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Journal of Generalized Lie Theory and Applications
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From the Kadomtsev-Petviashvili equation halfway to Ward’s chiral model 1

Aristophanes DIMAKIS1, Folkert M¨ ULLER-HOISSEN2

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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Abstract

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.

Relation between KP and Ward’s chiral model

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

equation   (1.1)

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

equation   (1.2)

this simplifies to equation 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

equation   (1.3)

The Ward equation is completely integrable2 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 equation for the real scalar function equation in terms of independent variables t1, t2 (spatial coordinates) and t3 (time), the (potential) KP equation is given by

equation

with equation 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,

equation

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

equation

Here equation 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 equation 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,

equation   (1.4)

obtained by introducing equation with a parameter ε, and lettingequation (assuming an appropriate dependence of the KP variable equation) [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 equation solves

equation   (1.5)

if equation 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

equation   (1.6)

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

equation   (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

equation   (1.8)

and the integrability condition of the latter system is the hierarchy (1.3). Rewriting (1.8) as equation 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 equation must have values in the Lie algebra su(m), so has to be traceless and anti-Hermitian. Corresponding conditions have to be imposed on equation to achieve this. The KP-I counterpart is the reality condition for the scalar Á.

Exact solutions of the pdCM hierarchy

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 Xxn+1 = Xx1 Pn, n = 1, 2, . . .. Then equation := −V †T(X −K)−1V solves the su(m) pdCM hierarchy.

Example 2.1. Let m = 2, N = 2, and

equation

with complex parameters a, b, c, d, p and a function f (with complex conjugate f*). Then equation is satisfied if f is an arbitrary holomorphic function of

equation   (2.1)

Furthermore, [P,K] = −V V †T has a solution iff ac* + bd* = 0 and equation (where equation 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 equation. We obtain the following components of equation,

equation

where

equation

If f is a non-constant polynomial in equation, the solution is regular, rational and localized. It describes a simple lump if f is linear in equation. Otherwise it attains a more complicated shape (see [4] 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

equation

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

equation   (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

equation

The diagonal blocks of the new big matrix K will be K1 and K2. It only remains to solve

equation

for the upper off-diagonal block of K and set equation 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

equation

where equation with arbitrary holomorphic functions equation (where w1 is (2.1) built with p1), respectively equation

equation

This solution is again regular if equation we recover the single lump solution (2.1) with V = I2 and f replaced by f2 (resp. f1).

Choosing equation (or correspondingly with i replaced by −i) with equation and f1, f2 linear in equation respectively equation one observes scattering at right angle (cf. [16] for the analogous case in the Ward model).

If equation 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 equation a vanishes and equation becomes constant (assuming f1, f2 independent of equation), 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 equation and f2 quadratic in equation 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 equation, the larger the range of the interaction.

generalized-theory-applications-Plots-solution-Example

Figure 1: Plots of ε at times t = −90,−55,−53, 0, 30, 80 for the solution in Example 2.2 with p1 = −i(1 − equation) and p2 = i(1 + equation) where equation = 1/20, equation

generalized-theory-applications-Origin-fate-lump

Figure 2: Origin and fate of the lump pair parts appearing in Fig. 1 (to the right in the first plot and to the left in the last). Plots of ε at t = −20000,−2000, 2000, 20000.

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) equation 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.

Acknowledgement

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.

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

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 [10] for instance.

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

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

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

7We 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.

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

References

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