From the Kadomtsev-Petviashvili equation halfway to Wards chiral model

F rom the Kadom tsev-P et viash vili equation halfw a y to W ard’s c hiral mo de l Aristo phanes Dimaki s a and F olkert M ¨ uller-H oissen b a Departmen t of Financial and Managemen t Engineering, Un iv ersity of the Aegean 31 F ostini Str., GR-821 00 Chios, Greece b Max-Planc k-Institute for Dynamics and Self-Organization Bunsenstrasse 10, D-37073 G¨ ottingen, German y E-mails: dimakis@aegean.gr, fo lk ert.m ueller-hoissen@ds.mpg.de Abstract The “pseudo dual” of W ard’s mo dified chiral model is a dispers ionless limit of the matrix Kadomtsev-Petviash vili (KP ) equation. This relation allows to carry solution tec hniques from K P ov e r to the former mo del. In particular, lump solutions o f the su ( m ) mo del with rather complex in teractio n patterns ar e reached in this way . W e present a new example. 2000 MSC: 3 7Kxx 70Hxx W ard’s c h iral m o del in 2 + 1 d imensions [1] (see [2] for fur ther references) is give n b y ( J − 1 J t ) t − ( J − 1 J x ) x − ( J − 1 J y ) y + [ J − 1 J x , J − 1 J t ] = 0 (1) for an S U ( m ) m atrix J , where J t = ∂ J /∂ t , etc. In terms of the new v ariables x 1 := ( t − x ) / 2 , x 2 := y , x 3 := ( t + x ) / 2 , (2) this simp lifies to ( J − 1 J x 3 ) x 1 − ( J − 1 J x 2 ) x 2 = 0, w hic h extends to the h ierarc hy ( J − 1 J x n +1 ) x m − ( J − 1 J x m +1 ) x n = 0 , m, n = 1 , 2 , . . . . (3) The W ard equ ation is completely integrable 1 and adm its soliton-lik e s olutions, often called “lumps”. It wa s sho wn n u merically [3] and later analytically [4, 5, 6] that such lumps can in teract in a nont rivial w ay , unlik e u sual solitons. In p articular, they can scatte r at righ t angles, a phenomenon sometimes referred to as “anomalous scattering”. 2 Also the integrable KP equation, more precisely KP-I (“ p ositiv e disp ersion”), p ossesses lump solutions with anomalous scattering [8, 9, 10] (b esides th ose with trivial scattering [11]). I ntro ducing a p otenti al φ for the real scalar fu nction u via u = φ x , in terms of indep enden t v ariables t 1 , t 2 (spatial co ordinates) and t 3 (time), th e (p oten tial) KP equation is giv en b y (4 φ t 3 − φ t 1 t 1 t 1 − 6 φ t 1 φ t 1 ) t 1 − 3 σ 2 φ t 2 t 2 = 0 , (4) with σ = i in case of KP-I and σ = 1 for KP-I I. Could it b e that this equation has a closer relation with the W ard equation? W e are trying to compare an equ ation for a scalar with a 1 In the sense of th e inv erse scattering metho d, t h e existence of a hierarch y , and v arious other chara cterisations of complete integrabilit y . In the follo wing “in tegrable” loosely refers t o any of th em. 2 See also the references cited above fo r related wo rk . Anomalous scattering has also b een found in some related non-integrable systems, like sigma mo d els, Y ang-Mills-Higgs equation (monop oles) and the Ab elian Higgs model or Ginzburg-Landau equ ation (vortices), see [7] for instance. matrix equ ation, and in [4] the app earance of nontrivial lump inte r actions in the W ard mo del had b een attributed to the p resence of the “in tern al degrees of fr eedom” of the latter. A t fi rst sigh t this do es n ot matc h at all. Ho wev er, the resolution lies in the fact that the KP equation p ossesses an in tegrable extension to a (complex) matrix v ersion,  4 Φ t 3 − Φ t 1 t 1 t 1 − 6 Φ t 1 Q Φ t 1  t 1 − 3 σ 2 Φ t 2 t 2 = − 6 σ [Φ t 1 , Φ t 2 ] Q , (5) where we mo d ified the p ro du ct by intro d ucing a constant N × M matrix Q , and the comm utator is mo d ified accordingly , so that [Φ t 1 , Φ t 2 ] Q = Φ t 1 Q Φ t 2 − Φ t 2 Q Φ t 1 . Here Φ is an M × N matrix. If r an k ( Q ) = 1, and thus Q = V U † with v ectors U and V , then an y s olution of this (p oten tial) matrix KP equation determines a solution φ := U † Φ V of the scalar KP equation. 3 More generally , this extend s to the corresp ond ing (p oten tial) KP hierarchies. Next w e lo ok for a relation b et ween the matrix KP and th e W ard equation. Indeed, there is a disp ersionless (m u ltiscaling) limit of the ab ov e “noncomm utativ e” (i.e. matrix) KP equatio n, Φ x 1 x 3 − σ 2 Φ x 2 x 2 = − σ [Φ x 1 , Φ x 2 ] Q , (6) obtained b y in tro d ucing x n = n ǫ t n with a p arameter ǫ , and letting ǫ → 0 (assumin g an appropriate d ep enden ce of the KP v ariable Φ on ǫ ) [2]. If rank( Q ) = m , and th us Q = V U † with an M × m matrix U and an N × m matrix V , th en the m × m matrix ϕ := σ U † Φ V solves ϕ x 1 x 3 − σ 2 ϕ x 2 x 2 = − [ ϕ x 1 , ϕ x 2 ] , (7) if Φ solv es (6). In terms of the v ariables x, y , t , this b ecomes 4 ϕ tt − ϕ xx − σ 2 ϕ y y + [ ϕ t − ϕ x , ϕ y ] = 0 . (8) No w w e note that the cases σ = i and σ = 1 are related b y exc hanging x and t , hence they are equiv alent. 5 W e c h o ose σ = 1 in th e follo wing. Then (7 ) extends to the h ierarc hy ϕ x m x n +1 − ϕ x m +1 x n = [ ϕ x n , ϕ x m ] , m, n = 1 , 2 , . . . . (9) The circle closes b y observing that this is “pseudo dual” to the h ierarc hy (3) of W ard’s c hiral mo del in the follo w ing sense. (9 ) is solv ed b y ϕ x n = − J − 1 J x n +1 , n = 1 , 2 , . . . , (1 0) and the integrabilit y condition of the latter system is the hierarc hy (3). Rewriting (10 ) as J x n +1 = − J ϕ x n , the in tegrabilit y condition is the hierarch y (9). All this indeed connects the W ard mo del with the KP equation, but more closely with its matrix v ersion, and not quite on a lev el whic h w ould allo w a closer comparison of solutions. Note that the only nonlinearit y that survive s in the d isp ersionless limit is the comm utator term, but this drop s out in the “pro jection” to scalar KP . On the other hand, we established relati ons b etw een hier ar chies , whic h somewhat ties their solution stru cture together. 6 In th e W ard mo del, J has v alues in S U ( m ), th us ϕ must ha ve v alues in the Lie algebra su ( m ), so has to b e traceless and an ti-Hermitian. Suitable cond itions ha v e to b e imp osed on Φ to ac h iev e this. Via the disp ersionless limit, metho ds of constru cting exact s olutions can b e transfered from the (matrix) KP hierarc hy to the pseudo dual chir al mo del (p d C M) hierarc h y (9). F rom [2 ] w e reca ll the follo win g result. I t d etermin es in p articular v arious classes of (multi- ) lump solutions of the su ( m ) p dCM hierarc hy . 3 See e.g. [12, 13] for related ideas. 4 This Lezno v equation [14] and the W ard equ ation arise by gauge-fix ing of the hyperb olic Bogomoln y equation, see e.g. [15]. 5 Note also th at this transformation leav es t he conserved d ensity (14) inv arian t. 6 W e note, h ow ever, that e.g. the singular sho ck w av e solutions of the disp ersionless limit of t he sc alar KdV equation hav e little in common with KdV solitons. 2 Theorem 1. L et P , T b e c onstant N × N matric es such that T † = − T and P † = T P T − 1 , and V a c onstant N × m matrix. Supp ose ther e is a c onstant solution K of [ P , K ] = − V V † T ( = Q ) such that K † = T K T − 1 . L et X b e an N × N matrix solving [ X , P ] = 0 , X † = T X T − 1 and X x n +1 = X x 1 P n , n = 1 , 2 , . . . . Then ϕ := − V † T ( X − K ) − 1 V solves the s u ( m ) p dCM hier ar chy. Example 1. Let m = 2, N = 2, and P =  p 0 0 p ∗  , T =  0 − 1 1 0  , X =  f 0 0 f ∗  , V =  a b c d  , (11) with complex parameters a, b, c, d, p and a function f (with complex conju gate f ∗ ). Then X x n +1 = X x 1 P n is satisfied if f is an arbitrary holomorphic fu nction of ω := X n ≥ 1 x n p n − 1 . (12) F ur thermore, [ P , K ] = − V V † T has a solution iff ac ∗ + bd ∗ = 0 and β := 2 ℑ ( p ) 6 = 0 (where ℑ ( p ) denotes the imaginary p art of p ). Without restriction of generalit y w e can set the d iagonal part of K to zero, since it can b e absorb ed b y redefinition of f in the formula for ϕ . W e obtain the follo wing comp onents of ϕ , ϕ 11 = − ϕ 22 = i β D  | bc | 2 − | ad | 2 + 2 β ℑ ( a ∗ cf )  , ϕ 12 = − ϕ ∗ 21 = β D  − 2 i ( | c | 2 + | d | 2 ) a ∗ b + β ( a ∗ d f − bc ∗ f ∗ )  , (13) where D := ( | a | 2 + | b | 2 )( | c | 2 + | d | 2 ) + β 2 | f ( ω ) | 2 > 0 if det( V ) 6 = 0. If f is a non-constan t p olynomial in ω , the solution is regular, rational and lo calized. It describ es a simple lump if f is linear in ω . Otherwise it attains a more complicated shap e (see [2] for some examples).  Fixing the v alues of x 4 , x 5 , . . . , we concentrat e on the first p dCM hierarc h y equation. In terms of the v ariables x, y , t giv en b y (2), w e then ha v e ω = 1 2 ( t − x + 2 py + p 2 ( t + x )), subtracting a constan t that can b e absorb ed by redefinition of the fu n ction f in the solution in example 1. This solution b ecomes statio nary , i.e. t -indep end en t, if p = ± i . The conserv ed densit y E := − tr[( ϕ t − ϕ x ) 2 + ϕ y 2 ] / 2 . (14) of (8) is non-n egativ e and will b e u sed b elo w to display the b eha viour of s ome solutions. More complicated solutions are obtained by sup erp osition in the follo wing sense. Giv en data ( X 1 , P 1 , T 1 , V 1 ) and ( X 2 , P 2 , T 2 , V 2 ) that determine s olutions according to theorem 1, we build P =  P 1 0 0 P 2  , X =  X 1 0 0 X 2  , T =  T 1 0 0 T 2  , V =  V 1 V 2  . (15) The d iagonal blo cks of the new big matrix K will b e K 1 and K 2 . It only remains to solv e P 1 K 12 − K 12 P 2 = − V 1 V † 2 T 2 (16) for the upp er off-diagonal blo c k of K and set K 21 = T − 1 2 K † 12 T 1 . In particular, one can sup erp ose lump solutions as giv en in the preceding example. Example 2. Su p erp osition of t wo single lumps with V 1 = V 2 = I 2 , the 2 × 2 un it matrix, yields ϕ 11 = − ϕ 22 = − i D  β 2 | ah 1 | 2 + β 1 | ah 2 | 2 + 2 β 1 β 2 ℑ ( a ∗ h 1 h ∗ 2 ) + ( β 1 + β 2 ) | b | 4  , ϕ 12 = − ϕ ∗ 21 = 1 D  a | h 1 | 2 β 2 h 2 + a ∗ β 1 h 1 | h 2 | 2 + ( b ∗ ) 2 ( aβ 1 h 1 + a ∗ β 2 h 2 )  , (17) 3 Figure 1: Plot s of E at times t = − 90 , − 55 , − 53 , 0 , 3 0 , 80 for the solution in example 2 w ith p 1 = − i (1 − ǫ ) and p 2 = i (1 + ǫ ) where ǫ = 1 / 20, f 1 ( ω 1 ) = 4 i ω 1 , f 2 ( ω 2 ) = i ω 2 2 . Figure 2: Origin an d fate of the lump p air parts app earing in Fig. 1 (to the right in the fir st plot and to th e left in the last). Plots of E at t = − 200 00 , − 2000 , 2000 , 2 0000. where β i := 2 ℑ ( p i ), a := p 1 − p ∗ 2 , b := p 1 − p 2 , h 1 := aβ 1 f 1 , h 2 := a ∗ β 2 f 2 with arbitrary holomorphic functions f 1 ( ω 1 ) (where ω 1 is (12) built with p 1 ), r esp ectiv ely f 2 ( ω 2 ), and D := ( | b | 2 + | h 1 | 2 )( | b | 2 + | h 2 | 2 ) + β 1 β 2 | h 1 − h 2 | 2 . This solution is aga in regular if p 1 6 = p 2 [2]. F or | f 1 | → ∞ (resp. | f 2 | → ∞ ) w e reco v er the single lump solution (13 ) with V = I 2 and f r eplaced b y f 2 (resp. f 1 ). Cho osing p 1 = i (1 − ǫ ) and p 2 = i (1 + ǫ ) (or corresp ondingly with i r ep laced by − i ) with 0 < ǫ ≪ 1, and f 1 , f 2 linear in ω 1 , resp ectiv ely ω 2 , one observes scat terin g at right angle (cf. [4] for the analogous case in th e W ard mod el). If p 1 = − i (1 − ǫ ) and p 2 = + i (1 + ǫ ), one obser ves the follo wing phenomenon: tw o lumps approac h one another, meet, then separate in the orthogonal direction up to some maximal distance, reproac h , merge ag ain, and then separate again while mo ving in the original direction [2]. 7 In the limit ǫ → 0, a v anishes and ϕ b ecomes constan t (assuming f 1 , f 2 indep en d en t of ǫ ), so th at E v anishes. F or other c hoices of f 1 and f 2 more complex phenomena occur , including a kind of “exc hange pro cess” describ ed in the follo wing. Fig. 1 sh o ws plots of E at successiv e times t for th e abov e solution with f 1 linear in ω 1 and f 2 quadratic in ω 2 . The latter fu nction th en corresp onds to a b o wl-shap ed lump (see the left of the plots in Fig. 2) w h ic h, at early times, mo ve s to the left along the x -axis, deforming in to the lu m p p air , sh own on the righ t hand side of 7 See also [16] fo r an analogous phenomenon in case of K P- I lumps. 4 the first plot in Fig. 1, u nder the increasing influence of the s imple lump (corresp onding to the linear function f 1 ) that mo ves to the righ t. When the latter meets the fi rst partner of the lump pair, they merge, separate in y -direction to a maximal distance, mo ve back to ward eac h other and then co n tinue mo ving as a lump pair (shown on the left hand side of the last plot in Fig. 1) in to th e negativ e x -direction. Mean wh ile the remaining partn er of the lump pair, th at ev olv ed from the original b o wl-lump , retreats into the (p ositiv e) x -direction, with diminishing influen ce on the new lump pair, wh ich then finally ev olv es into a b o wl-shap ed lump (see the righ t of the plots in Fig. 2). Th e smaller the v alue of ǫ , the larger th e range of the in teraction.  Other classes of solutions are obtained b y taking for P matrices of Jordan norm al form , generalizing T approp r iately , and bu ilding sup erp ositions in the aforementio n ed sense. Some examples in th e su (2) case ha ve b een wo r k ed out in [2]. This includes examples exhibiting (asymptotic) π /n scattering of n -lu m p configur ations. The p dCM (and also th e W ard mo del) th u s exhibits su rprisingly complex lu mp in teraction patterns, wh ic h are comparativ ely well ac- cessible via the ab o ve theorem, though a kind of systematic classification is b y far out of r eac h. Ac kno w ledgemen t. F M-H would like to thank the German Researc h F oundation for financial supp ort to attend the workshop Algebr a, Ge ometry, and M athematic al Physics in G¨ oteb org. References [1] R.S. W ard. Soliton solutions in an in tegrable c hiral mo del in 2 + 1 dimensions. J. Math. Phys. 29 (198 8), 386-38 9. [2] A. Dimakis and F. M ¨ uller-Hoissen. Disp ersionless limit of the noncommuta tiv e p oten tial K P hierarc hy and solutions of the pseu do du al chiral mo d el in 2 + 1 dimensions. . [3] P .M. Su tcliffe. Non-trivial soliton scattering in an in tegrable c hiral mo del in (2 + 1)- dimensions. J. Math. Phys. 33 (199 2), 2269-2278. [4] R.S. W ard. Nontrivial scattering of lo calized s olitons in a (2 + 1)-dimensional in tegrable system. Phys. L ett. A 208 (1995 ), 203-2 08. [5] T. Ioannidou . Soliton solutions and non trivial scatte ring in an in tegrable chiral mo del in (2 + 1) dimensions. J. Math. Phys. 37 (199 6), 3422- 3441. [6] B. Dai and C.-L. T erng. B¨ ac klund transformations, W ard solitons, and unitons. J. Diff. Ge om. 75 (2007), 57-108 . [7] N. Man ton and P . Sutcliffe. T op olo gic al Solitons . (Cam brid ge Universit y Press, 2004). [8] K.A. Gorsho v, D.E. P elinovsky , and Y u.A. S tepan yan ts. Normal and anomalous scatter- ing, formation and d eca y of b ound states of t wo- dimensional solitons describ ed by th e Kadom tsev-Pe tviash vili equation. JE TP 77 (199 3), 237-245. [9] J. Villarroel and M.J. Ablo w itz. On the discrete sp ectrum of the nons tationary Schr¨ odin ger equation and m ultip ole lumps of the Kadom tsev-P etviash vili I equation. Comm. Math. Phys. 207 (199 9), 1-47. [10] M.J. Ab lo witz, S. Chakra v art y , A.D. T rub atc h, and J. Villarro el. A n o ve l class of solutions of the non-statio nary Sc hr¨ odinger and the Kadom tsev-P etviash vili I equ ations. Phys. L ett. A 267 (2000 ), 132-14 6. [11] S .V. Manak ov, V.E. Zakharo v, L.A. Bordag, A.R. Its, and V.B. Matv eev. Tw o-dimensional solitons of the K adom tsev-Pe tviashvili equation and their interacti on. P hys. L ett. A 63 (1977 ), 205-206 . 5 [12] V.A. Marc henko. Nonline ar E q uations and Op er ator Algebr as . (Reidel, Dordrech t, 1988 ). [13] B. Carl an d C . Sc h ieb old. Nonlinear equations in solito n physics and oper ator ideals. Non- line arity 12 (1999), 333-36 4. [14] A.N. Leznov. Eq u iv alence of four -dimensional self-dualit y equations and the cont in u um analog of the p rincipal chiral field problem. The or. Math. P hys. 73 (198 7), 1233 -1237 . [15] M. Duna jski and S . Man ton. Reduced dynamics of W ard solitons. Nonline arity 18 (2005), 1677- 1689. [16] Z . Lu, E.M. Tian, and R. Grimshaw. Int eraction of t wo lu mp solito n s describ ed b y the Kadom tsev-Pe tviash vili I equation. Wave Motion 40 (2004) , 123-135 . 6