From the Kadomtsev-Petviashvili equation halfway to Wards chiral model

for an SU (m) matrix J, where J t = ∂J/∂t, etc. In terms of the new variables

this simplifies to (J -1 J x 3 ) x 1 -(J -1 J x 2 ) x 2 = 0, which extends to the hierarchy

The Ward equation is completely integrable 1 and admits soliton-like solutions, often called “lumps”. It was shown numerically [3] and later analytically [4,5,6] that such lumps can interact in a nontrivial way, unlike usual solitons. In particular, they can scatter at right angles, a phenomenon sometimes referred to as “anomalous scattering”. 2 Also the integrable KP equation, more precisely KP-I (“positive dispersion”), possesses lump solutions with anomalous scattering [8,9,10] (besides those with trivial scattering [11]). Introducing a potential φ for the real scalar function u via u = φ x , in terms of independent variables t 1 , t 2 (spatial coordinates) and t 3 (time), the (potential) KP equation is given by

with σ = i in case of KP-I and σ = 1 for KP-II. Could it be that this equation has a closer relation with the Ward equation? We are trying to compare an equation for a scalar with a matrix equation, and in [4] the appearance of nontrivial lump interactions in the Ward model had been attributed to the presence of the “internal degrees of freedom” of the latter. At first sight this does not match at all. However, the resolution lies in the fact that the KP equation possesses an integrable extension to a (complex) matrix version,

where we modified the product by introducing a constant N × M matrix Q, and the commutator is modified accordingly, so that

Here Φ is an M × N matrix.

If rank(Q) = 1, and thus Q = V U † with vectors U and V , then any solution of this (potential) matrix KP equation determines a solution φ := U † ΦV of the scalar KP equation. 3 More generally, this extends to the corresponding (potential) KP hierarchies.

Next we look for a relation between the matrix KP and the Ward equation. Indeed, there is a dispersionless (multiscaling) limit of the above “noncommutative” (i.e. matrix) KP equation,

obtained by introducing x n = n ǫ t n with a parameter ǫ, and letting ǫ → 0 (assuming an appropriate dependence of the KP variable Φ on ǫ) [2]. If rank(Q) = m, and thus Q = V U † with an M × m matrix U and an N × m matrix V , then the m × m matrix ϕ := σ U † ΦV solves

if Φ solves (6). In terms of the variables x, y, t, this becomes4

Now we note that the cases σ = i and σ = 1 are related by exchanging x and t, hence they are equivalent. 5 We choose σ = 1 in the following. Then (7) extends to the hierarchy

The circle closes by observing that this is “pseudodual” to the hierarchy (3) of Ward’s chiral model in the following sense. ( 9) is solved by

and the integrability condition of the latter system is the hierarchy (3). Rewriting (10) as J x n+1 = -J ϕ xn , the integrability condition is the hierarchy (9). All this indeed connects the Ward model with the KP equation, but more closely with its matrix version, and not quite on a level which would allow a closer comparison of solutions. Note that the only nonlinearity that survives in the dispersionless limit is the commutator term, but this drops out in the “projection” to scalar KP. On the other hand, we established relations between hierarchies, which somewhat ties their solution structure together. 6In the Ward model, J has values in SU (m), thus ϕ must have values in the Lie algebra su(m), so has to be traceless and anti-Hermitian. Suitable conditions have to be imposed on Φ to achieve this. Via the dispersionless limit, methods of constructing exact solutions can be transfered from the (matrix) KP hierarchy to the pseudodual chiral model (pdCM) hierarchy (9). From [2] we recall the following result. It determines in particular various classes of (multi-) lump solutions of the su(m) pdCM hierarchy.

Theorem 1. Let P, T be constant N × N matrices such that T † = -T and P † = T P T -1 , and V a constant N × m matrix. Suppose there is a constant solution K of

Example 1. Let m = 2, N = 2, and

with complex parameters a, b, c, d, p and a function f (with complex conjugate f * ). Then X x n+1 = X x 1 P n is satisfied if f is an arbitrary holomorphic function of

Furthermore, [P, K] = -V V † T has a solution iff ac * + bd * = 0 and β := 2ℑ(p) = 0 (where ℑ(p) denotes the imaginary part of p). Without restriction of generality we can set the diagonal part of K to zero, since it can be absorbed by redefinition of f in the formula for ϕ. We obtain the following components of ϕ,

where

polynomial in ω, the solution is regular, rational and localized. It describes a simple lump if f is linear in ω. Otherwise it attains a more complicated shape (see [2] for some examples).

Fixing the values of x 4 , x 5 , . . ., we concentrate on the first pdCM hierarchy equation. In terms of the variables x, y, t given by (2), we then have ω = 1 2 (t -x + 2py + p 2 (t + x)), subtracting a constant that