A New Reduction of the Self-Dual Yang–Mills Equations and its Applications

1 Introduction

The self-dual Yang–Mills (SDYM) equations are of great importance in their own right and have found a remarkable number of applications in both physics and mathematics. These equations arise in the context of gauge theory, in the classical general relativity, and can be used as a powerful tool in the analysis of four-manifolds [1]. SDYM equations are originated from the non-perturbative approach to the quantum theory of gauge fields [2]. Actually, the SDYM equations have given rise to the self-dual Einstein equations and many other equations of physical interest, including the (1+1)-dimensional Korteweg de Vries (KdV) equation which governs a long one-dimensional, small-amplitude, and surface gravity wave propagating in a shallow channel of water, and the (1+1)-dimensional nonlinear Schrödinger equation which is a partial differential equation in quantum mechanics that describes how the quantum state of a quantum system changes with time. The SDYM equations have also led to (2+1)-dimensional nonlinear equations, such as the Kadomtsev–Petviasgvili (KP) equation which can be thought of as a two-spatial-dimensional analogue of the KdV equation, and it is one of the classical prototype problems in the field of exactly solvable equations and arises generically in physical contexts such as the plasma physics and the surface water wave, and the (2+1)-dimensional Davey–Stewartson (DS) equation which is used to describe the evolution of a three-dimensional wave-packet on water of finite depth in fluid dynamics [3]. In the aspect of mathematical applications, by symmetric reductions of the SDYM equations, some different Lax pairs have been introduced, whose compatibility conditions not only generate some known (1+1)- and (2+1)-dimensional physical equations, but also help us study their some properties, such as Hamiltonian structures, Darboux transformations, and symmetries. Ablowitz et al. [1] have shown us some reductions of the SDYM equations, derived the KdV equation and the (2+1)-dimensional KP equation, and studied their Lax pairs and Painlevé properties. Based on this, Chakravarty et al. [2] introduced a reduction of the SDYM equations in the aspect of mathematics whose compatibility condition leads to a set of integrable equations. By choosing an operator Lie algebra, here the operator is ∂_y, whose coefficients of different powers are 2×2 matrices, a forced Burgers equation, the KP equation, and an mKP equation were re-obtained. Zhang and Hon [4] developed an approach for constructing Lie algebras by the reduction of the SDYM equations for which a (2+1)-dimensional expanding integrable model of the Giachetti Johnson (GJ) hierarchy was obtained; furthermore, a few Hamiltonian equations were generated in the linear space R³. In [5], we introduced a reduction of the SDYM equations whose compatibility condition admits a variable-coefficient Burgers equation and a (2+1)-dimensional integrable coupling system. In this paper, we shall construct a new reduction of the SDYM equations by imposing a space–time symmetry whose compatibility condition gives rise to a set of (2+1)-dimensional integrable systems containing multi-potential functions which are similar to those in [2, 5], but they are different from each other. It is remarkable that the variables presented in this paper are different from those in [2]. Therefore, the differential equations obtained by us are not the same as those in [2]. In [2] and [5], the chosen Lie algebras are 2×2 matrix coefficients; however, in this paper we choose 3×3 coefficients of the various powers of the operator ∂_y so that some new (2+1)-dimensional integrable systems with mathematical structures or physical backgrounds could be generated. Hence, when the central matrix is degenerated, some new (2+1)-dimensional integrable systems containing six potential functions are obtained, which can be reduced to a (2+1)-dimensional integrable coupling of a generalised KP equation which enriches the theory on integrable couplings, where such (2+1)-dimensional integrable couplings could describe models of atmospheric turbulence. In particular, we obtain a (2+1)-dimensional Burgers equation which could be the equation of motion governing the surface perturbations of shallow viscous fluid. When the central matrix is nonsingular, a (2+1)-dimensional integrable system containing six potential functions is obtained, which can be reduced to a (2+1)-dimensional generalised DS system; in particular, it is again reduced to the DS integrable system in [2]. Because the KP equation has the known physical applications stated as above, the (2+1)-dimensional integrable system obtained in this paper also describes some physical phenomena just like the surface water waves.

2 A New Reduction of the SDYM Equations

Set M to be a complex four-dimensional manifold whose complex coordinates are denoted by (ω, ω¯, z, z¯). Then the SDYM equations are the compatibility condition of the following isospectral problem [1, 2]:

where

D1 = ∂ω + λ∂z¯, D2 = ∂z − λ∂ω¯, A1 = Aω + λAz¯, A2 = Az − λAω¯,

and Aa(a = ω, ω¯, z, z¯) stand for Yang–Mills potentials; the variable λ ∈ C is referred to as the spectral parameter. The compatibility condition of (1) possesses the form

In [1], it is shown that the components Aω¯ and Az¯ can be represented by two commuting elements in a Lie algebra g through a suitable gauge. Hence, we assume that

By following the way presented in [2], we also impose space–time symmetry on the components A₁ and A₂ in (2), but the variables chosen by us are different from those in [2], which lead to the similar differential equations. Suppose the Yang–Mills potentials are independent of the variables ω + ω¯ and z¯. We redefine the remaining coordinates as ω − ω¯ = x and z=t; then one gets that

which is slightly different from that in [2]. In terms of (4), (1) becomes

We assume that the potentials in (5) are elements of an infinite-dimensional Lie algebra,

where a₀, a₁, a₂ are n×n matrices, which belong to a ring of matrix functions of y, x, and t, and the Lie bracket on g is closed, that is, [A, B] ∈ g, A, B ∈ g. Reference [2] takes the 2×2 matrices in (6) to generate some (2+1)-dimensional integrable systems. In this paper, we want to take 3×3 matrices in (6) to deduce more richer (2+1)-dimensional integrable dynamical systems to further supplement the results in [2, 4, 5].

Assume that

Aω = U0 + U1∂y + U2∂y2, Aω¯ = B0 + B1∂y + B2∂y2, Az¯ = A,Az = V0 + V1∂y + V2∂y2,

and substitute them into (2); we have

Equation (3) is equivalent to

To eliminate the spectral parameter λ in (7), we make a transformation

ψ = φeλy

and substitute it into (7); we get

φx = (U0 + U1∂y + U2∂y2)φ + λ(U1 + 2U2 + A)φ + λ2U2φ,φt = (V0 + V1∂y + V2∂y2)φ + λ[V1 + 2V2∂y − (U0 + B0)  − (U1 + B1)∂y − (U2 + B2)∂y2]φ + λ2[V2 − U1 − B1 − A  − 2(U2 + B2)∂y − λ3(U2 + B2)φ,

which give the following equations by setting the coefficients of λ^j (j=3, 2, 1) to be zero that

and

Solving (9) yields

Substituting (12) into (10) and dropping the subscripts of U₀, V₀, and B₀, one gets that

The compatibility condition of (13) reads

Assume that A is independent of x, y, and t; then (14) reduces to

Equation (11) can be written as

We call the matrix A in (15) and (16) a central matrix.

3 A Few (2+1)-Dimensional Integrable Systems

In the section, we shall deduce a few (2+1)-dimensional integrable systems according to (15) and (16) as well as the operator Lie algebra g with 3×3 matrix coefficients of the powers of the operator ∂_y. A simple case is that U, V, A in (15) and (16) are scalar functions. With no loss of generality, we take A=1; (15) becomes

A special solution to (17) is given by

where f is a derivative function in x, y, and t. Substituting (19) into (18) yields that

Set Y=y – x, X=x; then (20) becomes

Integrating (21) on the variable X gives

which is the same form except for negative sign as that in [2]. When V=0, (22) is a form of Burgers’ equation.

Next, we take some explicit matrices in (14) to deduce a type of new KP equation. Set

U = (01u0),  A = (00−10),  B = − u2A, V = 12(f(x, y) +ux + γuy−3uuxx + 3u2 + uyf(x, y) − ux − γyy),

where f(x, y) is a derivative function and γ is an arbitrary parameter. Then (14) admits the following equations:

ux = 2γuy,6u2 + uy − fx − γuxy − 2uy = 0,ut + 12(2uux + 2γuuy) + 12(fy + γuyy − uxxx − 6uux) = 0,6u2 + uy − fx − γuxy − 2uy = 0.

Solving the above four equations yields that

ut − uux + 6∂x−1uux − 14uxx − 12∂x−1uyy − 12uxxx = 0,

which could describe the surface water waves in fluid dynamics. It is a generalised form given by Ablowtiz et al. in [1]. In what follows, we consider only two cases of the matrix A; the first case is degenerative, while another case is non-degenerate.