Matrix extension of the Manakov-Santini system and integrable chiral model on Einstein-Weyl background

TL;DR

This paper introduces a matrix extension of the Manakov-Santini system, linking it to integrable chiral models on Einstein-Weyl backgrounds, and develops associated hierarchies and dressing schemes.

Contribution

It presents a novel matrix extension of the Manakov-Santini system and connects it to (2+1)-dimensional integrable chiral models on Einstein-Weyl spaces.

Findings

Developed a dressing scheme for the extended system

Defined an extended hierarchy of integrable equations

Connected matrix extensions to Einstein-Weyl geometry

Abstract

It was demonstrated recently [Dunajski, Ferapontov and Kruglikov (2014)] that the Manakov-Santini system describes a local form of general Lorentzian Einstein-Weyl geometry. We introduce integrable matrix extension of the Manakov-Santini system and show that it describes (2+1)-dimensional integrable chiral model in Einstein-Weyl space. We develop a dressing scheme for the extended MS system and define an extended hierarchy. Matrix extension of Toda type system connected with another local form of Einstein-Weyl geometry is also considered.

Equations

$$\frac{1}{2}g=dwdx-dzdy-F_{y}d{w}^2-(F_x-G_y)dwdz+G_{x}d{z}^2,$$

$$X_1={\partial }_z-\lambda {\partial }_x+F_x{\partial }_x+G_x{\partial }_y+f_1{\partial }_{\lambda },$$

$$X_2={\partial }_w-\lambda {\partial }_y+F_y{\partial }_x+G_y{\partial }_y+f_2{\partial }_{\lambda },$$

$${\partial }_x(Q(F))+{\partial }_y(Q(G))=0,$$

$$({\partial }_w+F_y{\partial }_x+G_y{\partial }_y)Q(G)+({\partial }_z+F_x{\partial }_x+G_x{\partial }_y)Q(F)=0,$$

$$Q={\partial }_w{\partial }_x-{\partial }_z{\partial }_y+F_y{{\partial }_x}^2-G_x{{\partial }_y}^2-(F_x-G_y){\partial }_x{\partial }_y,$$

$$f_1=-Q(G),f_2=Q(F).$$

$$\begin{array}{rl}& {\nabla }_{X_1}={\partial }_z-\lambda {\partial }_x+F_x{\partial }_x+G_x{\partial }_y+f_1{\partial }_{\lambda }+A_1,\\ & {\nabla }_{X_2}={\partial }_w-\lambda {\partial }_y+F_y{\partial }_x+G_y{\partial }_y+f_2{\partial }_{\lambda }+A_2,\end{array}$$

$$\begin{array}{rl}& {\partial }_x{A}_2={\partial }_y{A}_1,\\ & ({\partial }_z+F_x{\partial }_x+G_x{\partial }_y)A_2-({\partial }_w+F_y{\partial }_x+G_y{\partial }_y)A_1+[A_1,A_2]=0,\end{array}$$

$$\begin{array}{rl}{u}_{xt}& =u_{yy}+(u{u}_x{)}_x+v_x{u}_{xy}-u_{xx}{v}_y,\\ v_{xt}& =v_{yy}+u{v}_{xx}+v_x{v}_{xy}-v_{xx}{v}_y\end{array}$$

$$\begin{array}{rl}{X}_1& ={\partial }_y-(\lambda -v_x){\partial }_x+u_x{\partial }_{\lambda },\\ X_2& ={\partial }_t-(({\lambda }^2-v_x\lambda +u-v_y){\partial }_x+(u_x\lambda +u_y){\partial }_{\lambda }\end{array}$$

$$\begin{array}{rl}& {\nabla }_{X_1}={\partial }_y-(\lambda -v_x){\partial }_x+u_x{\partial }_{\lambda }+A,\\ & {\nabla }_{X_2}={\partial }_t-({\lambda }^2-v_x\lambda +u-v_y){\partial }_x+(u_x\lambda +u_y){\partial }_{\lambda }+\lambda A+B,\end{array}$$

$$\begin{array}{rl}& {\partial }_{y}A-{\partial }_{x}B=0,\\ & ({\partial }_y+v_x{\partial }_x)B-({\partial }_t+(v_y-u){\partial }_x)A+u_{x}A+[A,B]=0\end{array}$$

$$K_{tx}-K_{yy}-[K_x,K_y]-{\partial }_x(u{K}_x)+v_y{K}_{xx}-v_x{K}_{xy}=0,$$

$$\begin{array}{rl}({\mathrm{e}}^{-\varphi }{)}_{tt}& =m_t{\varphi }_{xy}-m_x{\varphi }_{ty},\\ m_{tt}{\mathrm{e}}^{-\varphi }& =m_{ty}{m}_x-m_{xy}{m}_t\end{array}$$

$$\begin{split}X_{1}&=\partial_{x}-\bigl{(}\lambda+\frac{m_{x}}{m_{t}}\bigr{)}\partial_{t}+\lambda\bigl{(}\phi_{t}\frac{m_{x}}{m_{t}}-\phi_{x}\bigr{)}\partial_{\lambda},\\ X_{2}&=\partial_{y}-\frac{1}{\lambda}\frac{\mathrm{e}^{-\phi}}{m_{t}}\partial_{t}-\frac{(\mathrm{e}^{-\phi})_{t}}{m_{t}}\partial_{\lambda}.\end{split}$$

$$({\mathrm{e}}^{-\varphi }{)}_{tt}={\varphi }_{xy},$$

$$\begin{array}{rl}{\nabla }_{X_1}& ={\partial }_x-(\lambda +\frac{m_x}{m_t}){\partial }_t+\lambda ({\varphi }_t\frac{m_x}{m_t}-{\varphi }_x){\partial }_{\lambda }+A\\ {\nabla }_{X_2}& ={\partial }_y-\frac{1}{\lambda }\frac{{\mathrm{e}}^{-\varphi }}{m_t}{\partial }_t-\frac{({\mathrm{e}}^{-\varphi }{)}_t}{m_t}{\partial }_{\lambda }+\frac{1}{\lambda }B,\end{array}$$

$$\begin{array}{rl}& {\partial }_{t}B+{\partial }_{y}A=0,\\ & ({\partial }_x-\frac{m_x}{m_t}{\partial }_t)B+\frac{{\mathrm{e}}^{-\varphi }}{m_t}{\partial }_{t}A-({\varphi }_t\frac{m_x}{m_t}-{\varphi }_x)B+[A,B]=0,\end{array}$$

$${\mathrm{e}}^{-\varphi }{K}_{tt}+m_x{K}_{yt}-m_t{K}_{xy}+({\varphi }_t{m}_x-{\varphi }_x{m}_t)K_y+m_t[K_y,K_t]=0.$$

$$\mathbb{D}g=\omega \otimes g$$

$$\begin{array}{rl}g& =-(dy+v_{x}dt{)}^2+4(dx+(u-v_y)dt)dt,\\ \omega & =v_{xx}dy+(-4u_x+2v_{xy}+v_x{v}_{xx})dt,\end{array}$$

$$P(u)-u_x^2=0,P(v)=0,$$

$$P={\partial }_x{\partial }_t-{\partial }_y^2+(v_y-u){{\partial }_x}^2-v_x{\partial }_x{\partial }_y.$$

$$D\Phi +\frac{1}{2}\omega \Phi =\ast F,$$

$$F(u,v)={\nabla }_u{\nabla }_v-{\nabla }_v{\nabla }_u-{\nabla }_{[u,v]}$$

$$\begin{array}{rl}{\mathbf{e}}_1& ={\partial }_x,\\ {\mathbf{e}}_2& ={\partial }_y,\\ {\mathbf{e}}_3& ={\partial }_t+(v_y-u){\partial }_x-v_x{\partial }_y,\end{array}$$

$$\begin{array}{rl}{\mathbf{e}}^1& =dx-(v_y-u)dt,\\ {\mathbf{e}}^2& =dy+v_{x}dt,\\ {\mathbf{e}}^3& ={\partial }_t,\end{array}$$

$$g=-({\mathbf{e}}^2{)}^2+4{\mathbf{e}}^1{\mathbf{e}}^3,$$

$$P=-({\mathbf{e}}_2{)}^2+{\mathbf{e}}_1{\mathbf{e}}_3.$$

$$[{\partial }_x+A_x,{\partial }_y+A_y+\Phi ]=0,$$

Full text

1 Introduction

It was demonstrated recently [1] that local form of important integrable geometric structures connected with conformal self-duality is described in general case by dispersionless integrable systems (twistor integrability of these geometric structures was established in [2], [3]). A crucial observation made in [1] is that any real conformal anti-self-dual (ASD) structure in signature (2,2) (or general complex analytic conformal ASD structure) can be locally represented by a metric

[TABLE]

where the functions $F$, $G$ satisfy the compatibility conditions $[X_{1},X_{2}]=0$ for the dispersionless Lax pair represented by vector fields containing the derivative over spectral parameter $\lambda$ (introduced in [4] in the context of dispersionless integrable hierarchies, see also [5])

[TABLE]

which can be written as a coupled system of third order PDEs

[TABLE]

where

[TABLE]

and, due to compatibility conditions, $f_{1}$ and $f_{2}$ are expressed through $F$ and $G$,

[TABLE]

In [6] we have introduced a matrix extension of Lax pair (2),

[TABLE]

where gauge field components $A_{1}$, $A_{2}$ do not depend on $\lambda$ and take their values in some (matrix) Lie algebra. Lax pairs of this structure (without derivative over $\lambda$) were already present in the seminal work of Zakharov and Shabat [7] (1979). The commutation relation splits into (scalar) vector field part, which is the same as for unextended Lax pair, and matrix part

[TABLE]

which contains the coefficients of the metric as some kind of ‘background’.

The following statement proved in [6] demonstrates that scalar background in equations (5) has a direct geometric sense.

Proposition 1**.**

Equations (5) represent anti-self-dual Yang-Mills (ASDYM) equations for the background conformal structure (1) (in a special gauge).

Different reductions of ASDYM equations give rise to integrable background geometries which are connected to dispersionless integrable systems. The current picture of the field and many examples are provided in [8].

In (2+1)-dimensional case conformal ASD structure reduces to Einstein-Weyl geometry, and it was proved in [1] that for Lorentzian signature it is locally described by the Manakov-Santini system [9, 10]

[TABLE]

with the Lax pair

[TABLE]

In [6] we presented matrix extension of the Lax pair (7),

[TABLE]

leading to the matrix system on the background of the Manakov-Santini system

[TABLE]

For the potential $K$, $A=K_{x}$, $B=K_{y}$ we have

[TABLE]

where $u$, $v$ satisfy Manakov-Santini system describing Einstein-Weyl geometry. For trivial background $u=v=0$ this equation represents one of the forms of integrable chiral model [11], [12].

In Section 2 we shall demonstrate that system (9) represents the first-order Yang-Mills-Higgs system introduced by Ward [11], defining integrable chiral model, on Einstein-Weyl background in the form described by the Manakov-Santini system.

In Section 3 we develop the techique of matrix extension of dispersionless integrable hierarchies introduced in [6] for the case of the Manakov-Santini hierarchy, formulate the dressing scheme and derive Lax-Sato equations for the extended hierarchy.

Another general local form of Lorentzian Einstein-Weyl structure [1] is given by two-component generalization of the dispersionless 2DTL equation [13]

[TABLE]

with the Lax pair

[TABLE]

For $m=t$ the system (11) reduces to the dispersionless 2DTL equation

[TABLE]

System (11) doesn’t preserve the symmetry of the dispersionless 2DTL equation with respect to $x$, $y$ variables, however, it is possible to introduce symmetric generalizations of the d2DTL equation, including elliptic case with complex variables $z$, $\bar{z}$ instead of $x$, $y$ [13]. It is an interesting question whether it is possible to represent Einstein-Weyl geometry in Euclidean signature in this way.

Lax pair (12) posesses a natural matrix extension

[TABLE]

leading to matrix equations

[TABLE]

or, in terms of potential $K$, $A=K_{t}$, $B=-K_{y}$,

[TABLE]

This equation on trivial background $m=t$, $\phi=1$ coincides (up to a change of variables) with equation (10) on trivial background, representing integrable chiral model [12].

It is a natural conjecture that system (15), similar to system (9), represents an integrable chiral model on Einstein-Weyl background. Though we will not provide a proof in the present work, it should be completely analogous to the case of Manakov-Santini system. In Section 4 we develop the techique of matrix extension of dispersionless integrable hierarchies for the case of Toda-type hierarchy connected with the system (11), formulate the dressing scheme and derive Lax-Sato equations for the extended hierarchy.

2 Integrable chiral model on the background

of Einstein-Weyl geometry

Let us recall the results relating Einstein-Weyl geometry and the Manakov-Santini system [1].

EW geometry on a three-dimensional manifold $M^{3}$ consists of a conformal structure $[g]$ and a symmetric connection ${\mathbb{D}}$ compatible with $[g]$ in the sense that, for any $g\in[g]$,

[TABLE]

for some covector $\omega$, and such that the trace-free part of the symmetrized Ricci tensor of ${\mathbb{D}}$ vanishes.

Theorem 1** (Dunajski, Ferapontov and Kruglikov (2014)).**

There exists a local coordinate system $(x,y,t)$ on $M^{3}$ such that any Lorentzian Einstein-Weyl structure is locally of the form

[TABLE]

where the functions $u$ and $v$ on $M^{3}$ satisfy a coupled system of second-order PDEs

[TABLE]

where

[TABLE]

System (18) coincides with Manakov-Santini system (6). It was shown in [14] that any solution to (18) gives rise to an EW structure of the form (17), but the question whether all EW structures arise in that way has remained open.

Let us introduce a gauge field (potential) ${A}$, which is a one-form taking its values in some (matrix) Lie algebra, and the two-form ${F}=\mathrm{d}{A}+{A}\wedge{A}$ (connection curvature, field intensity) and consider the equation

[TABLE]

where $D\Phi=\mathrm{d}\Phi+[{A},\Phi]$, $\Phi$ is a function taking values in the Lie algebra (Higgs field, [12]). This equation for Minkowski metric coincides the Yang-Mills-Higgs system introduced by Ward [11], [12], leading to integrable chiral model. Equation (20) represents an integrable Ward system on Einstein-Weyl geometry background, the term $\tfrac{1}{2}\omega\Phi$ is responsible for correct behavior of equation (20) under conformal transformation $g\rightarrow fg$, $\Phi\rightarrow f^{-\frac{1}{2}}\Phi$.

Proposition 2**.**

There exists a gauge for which equation (20) for Einstein-Weyl structure (17) takes the form of matrix extension of MS system (9), (10). For this gauge $\Phi=-A_{y}$, $A_{x}=0$, $A=2A_{y}$, $B=A_{t}-v_{x}A_{y}$.

Proof.

This proposition is proved directly, introducing a basis (frame, vielbein) of vector fields and dual basis of forms in which the metric takes a simple form and writing down components of equation (20) with respect to this basis, using a standard formula

[TABLE]

valid for arbitrary vector fields ${u}$, ${v}$.

Let us introduce a basis of vector fields

[TABLE]

and dual basis of forms

[TABLE]

Then metric (17) takes the form

[TABLE]

and symmetric bivector (17) (inverse metric) is

[TABLE]

First component of equation (20) gives

[TABLE]

that implies the existence of gauge in which $A_{x}=0$, $A_{y}+\Phi=0$.

Second component reads

[TABLE]

where $A_{3}=A_{t}-v_{y}A_{y}$, corresponding to first equation of system (9) and leading to the existence of potential $K$, $\partial_{x}K=2A_{y}$, $\partial_{y}K=A_{3}.$

Third component

[TABLE]

after some transformations takes the form of second equation of system (9) ∎

Corollary**.**

There exist local coordinates and a gauge such that for any Lorentzian Einstein-Weyl structure equation (20) is locally of the form (9), (10).

3 Manakov-Santini hierarchy and its

matrix extension

The crucial object for matrix extension of involutive distribution of polynomial (meromorphic) in spectral parameter vector fields with the basis $X_{i}$ is the matrix function $\Psi$ possessing the property that all the functions $\Psi^{-1}X_{i}\Psi$ are polynomial (meromorphic). The function $\Psi$ is suggested to be bounded and without zeroes in the spectral plane and analytic in some neighborhood of poles of vector fields. Extended linear problems then read

[TABLE]

where $(\Psi^{-1}X_{i}\Psi)_{+}$ is polynomial part of the function represented as Laurent series containing finite number of matrix fields (meromorphic part in the case of multiple poles). This function can be constructed using a matrix Riemann-Hilbert (RH) problem of the form

[TABLE]

defined on some oriented curve $\gamma$ in the complex plane, or matrix $\bar{\partial}$ problem

[TABLE]

defined in some region $G$, where $\psi_{k}(\lambda,\mathbf{t})$ are wave functions for the distribution,

[TABLE]

defined on $\gamma$ or in $G$. We suggest that solution $\Psi$ of RH (or $\bar{\partial}$) problem is bounded and has no zeroes, normalisation by 1 at infinity fixes the gauge and leads to closed systems of equations for matrix coefficients of extended Lax pairs. The dispersionless hierarchy corresponding to integrable distribution with the basis $X_{i}$ plays the role of the background.

Manakov-Santini hierarchy is defined by Lax-Sato equations [15]

[TABLE]

where $L$, $M$, corresponding to Lax and Orlov functions of dispersionless KP hierarchy, are the series

[TABLE]

and $x=t_{0}$, $(\sum_{-\infty}^{\infty}u_{n}\lambda^{n})_{+}=\sum_{n=0}^{\infty}u_{n}\lambda^{n}$, $\{L,M\}=L_{\lambda}M_{x}-L_{x}M_{\lambda}$. A more standard choice of times for dKP hierarchy corresponds to $M_{0}$=${\sum_{n=0}^{\infty}(n+1)t_{n}L^{n}}$, it is easy to transfer to it by rescaling of times.

Lax-Sato equations (30) are equivalent to the generating relation [4, 15]

[TABLE]

where differential takes into account all times $\mathbf{t}$ and variable $\lambda$.

A dressing scheme for Manakov-Santini hierarchy can be formulated in terms of two-component nonlinear Riemann problem on the unit circle $S$ in the complex plane of the variable $\lambda$,

[TABLE]

where the functions $L_{\text{in}}(\lambda,\mathbf{t})$, $M_{\text{in}}(\lambda,\mathbf{t})$ are analytic inside the unit circle, the functions $L_{\text{out}}(\lambda,\mathbf{t})$, $M_{\text{out}}(\lambda,\mathbf{t})$ are analytic outside the unit circle and have an expansion of the form (31), (32). It is straightforward to demonstrate that relation (34) implies analyticity of the differential form

[TABLE]

in the complex plane and generating relation (33), thus defining a solution of Manakov-Santini hierarchy. Considering a reduction to area-preserving diffeomorphisms SDiff(2), we obtain the dKP hierarchy.

To obtain a gauge field extension of the hierarchy, we introduce also a matrix Riemann-Hilbert problem

[TABLE]

$\Psi$ is normalised by 1 at infinity and analytic inside and outside the unit circle,

[TABLE]

Expansions of $\Psi$, $L$, $M$ give coefficients for extended vector fields, $\Psi$ is a wave function. A general wave function is given by the expression $\Psi F(L,M)$, $F$ is an arbitrary complex-analytic matrix-valued function. From matrix RH problem we get analyticity of the matrix-valued form

[TABLE]

and of the functions $\Psi^{-1}X_{i}\Psi$, leading to Lax-Sato equations for the series $\Psi$ (35), $L$ (31), $M$ (32), defining the evolution of these series,

[TABLE]

where vector fields $V_{n}(\lambda)$ are defined by formula (30). First flows give exactly extended Lax pair (8), if we identify $y=t_{1}$, $t=t_{2}$.

4 Toda type hierarchy and its

matrix extension

Let us recall a picture of the hierarchy connected with system (11) [13]. A complete set of Lax-Sato equations reads

[TABLE]

where the definition of the Poisson bracket is $\{f,g\}=\lambda(f_{\lambda}g_{t}-f_{t}g_{\lambda})$, and we consider formal series

[TABLE]

where $\lambda$ is a spectral variable. Usually we suggest that ‘out’ and ‘in’ components of the series define the functions outside and inside the unit circle in the complex plane of the variable $\lambda$ respectively, with $\Lambda^{\text{in}}-\ln\lambda$, $M^{\text{in}}-M_{0}^{\text{in}}$ analytic in the unit disc, and $\Lambda^{\text{out}}-\ln\lambda$, $M^{\text{out}}-M_{0}^{\text{out}}$ analytic outside the unit disc and decreasing at infinity. For a function on the complex plane, having a discontinuity on the unit circle, by ‘in’ and ‘out’ components we mean the function inside and outside the unit disc.

A dressing scheme for the hierarchy (36) can be formulated in terms of the two-component nonlinear Riemann-Hilbert problem on the unit circle $S$,

[TABLE]

where the functions $\Lambda^{\text{out}}(\lambda,\mathbf{x},\mathbf{y},t)$, $M^{\text{out}}(\lambda,\mathbf{x},\mathbf{y},t)$ are defined outside the unit circle, the functions $\Lambda^{\text{in}}(\lambda,\mathbf{x},\mathbf{y},t)$, $M^{\text{in}}(\lambda,\mathbf{x},\mathbf{y},t)$ inside the unit circle by the series of the form (37), (38), with $\Lambda^{\text{in}}-\ln\lambda$, $M^{\text{in}}-M_{0}^{\text{in}}$ analytic in the unit disc, and $\Lambda^{\text{out}}-\ln\lambda$, $M^{\text{out}}-M_{0}^{\text{out}}$ analytic outside the unit disc and decreasing at infinity.

Lax-Sato equations for the times $x=x_{1}$, $y=y_{1}$

[TABLE]

$m=m_{0}+t$, correspond to the Lax pair (12), where the coefficients in the first Lax-Sato equation can be transformed to the form (12) by taking its expansion at $\lambda=0$, and the system (11) arises as a compatibility condition.

To obtain a matrix extension of the hierarchy, we introduce also a matrix Riemann-Hilbert problem

[TABLE]

$\Psi$ is normalised by 1 at infinity and analytic inside and outside the unit circle,

[TABLE]

Lax-Sato equations for the series (41) define the evolution of these series on the background defined by Lax-Sato equations (36)

[TABLE]

where vector fields $V^{+}_{n}(\lambda)$, $V^{-}_{n}(\lambda)$ are defined by formula (36) and have coefficients polynomial respectively in $\lambda$ and $\lambda^{-1}$.

Acknowledgements

This work was performed in the framework of State assignment topic 0033-2019-0006 (Integrable systems of mathematical physics).