficients ofz −1 we have (5.3), which implies (5.2) from (5.6). The proof of (ii) is similar computation of a Laurent series and derivation. The point is that the conditions (5.4) and (5.5) lead to the equation (5.3).

Example
Here, as an application of Theorem 4.1 (N = 1) and Proposition 5.1, we construct an example of a solution of the dBKP hierarchy, starting from a solution of the chordal Löwner equation.