1Department of Financial and Management Engineering, University of the Aegean, 31 Fostini Street, GR-82100 Chios, Greece, E-mail: [email protected]
2Max-Planck-Institute for Dynamics and Self-Organization, Bunsenstrasse 10, D-37073 G¨ottingen, Germany, E-mail: [email protected]
Received date: December 12, 2007; Accepted date: March 18, 2008
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The “pseudodual” of Ward’s modified chiral model is a dispersionless limit of the matrix Kadomtsev-Petviashvili (KP) equation. This relation allows to carry solution techniques from KP over to the former model. In particular, lump solutions of the su(m) model with rather complex interaction patterns are reached in this way. We present a new example.
for an SU(m) matrix J, where Jt = ∂J/∂t, etc. In terms of the new variables(1.2)
this simplifies to which is a straight reduction of Yang’s equation, one of the potential forms of the self-dual Yang-Mills equation . It extends to the hierarchy(1.3)
The Ward equation is completely integrable2 and admits soliton-like solutions, often called “lumps”. It was shown numerically  and later analytically [16, 6, 3] 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”.3 Also the integrable KP equation, more precisely KP-I (“positive dispersion”), possesses lump solutions with anomalous scattering [5, 14, 1] (besides those with trivial scattering ). Introducing a potential for the real scalar function in terms of independent variables t1, t2 (spatial coordinates) and t3 (time), the (potential) KP equation is given by
with 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  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 is an M×N matrix. If rank(Q) = 1, and thus = V U† with vectors U and V , then any solution of this (potential) matrix KP equation determines a solution of the scalar KP equation.4 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,(1.4)
obtained by introducing with a parameter ε, and letting (assuming an appropriate dependence of the KP variable ) . 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 solves(1.5)
Now we note that the cases σ = i and σ = 1 are related by exchanging x and t, hence they are equivalent.6 We choose σ = 1 in the following. Then (1.5) extends to the hierarchy(1.7)
The circle closes by observing that this is “pseudodual” to the hierarchy (1.3) of Ward’s chiral model in the following sense. (1.7) is solved by(1.8)
and the integrability condition of the latter system is the hierarchy (1.3). Rewriting (1.8) as the integrability condition is the hierarchy (1.7). 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.7
In the Ward model, J has values in SU(m), thus must have values in the Lie algebra su(m), so has to be traceless and anti-Hermitian. Corresponding conditions have to be imposed on to achieve this. The KP-I counterpart is the reality condition for the scalar Á.
Via the dispersionless limit, methods of constructing exact solutions can be transfered from the (matrix) KP hierarchy to the pseudodual chiral model (pdCM) hierarchy (1.7). From  we recall the following result. It determines in particular various classes of (multi-) lump solutions of the su(m) pdCM hierarchy.
Theorem 2.1. Let P, T be constant N ×N matrices such that T† = −T and P† = TPT−1, and V a constant N ×m matrix. Suppose there is a constant solution K of [P,K] = −V V †T (= Q) such that K† = TKT−1. Let X be an N × N matrix solving [X, P] = 0, X† = TXT−1 and Xxn+1 = Xx1 Pn, n = 1, 2, . . .. Then := −V †T(X −K)−1V solves the su(m) pdCM hierarchy.
Example 2.1. Let m = 2, N = 2, and
with complex parameters a, b, c, d, p and a function f (with complex conjugate f*). Then is satisfied if f is an arbitrary holomorphic function of(2.1)
Furthermore, [P,K] = −V V †T has a solution iff ac* + bd* = 0 and (where 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 . We obtain the following components of ,
If f is a non-constant polynomial in , the solution is regular, rational and localized. It describes a simple lump if f is linear in . Otherwise it attains a more complicated shape (see  for some examples).
Fixing the values of x4, x5, . . ., we concentrate on the first pdCM hierarchy equation. In terms of the variables x, y, t given by (1.2), we then have
subtracting a constant that can be absorbed by redefinition of the function f in the solution in Example 2.1. This solution becomes stationary, i.e. t-independent, if p = ±i. The conserved density(2.2)
of (1.6) is non-negative and will be used below to display the behaviour of some solutions.
More complicated solutions are obtained by superposition in the following sense. Given data (X1, P1, T1, V1) and (X2, P2, T2, V2) that determine solutions according to theorem 2.1, we build
The diagonal blocks of the new big matrix K will be K1 and K2. It only remains to solve
for the upper off-diagonal block of K and set In particular, one can superpose lump solutions as given in the preceding example.
Example 2.2. Superposition of two single lumps with V1 = V2 = I2, the 2 × 2 unit matrix, yields
where with arbitrary holomorphic functions (where w1 is (2.1) built with p1), respectively
This solution is again regular if we recover the single lump solution (2.1) with V = I2 and f replaced by f2 (resp. f1).
Choosing (or correspondingly with i replaced by −i) with and f1, f2 linear in respectively one observes scattering at right angle (cf.  for the analogous case in the Ward model).
If one observes the following phenomenon: two lumps approach one another, meet, then separate in the orthogonal direction up to some maximal distance, reproach, merge again, and then separate again while moving in the original direction .8 In the limit a vanishes and becomes constant (assuming f1, f2 independent of ), so that ε vanishes. For other choices of f1 and f2 more complex phenomena occur, including a kind of “exchange process” described in the following. Fig. 1 shows plots of ε at successive times t for the above solution with f1 linear in and f2 quadratic in The latter function then corresponds to a bowl-shaped lump (see the left of the plots in Fig. 2) which, at early times, moves to the left along the x-axis, deforming into the lump pair, shown on the right hand side of the first plot in Fig. 1, under the increasing influence of the simple lump (corresponding to the linear function f1) that moves to the right. When the latter meets the first partner of the lump pair, they merge, separate in y-direction to a maximal distance, move back toward each other and then continue moving as a lump pair (shown on the left hand side of the last plot in Fig. 1) into the negative x-direction. Meanwhile the remaining partner of the lump pair, that evolved from the original bowl-lump, retreats into the (positive) x-direction, with diminishing influence on the new lump pair, which then finally evolves into a bowl-shaped lump (see the right of the plots in Fig. 2). The smaller the value of , the larger the range of the interaction.
Other classes of solutions are obtained by taking for P matrices of Jordan normal form, generalizing T appropriately, and building superpositions in the aforementioned sense. Some examples in the su(2) case have been worked out in . This includes examples exhibiting (asymptotic) scattering of n-lump configurations. The pdCM (and also the Ward model) thus exhibits surprisingly complex lump interaction patterns, which are comparatively well accessible via the above theorem, though a kind of systematic classification is by far out of reach.
F M-H would like to thank the German Research Foundation for financial support to attend the workshop Algebra, Geometry, and Mathematical Physics in G¨oteborg.
2In the sense of the inverse scattering method, the existence of a hierarchy, and various other characterisations of complete integrability. In the following “integrable” loosely refers to any of them.
3See also the references cited above for related work. Anomalous scattering has also been found in some related non-integrable systems, like sigma models, Yang-Mills-Higgs equation (monopoles) and the Abelian Higgs model or Ginzburg-Landau equation (vortices), see  for instance.