Medical, Pharma, Engineering, Science, Technology and Business

^{1}Department of Financial and Management Engineering, University of the Aegean,
31 Fostini Street, GR-82100 Chios, Greece, **E-mail:** [email protected]

^{2}Max-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

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

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.

Ward’s chiral model in 2 + 1 dimensions [15] (see [4] for further references) is given by

(1.1)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 [12]. It extends to the hierarchy

(1.3)The Ward equation is completely integrable^{2} and admits soliton-like solutions, often called
“lumps”. It was shown numerically [13] 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 [9]). Introducing a potential for the real scalar
function in terms of independent variables *t*_{1}, *t*_{2} (spatial coordinates) and *t*_{3} (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 [16] 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 ) [4]. 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)if solves (1.4).^{5} This is a straight reduction of another potential form of the self-dual Yang-
Mills equation [12]. In terms of the variables *x, y, t*, it becomes

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

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 [4] 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
X_{x}_{n+1} = X_{x1} P^{n}, n = 1, 2, . . .. Then := −V †T(X −K)^{−1}V 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

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 ,

where

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 [4] for some examples).

Fixing the values of *x _{4}, x_{5}*, . . ., we concentrate on the first pdCM hierarchy equation. In
terms of the variables

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

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
(X_{1}, P_{1}, T_{1}, V_{1}) and (X_{2}, P_{2}, T_{2}, V_{2}) that determine solutions according to theorem 2.1, we build

The diagonal blocks of the new big matrix K will be K_{1} and K_{2}. 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 V_{1} = V_{2} = I_{2}, the 2 × 2 unit matrix,
yields

where with arbitrary
holomorphic functions (where *w*_{1} is (2.1) built with p1), respectively

This solution is again regular if we recover the single
lump solution (2.1) with V = *I*_{2} and *f* replaced by *f*_{2} (resp. *f*_{1}).

Choosing (or correspondingly with *i* replaced by −*i*) with and *f*_{1}, *f*_{2} linear in respectively one observes scattering at right angle (cf.
[16] 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
[4].^{8} In the limit *a* vanishes and becomes constant (assuming *f*_{1}, *f*_{2} independent of ),
so that ε vanishes. For other choices of *f*_{1} and *f*_{2} 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 *f*_{1} linear in and *f*_{2} 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 *f*_{1}) 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 [4]. 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.

^{1}Presented at the 3rd Baltic-Nordic Workshop “Algebra, Geometry, and Mathematical Physics“, G¨oteborg,
Sweden, October 11–13, 2007.

^{2}In 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.

^{3}See 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 [10] for instance.

^{4}See e.g [11, 2] for related ideas.

^{5}(1.5) and its four-dimensional extension are sometimes called “Leznov equation”, see [7].

^{6}Note also that this transformation leaves the conserved density (2.2) invariant.

^{7}We note, however, that e.g the singular shock wave solutions of the dispersionless limit of the scalar KdV
equation have little in common with KdV solitons.

^{8}See also [8] for an analogous phenomenon in case of KP-I lumps.

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