Resonance and web structure in discrete soliton systems: the two-dimensional Toda lattice and its fully discrete and ultra-discrete analogues

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1 INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND GENERAL J. Phys. A: Math. Ge. 37 (24) PII: S35-447(4) Resoace ad web structure i discrete solito systems: the two-dimesioal Toda lattice ad its fully discrete ad ultra-discrete aalogues Ke-ichi Maruo 1 ad Gio Biodii 2 1 Faculty of Mathematics, Kyushu Uiversity, Hakozaki, Higashi-ku, Fukuoka, , Japa 2 Departmet of Mathematics, State Uiversity of New York, Buffalo, NY , USA maruo@math.kyushu-u.ac.jp Received 2 Jue 24, i fial form 21 September 24 Published 24 November 24 Olie at stacks.iop.org/jphysa/37/11819 doi:1.188/35-447/37/49/5 Abstract We preset a class of solutios of the two-dimesioal Toda lattice equatio, its fully discrete aalogue ad its ultra-discrete limit. These solutios demostrate the existece of solito resoace ad web-like structure i discrete itegrable systems such as differetial-differece equatios, differece equatios ad cellular automata (ultra-discrete equatios). PACS umbers: 2.3.Jr, 5.45.Yv 1. Itroductio The discretizatio of itegrable systems is a importat issue i mathematical physics. The most commo situatio is that i which some or all of the idepedet variables are discretized. A discretizatio process i which the depedet variables are also discretized i additio to the idepedet variables is kow as ultra-discretizatio. Oe of the most importat ultradiscrete solito systems is the so-called solito cellular automato (SCA) [12, 16, 17]. A geeral method to obtai the SCA from discrete solito equatios was proposed i [6, 18] ad ivolves usig a appropriate limitig procedure. Aother issue which has received reewed iterest i recet years is the pheomeo of solito resoace, which was first discovered for the Kadomtsev Petviashvili (KP) equatio [8] (see also [7, 11]). More geeral resoat solutios possessig a web-like structure have recetly bee observed i a coupled KP (ckp) system [4, 5] ad for the KP equatio itself [1]. I particular, the Wroskia formalism was used i [1] to classify a class of resoat solutios of KP which also satisfy the Toda lattice hierarchy. It was also cojectured i [1] that resoace ad web structure are ot limited to KP ad ckp, but rather they are a geeric feature of itegrable systems whose solutios ca be expressed i terms of Wroskias /4/ $3. 24 IOP Publishig Ltd Prited i the UK 11819

2 1182 K-i Maruo ad G Biodii The aim of this paper is to study solito resoace ad web structure i discrete solito systems. I particular, by studyig a class of solito solutios of the two-dimesioal Toda lattice (2DTL) equatio, of its fully discrete versio, ad of their ultra-discrete aalogue which was recetly itroduced by Nagai et al [9, 1], we show that a aalogue to the class of solutios studied i [1] ca be defied for all three of these systems, ad that a similar type of resoat solutios with web-like structure is produced as a result i all three of these systems. To our kowledge, this is the first time that resoat behaviour ad web structure have bee observed i discrete solito systems. These results also cofirm that solito resoace ad web-like structure are geeral features of two-dimesioal itegrable systems whose solutios ca be expressed via the determiat formalism. 2. The two-dimesioal Toda lattice equatio We start by cosiderig the two-dimesioal Toda lattice (2DTL) equatio, 2 x t Q = V +1 2V + V 1, (2.1) with Q (x, t) = log[1 + V (x, t)]. Equatio (2.1) ca be writte i biliear form 2 τ x t τ τ τ t x = τ +1τ 1 τ 2 (2.2) through the trasformatio V (x, t) = 2 x t log τ (x, t). (2.3) It is well kow that some solutios of the 2DTL equatio ca be writte via the Casorati determiat form τ = τ (N) [3], with τ (N) = f (N) f (N) f (1) f (1) N 1 +N 1, (2.4) where { f (1)(x,t),...,f(N) (x, t) } is a set of N liearly idepedet solutios of the liear equatios f (i) = f +1, f (i) = f 1, (2.5) x t for 1 i N. (Note that the superscript (i) does ot deote differetiatio here.) For example, a two-solito solutio of the 2DTL is obtaied by the set {f (1),f (2) }, with f (i) = e θ (2i 1) +e θ (2i), i = 1, 2, (2.6) where the phases θ (j) are give by liear fuctios of (,x,t): θ (j) (x, t) = log p j + p j x 1 t + θ (j), j = 1,...,4, (2.7) p j with p 1 <p 2 <p 3 <p 4. Equatio (2.6) ca be exteded to the N-solito solutio by cosiderig {f (1),...,f (N) }, with each f (i) defied accordig to equatio (2.6). O the other had, solutios of the 2DTL equatio ca also be obtaied by the set of τ fuctios { } τ (N) N = 1,...,M with the choice of f -fuctios, f (i) = f +i 1, 1 <i N M, (2.8)

3 Resoace ad web structure i discrete solito systems: the 2D Toda lattice with f = M e θ (j), (2.9) ad with the phases θ (j), 1 j M, still give by equatio (2.7). (Note that the meaig of N ad M is the opposite of [1].) If the f -fuctios are chose accordig to equatio (2.8), the τ fuctio τ (N) is the give by the Hakel determiat f f +N 1 τ (N) = , (2.1) f +N 1 f +2N 2 for 1 N M. It should be oted that, eve whe the set of fuctios { } f (i) N i=1 is chose accordig to equatio (2.8), o derivatives appear i the τ fuctio, ad therefore equatio (2.1) caot be cosidered a Wroskia i the same sese as for the KP equatio (cf equatio (1.9) i [1]). Noetheless, this choice produces a similar outcome as i the KP equatio. Ideed, similar to [1], we have the followig: Lemma 2.1. Let f be give by equatio (2.9), with θ (j) (j = 1,...,M) give by equatio (2.7). The, for 1 N M 1 the τ fuctio defied by the Hakel determiat (2.1) has the form τ (N) = N (i 1,...,i N ) exp θ (i j ), (2.11) 1 i 1 < <i N M where (i 1,...,i N ) is the square of the va der Mode determiat, (i 1,...,i N ) = ( ) 2. pij p il 1 j<l N Proof. Apply the Biet Cauchy theorem to equatio (2.1), as i [1]. A immediate cosequece of lemma 2.1 is that the τ fuctio τ (N) is positive defiite, ad therefore all the solutios geerated by it are o-sigular. Like its aalogue i the KP equatio [1], the above τ fuctio produces solito solutios of resoat type with web structure. More precisely, i the ext sectio we show that, like its aalogue i the KP equatio, the above τ fuctio produces a (N,N + )-solito solutio, that is, a solutio with N = M N asymptotic lie solitos as ad N + = N asymptotic lie solitos as. Before we tur our attetio to resoat solutios, however, it is useful to take a look at the oe-solito solutio of the 2DTL equatio. Let us itroduce the fuctio w (x, t) = x log τ (x, t), (2.12) so that the solutio of the 2DTL equatio is give by V (x, t) = t w (x, t). (2.13) If τ = e θ (1) +e θ (2), with θ (i) give by equatio (2.7) ad p 1 <p 2, the w is give by w = 1 2 (p 1 + p 2 ) (p ( 1 p 2 ) tah 1 2 θ (1) θ (2) ) { p1 as x, p 2 as x,

4 11822 K-i Maruo ad G Biodii which leads to the oe-solito solutio of the 2DTL equatio: V = 1 ( 1 4 (p 1 p 2 ) 1 ) sech 2 1 ( θ (1) θ (2) ). (2.14) p 1 p 2 2 I the x plae, this solutio describes a plae wave u = (k x ωt) with x = (x, ), havig waveumber vector k = (k x,k ) ad frequecy ω give by k = (p 1 p 2, log p 1 log p 2 ) =: k 1,2, ω = 1 p 1 1 p 2 =: ω 1,2. The solito parameters (k,ω) satisfy the dispersio relatio ωk x + 2 cosh k 2 =. The above oe-solito solutio (2.14) is referred to as a lie solito, sice i the x plae it is localized aroud the (cotour) lie θ (1) = θ (2). Sice i this paper we are iterested i the patter of solito solutios i the x plae, we will refer to c = dx/d as the velocity of the lie solito i the x directio. (That is, c = idicates the directio of the positive -axis.) For the solito solutio i equatio (2.14), this velocity is c 1,2, where c i,j = (log p i log p j )/(p i p j ). (2.15) 3. Resoace ad web structure i the two-dimesioal Toda lattice equatio We first cosider (N, 1)-solito solutios, i.e., solutios obtaied whe N = 1. I particular, we start with (2, 1)-solito solutios (i.e., N = 1 ad M = 3), whose τ fuctio is give by τ = e θ (1) +e θ (2) +e θ (3), with p 1 <p 2 <p 3 without loss of geerality. The correspodig fuctio w describes the cofluece of two shocks: two shocks for (each correspodig to a lie solito for V ) with velocities c 1,2 ad c 2,3 merge ito a sigle shock for with velocity c 1,3, with c i,j give by equatio (2.15) i all cases. This Y-shape iteractio represets a resoace of three lie solitos. The resoace coditios for three solitos with waveumber vectors {k i,j 1 i<j 3} ad frequecies {ω i,j 1 i<j 3} are give by k 1,2 + k 2,3 = k 1,3, ad ω 1,2 + ω 2,3 = ω 1,3, (3.1) which are trivially satisfied by those lie solitos. We should poit out that this solutio is also the resoat case of the ordiary two-solito solutio of the 2DTL equatio. As metioed earlier, ordiary two-solito solutios are give by the N = 2τ fuctio (2.4) with (2.6). The explicit form of the τ (2) fuctio is τ (2) = (p 1 p 3 ) e θ (1,3) + (p 1 p 4 ) e θ (1,4) + (p 2 p 3 ) e θ (2,3) + (p 2 p 4 ) e θ (2,4), where for brevity we itroduced the otatio θ (i,j) equatio (2.7), as before. Note that if p 2 = p 3,theτ (2) = θ (i) + θ (j), ad where θ (i) fuctio ca be writte as is give by τ (2) = e θ (1) +θ (2) +θ [ (4) (4) (2) (1) θ θ θ (p 1 p 3 ) e + (p 1 p 4 ) e + (p 2 p 4 ) e ], where = exp ( θ (3) θ (2) ) = costat. Sice the expoetial factor e θ (1) +θ (2) +θ (4) gives zero cotributio to the solutio V = t x log τ (2) (2), the above τ fuctio is equivalet to a (2, 1)-solito solutio except for the sigs of the phases (more precisely, it is a (1, 2)-solito). Note also that the coditio p 2 = p 3 is othig else but the resoace coditio, ad it describes the limitig case of a ifiite phase shift i the ordiary two-solito solutio, where the phase shift betwee the solitos as ± is give by δ = (p 1 p 3 )(p 2 p 4 )/[(p 2 p 3 )(p 1 p 4 )]. (3.2)

5 Resoace ad web structure i discrete solito systems: the 2D Toda lattice The resoace process for the (N, 1)-solito solutios of the 2DTL equatio ca be expressed as a geeralizatio of the cofluece of shocks discussed above (cf [1]). We ext cosider more geeral (N,N + )-solito solutios. Followig [1], we ca describe the asymptotic patter of the solutio i the geeral case N 1 by itroducig a local coordiate frame (ξ, ) i order to study the asymptotics for large, with x = c + ξ. The phase fuctios θ (i) with θ (i) i f i equatio (2.7) the become = p i ξ + η i (c) + θ (i), for i = 1,...,M, η i (c) := p i (c + (1/p i ) log p i ). Without loss of geerality, we assume a orderig for the parameters {p i i = 1,...,M}: <p 1 <p 2 < <p M. The oe ca easily show that the lies η = η i (c) are i geeral positio; that is, each lie η = η i (c) itersects with all other lies at M 1 distict poits i the c η plae; i other words, oly two lies meet at each itersectio poit. The goal is ow to fid the domiat expoetial terms i the τ (N) fuctio (2.11) for ± as a fuctio of the velocity c. First ote that if oly oe expoetial is domiat, the w = x log τ (N) is just a costat, ad therefore the solutio V = t w is zero. The, otrivial cotributios to V arise whe oe ca fid two expoetial terms which domiate over the others. Note that because the itersectios of the η i are always pairwise, three or more terms caot make a domiat balace for large. I the case of (N, 1)-solito solutios, it is easy to see that at each c the domiat expoetial term for is provided by oly η 1 ad/or η M, ad therefore there is oly oe shock (N + = 1) movig with velocity c 1,M correspodig to the itersectio poit of η 1 ad η M. O the other had, as, each term η j ca become domiat for some c, ad at each itersectio poit η j = η j+1 the two expoetial terms correspodig to η j ad η j+1 give a domiat balace; therefore there are N = M 1 shocks movig with velocities c j,j+1 for j = 1,...,M 1. fuctio i (2.11) ivolves expoetial terms havig combiatios of phases. I this case the expoetial terms that make a domiat balace ca be foud usig the same methods as i [1]. Let us first defie the level of itersectio of the η i (c). Note that c i,j i equatio (2.15) idetifies the itersectio poit of η i (c) ad η j (c), i.e., η i (c i,j ) = η j (c i,j ). I the geeral case, N 1, the τ (N) Defiitio 3.1. We defie the level of itersectio, deoted by σ i,j, as the umber of other η l that are larger tha η i (c i,j ) = η j (c i,j ) at c = c i,j. That is, σ i,j := {η l η l (c i,j )>η i (c i,j ) = η j (c i,j )}. We also defie I()as the set of pairs (η i,η j ) havig the level σ i,j =, amely I() := {(η i,η j ) σ i,j =, for i<j}. The level of itersectio lies i the rage σ i,j M 2. Note also that the total umber of pairs (η i,η j ) is ( ) M = 12 M 2 2 M(M 1) = I(). =

6 11824 K-i Maruo ad G Biodii Oe ca show that Lemma 3.2. The set I()is give by I() ={(η i,η M +i 1 ) i = 1,...,+1}. Proof. From the assumptio q 1 <q 2 < <q M, we have the followig iequality at c = c i,j (i.e. η i = η j )fori<j, η i+1,...,η j 1 <η i = η j <η 1,...,η i 1,η j+1,...,η M. The takig j = M 1 leads to the assertio of the lemma. Now defie N + = N ad N = M N. The above lemma idicates that, for each itersectig pair (η i,η j ) with the level N 1 (N + 1), there are N + 1termsη l which are smaller (larger) tha η i = η j. The the sum of those N + 1 terms with either η i or η j provides two domiat expoets i the τ (N) fuctio for ( ) (see more detail i the proof of theorem 3.3). Note also that I(N ± 1) =N. Now we ca state our mai theorem: Theorem 3.3. Let w be defied by equatio (2.12), with τ (N) give by equatio (2.11). The w has the followig asymptotics for ± : (i) For ad x = c i,n+ +i + ξ for i = 1,...,N, { Ki (+, ) := N+i j=i+1 w p j as ξ, K i (, ) := N+i 1 j=i p j as ξ. (ii) For ad x = c i,n +i + ξ for i = 1,...,N +, { Ki (+, +) := i 1 w p j + N i+1 p M j+1 as ξ, K i (, +) := i p j + N 1 p M j+i as ξ. where c i,j is give by equatio (2.15). Proof. First ote that at the poit η i = η N+i, i.e., (η i,η N+i ) I(N 1), from lemma 3.2 we have the iequality, η i+1,η i+2,...,η i+n 1 <η i = η N+i. }{{} N 1 This implies that, for c = (log p N+i log p i )/(p N+i p i ), the followig two expoetial terms i the τ (N) fuctio i lemma 2.1, exp N+i 1 θ (j) j=i, exp N+i j=i+1 provide the domiat terms for. Note that the coditio η i = η N+i leads to c = c i,n+i = (log p N+i log p i )/(p N+i p i ). Thus the fuctio w ca be approximated by the followig form alog x = c i,n+i + ξ for : w ξ log( i(, ) e Ki(, )ξ + i (+, ) e Ki(+, )ξ ) = K i(, ) i (, ) e Ki(, )ξ + K i (+, ) i (+, ) e K i(+, )ξ, i (, ) e K i(, )ξ + i (+, ) e K i(+, )ξ θ (j), = K i(, ) i (, ) e (p i p N+i )ξ + K i (+, ) i (+, ), i (, ) e (p i p N+i )ξ + i (+, )

7 Resoace ad web structure i discrete solito systems: the 2D Toda lattice where i (, ) = (i,..., N + i 1) exp i (+, ) = (i +1,...,N + i)exp N+i 1 j=i N+i θ (j) θ (j) j=i+1 Now, from p i <p N+i it is obvious that w has the desired asymptotics as ξ ± for. Similarly, for the case of (η i,η N +i) I(N + 1) we have the iequality η i = η N +i <η 1,η 2,...,η i 1,η N +i+1...,η M. }{{} N 1 The the domiat terms i the τ (N) expoetial terms i exp θ (j) N i +. fuctio o x = c i,n +i + ξ for are give by the θ (M j+1), i 1 exp N i+1 θ (j) + θ (M j+1) The, followig the same argumet as before, we obtai the desired asymptotics as ξ ± for. For other values of c, that is for c c i,n+ +i ad c c i,n +i, just oe expoetial term is domiat, ad thus w approaches a costat as. This completes the proof. Theorem 3.3 determies the complete structure of asymptotic patters of the solutios V (x, t) give by (2.1). Ideed, theorem 3.3 ca be summarized as follows: as,the fuctio w has N jumps, movig with velocities c j,n+ +j for j = 1,...,N ;as,w has N + jumps, movig with velocities c i,n +i for i = 1,...,N +. Sice each jump represets a lie solito for V (x, t), the whole solutio therefore represets a (N,N + )-solito. The velocity of each of the asymptotic lie solitos i the (N,N + )-solito is determied from the c η graph of the levels of itersectios. As a example, i figure 1 we show a (2, 1)-solito solutio (also called a Y-shape solutio, or a Y-juctio), a (2, 2)-solito solutio, a (2, 3)-solito solutio ad a (3, 3)-solito solutio. Note that, give a set of M phases (as determied by the parameters p i for i = 1,...,M), the same graph ca be used for ay (N,N + )-solito with N + N + = M. I particular, if M = 2N, wehaven + = N = N, ad theorem 3.3 implies that the velocities of the N icomig solitos are equal to those of the N outgoig solitos. I the case of the ordiary multi-solito solutio of the 2DTL equatio, the τ fuctio (2.4) does ot cotai all the possible combiatios of phases ad therefore the theorem should be modified. However, the key idea of usig the levels of itersectio for the asymptotic aalysis is still applicable. I fact, by cosiderig the τ (N) fuctio give by the Casorati determiat (2.4) with f i = e θ (2i 1). +e θ (2i) for i = 1,...,N ad p 1 <p 2 < <p 2N, oe ca fid the asymptotic velocities for the ordiary N-solito solutios as c 2i 1,2i = (log p 2i log p 2i 1 )/(p 2i p 2i 1 ). Note that these velocities are differet from those of the resoat N-solito solutios. Note also that eve whe N + = N = N, the iteractio patter of resoat solito solutios differs from that of ordiary N-solito solutios. As see from figure 1, the resoat solutios of the 2DTL obtaied from equatio (2.1) are very similar to the solitos of the KP ad coupled KP equatio [1, 4, 5], where such solutios were called spider-web solitos.

8 11826 K-i Maruo ad G Biodii 1 (a) 15 (b) (c) 15 (d) Figure 1. Resoat solutios of the two-dimesioal Toda lattice: (a) (2, 1)-solito solutio (i.e., a Y-juctio) at t =, with N = 1,M = 3,p 1 = 1/4,p 2 = 1/2,p 3 = 2; (b) (2, 2)-solito solutio at t = 14, with N = 2,M = 4,p 1 = 1/8,p 2 = 1/2,p 3 = 1adp 4 = 4; (c) (3, 2)-solito solutio at t = 1, with N = 2,M = 5,p 1 = 1/1,p 2 = 1/5,p 3 = 1/2,p 4 = 1adp 5 = 6; (d) (3, 3)-solito solutio at t = 14, with N = 3,M = 6,p 1 = 1/1,p 2 = 1/4,p 3 = 1/2 ad p 4 = 1,p 5 = 2,p 6 = 4. I all cases the horizotal axis is ad the vertical axis is x, ad each figure is a plot of q(,x,t) i logarithmic greyscale. Note that the values of i the horizotal axis are discrete. (I cotrast, a ordiary N-solito solutio produces a simple patter of N itersectig lies.) The web structure maifests itself i the umber of bouded regios, the umber of vertices ad the umber of itermediate solitos, which are respectively (N 1)(N + 1),2N N + M ad 3N N + 2M for a (N,N + )-solito solutio [1]. (I cotrast, a ordiary N-solito solutio has (N 1)(N 2)/2 bouded regios ad N(N 1)/2 iteractio vertices.) Fially, it should be oted that, as i the KP equatio, oly the Y-shape solutio is a travellig wave solutio. All other resoat solutios (as well as ordiary N-solito solutios with N 3) have a time-depedet shape, as show i [1]. 4. The fully discrete 2D Toda lattice equatio The 2DTL equatio (2.1) is a differetial-differece evolutio equatio, sice oly oe of the idepedet variables is discrete, while the other two are cotiuous. Hereafter, we refer to equatio (2.1) as a semi-cotiuous case. We ow cosider a fully discrete aalogue of the

9 Resoace ad web structure i discrete solito systems: the 2D Toda lattice DTL equatio (2.1), amely + l m Q l,m, = V l,m 1,+1 V l+1,m 1, V l,m, + V l+1,m, 1, (4.1) V l,m, = (δκ) 1 log[1 + δκ(exp Q l,m, 1)], with l,m, Z,l ad m beig the discrete aalogues of the time t ad space x coordiates, respectively, ad where + l ad m are the forward ad backward differece operators defied by + l f l,m, = f l+1,m, f l,m,, (4.2) δ m f l,m, = f l,m, f l,m 1,. (4.3) κ Equatio (4.1), which is the discrete aalogue of equatio (2.1), ca be writte i biliear form [2] i a maer similar to equatio (2.2): ( + l m τ ) l,m, τl,m, ( + l τ ) l,m, m τ l,m, = τ l,m 1,+1 τ l+1,m, 1 τ l+1,m 1, τ l,m,, (4.4) with Q l,m, related to τ l,m, by the trasformatio V l,m, = + l m log τ l,m,, i.e., Q l,m, = log τ l+1,m+1, 1τ l,m,+1. (4.5) τ l+1,m, τ l,m+1, Note that Q l,m, = log[1 + (e δκv l,m, 1)/δκ]. Special solutios of equatio (4.4) (whichis the discrete aalogue of equatio (2.2)) are obtaied whe the τ fuctio τ l,m, is expressed i terms of a Casorati determiat τ l,m, = τ (N) l,m, as [2] f (1) l,m, f (1) l,m,+1 f (1) l,m,+n 1 τ (N) l,m, = f (2) l,m, f (2) l,m,+1 f (2) l,m,+n 1, (4.6)... f (N) l,m, f (N) l,m,+1 f (N) l,m,+n 1 where each of the fuctios { f (i) l,m,,i = 1, 2,...,N} satisfies the followig discrete dispersio relatios: + l f l,m, = f l,m,+1, (4.7) m f l,m, = f l,m, 1. (4.8) If we take as a solutio for equatios (4.7) ad (4.8) the fuctios f (i) l,m, = φ(p i) + φ(q i ), (4.9) with φ(p) = p (1+δp) l (1+κp 1 ) m, (4.1) the τ fuctio (4.6) yields a N-solito solutio for the discrete 2DTL equatio (4.4). As i the semi-cotiuous 2DTL, however, solutios of equatio (4.4) ca also be obtaied whe we cosider the τ fuctio defied by the Hakel determiat f l,m, f l,m,+1 f l,m,+n 1 τ (N) l,m, = f l,m,+1 f l,m,+2 f l,m,+n.., (4.11). f l,m,+n 1 f l,m,+n f l,m,+2n 2

10 11828 K-i Maruo ad G Biodii where f l,m, = M α i φ(p i ), (4.12) i=1 which correspods to choosig f (i) l,m, = f l,m,+i 1, (4.13) for i = 1,...,N. Without loss of geerality, we ca label the parameters p i so that <p 1 <p 2 < <p M 1 <p M. The, as i the semi-cotiuous 2DTL, we have the followig: Lemma 4.1. Let f l,m, be give by equatio (4.13). The, for 1 N M 1,theτ fuctio defied by the Hakel determiat (4.11) has the form N τ (N) = (i 1,...,i N ) α ij φ ( ) p ij (4.14) 1 i 1 < <i N M where (i 1,...,i N ) is the square of the va der Mode determiat, (i 1,...,i N ) = ( ) 2. pij p il (4.15) 1 j<l N Proof. Agai, the result follows by applyig the Biet Cauchy theorem to the Hakel determiat (4.11). Ulike its couterpart i the semi-cotiuous 2DTL equatio, the τ fuctio i equatio (4.11) caot be writte i terms of a Wroskia, sice o derivatives appear. However, as i the semi-cotiuous 2DTL equatio, the τ fuctio thus defied is positive defiite, ad therefore all the solutios geerated by it are o-sigular. I the ext sectio we show that, like its aalogue i the semi-cotiuous 2DTL equatio, the above τ fuctio produces solito solutios of resoat type with web structure, ad we cojecture that, like i the cotiuous case, a (N,N + )-solito with N = M N ad N + = N is created. Like with the semi-cotiuous 2DTL equatio, however, before discussig resoat solutios it is coveiet to first look at oe-solito solutios of the fully discrete 2DTL equatio. Let us itroduce the aalogue of equatio (2.12), amely the fuctio w l,m, = log τ l,m+1, 1, (4.16) τ l,m, so that the solutio of the discrete 2DTL equatio is give by Q l,m, = log τ l+1,m+1, 1τ l,m,+1 = w l+1,m, w l,m,+1. (4.17) τ l+1,m, τ l,m+1, It is also useful to rewrite the fuctio φ(p i ) i equatios (4.1), (4.12) as φ(p i ) = e θ (i) l,m,, where θ (i) l,m, = log p i m log ( 1+κp 1 ) i + l log(1+δpi ) + θ (i). (4.18) If τ = e θ (1) l,m, +e θ (2) l,m,, with p1 <p 2, the w l,m, is give by w l,m, = log 1 ( ( ) 2 p κp 1 1 ( ) 1 + p κp 1 1 ) 2 ( ( ) p κp 1 1 ( ) 1 p κp 1 1 ) ( 2 tah 1 2 θ (1) l,m, θ (2) ) l,m, { log p1 log ( 1+κp 1 ) 1 as, log p 2 log ( 1+κp 1 ) 2 as,

11 Resoace ad web structure i discrete solito systems: the 2D Toda lattice which leads to the oe-solito solutio of the discrete 2DTL equatio. I the m plae, this solutio describes a plae wave exp(q l,m, ) = (k x ωl) with x = (, m), havig waveumber vector k = (k,k m ) ad frequecy ω give by k = ( log p 1 log p 2, log ( 1+κp 1 ) ( )) 1 +log 1+κp 1 2 =: k1,2, ω = log(1+δp 1 ) +log(1+δp 2 ) =: ω 1,2. The solito parameters (k,ω) ow satisfy the discrete dispersio relatio (e ω 1) (1 e k m ) = δκ(e k m+k +e ω k e ω k m 1). The oe-solito solutio (4.17) is referred to as a lie solito sice, like its semi-cotiuous aalogue, it is localized aroud the (cotour) lie θ (1) l,m, = θ (2) l,m, i the m plae. Agai, we will refer to c = d/dm as the velocity of the lie solito i the directio. For the above lie solito solutio, this velocity is give by c 1,2, where ow c i,j = ( log ( 1+κp 1 ) ( ))/ i log 1+κp 1 j (log pi log p j ). (4.19) 5. Resoace ad web structure i the discrete 2D Toda lattice equatio As i the semi-cotiuous case, we first cosider (N, 1)-solito solutios, i.e., solutios obtaied i the case N = 1, ad i particular we start from (2, 1)-solito solutios (i.e., the case N = 1 ad M = 3), whose τ fuctio is give by τ = e θ (1) l,m, +e θ (2) l,m, +e θ (3) l,m,, with θ (i) l,m, give by equatio (4.18), ad where p 1 <p 2 <p 3 without loss of geerality. As i the cotiuous case, this solutio describes the cofluece of two shocks: two shocks for m (each correspodig to a lie solito for Q l,m, ) with velocities c 1,2 ad c 1,3 merge ito a sigle shock for m with velocity c 1,3, where c i,j is give by equatio (4.19) i all cases. This Y-shape iteractio represets a resoace of three lie solitos. The resoace coditios for three solitos with the waveumber vectors {k i,j 1 i < j 3} ad the frequecies {ω i,j 1 i < j 3} are still give by equatio (3.1), ad agai are trivially satisfied by those lie solitos. Furthermore, this solutio is also the resoat case of the ordiary two-solito solutio of the discrete 2DTL equatio, arisig i the limit of a ifiite phase shift. The resoace process for the (N, 1)- solito solutios of the discrete 2DTL equatio ca be expressed as a geeralizatio of the cofluece of shocks discussed earlier. Next we cosider more geeral (N,N + )-solito solutios. Followig [1] ad the semicotiuous case, we ca describe the asymptotic patter of the solutio by itroducig a local coordiate frame (ξ, m) i order to study the asymptotics for large m with = cm + ξ. The the phase fuctios θ (i) l,m, become θ (i) l,m, = ξ log p i + η i (c)m + θ (i), for i = 1,...,M, with ( ( )/ ) η i (c) := log p i c log 1+κp 1 i log pi. Without loss of geerality, we assume a orderig for the parameters {p i i = 1,...,M}: <p 1 <p 2 < <p M. The, as i the semi-cotiuous case, oe ca easily show that the lies η = η i (c) are i geeral positio. As before, the goal is the to fid the domiat expoetial terms i the τ (N) l,m, fuctio (4.14) form ± as a fuctio of

12 1183 K-i Maruo ad G Biodii the velocity c. First ote that if oly oe expoetial is domiat, the w l,m, = log ( τ (N) / l,m+1, 1 τ (N) l,m,) is just a costat, ad therefore the solutio Ql,m, = w l+1,m, w l,m,+1 is zero. The, as i the semi-cotiuous case, otrivial cotributios to Q l,m, arise whe oe ca fid two expoetial terms which domiate over the others. Also, as i the semicotiuous case, sice the itersectios of the η i are always pairwise, three or more terms caot make a domiat balace for large m. For (N, 1)-solito solutios, it is easy to see that at each c the domiat expoetial term for m is provided by oly η 1 ad/or η M, ad therefore there is oly oe shock (N = 1) movig with velocity c 1,M correspodig to the itersectio poit of η 1 ad η M. O the other had, as m, each term η j ca become domiat for some c, ad at each itersectio poit η j = η j+1 the two expoetial terms correspodig to η j ad η j+1 give a domiat balace; therefore there are N = M 1 shocks movig with velocities c j,j+1 for j = 1,...,M 1. I the geeral case, N 1, the τ (N) l,m, fuctio i (4.14) ivolves expoetial terms havig combiatios of phases, ad two expoetial terms that make a domiat balace, ca be foud i a similar way as i the semi-cotiuous case. We defie agai the level of itersectio of the η i (c). Agai, c i,j i equatio (4.19) idetifies the itersectio poit of η i (c) ad η j (c), i.e., η i (c i,j ) = η j (c i,j ). Defiitio 5.1. We defie the level of itersectio, deoted by σ i,j, as the umber of other η l that at c = c i,j are larger tha η i (c i,j ) = η j (c i,j ). That is, σ i,j := {η l η l (c i,j )>η i (c i,j ) = η j (c i,j )}. We also defie I()as the set of pairs (η i,η j ) havig the level σ i,j =, amely I() := {(η i,η j ) σ i,j =, for i<j}. As i the semi-cotiuous case, oe ca the show the followig: Lemma 5.2. The set I()is give by I() ={(η i,η M +i 1 ) i = 1,...,+1}. Proof. See the proof of lemma 3.2. As i the semi-cotiuous case, let N + = N ad N = M N. The above lemma idicates that, for each itersectig pair (η i,η j ) with the level N 1 (N 1), there are N + 1terms η l which are smaller (larger) tha η i = η j. The the sum of those N + 1 terms with either η i or η j provides two domiat expoets i the τ (N) l,m, fuctio for m (m ). We the have the followig: Theorem 5.3. Let w l,m, be a fuctio defied by equatio (4.16), with τ (N) l,m, give by equatio (4.14). The w l,m, has the followig asymptotics for m ± : (i) For m ad = c i,n+ +im + ξ for i = 1,...,N, K i (+, ) := N+i j=i+1 w l,m, log p j as ξ, K i (, ) := N+i 1 j=i log p j as ξ. (ii) For m ad = c i,n +im + ξ for i = 1,...,N +, K i (+, +) := i 1 w l,m, log p j + N i+1 log p M j+1 as ξ, K i (, +) := i log p j + N 1 log p M j+i as ξ, where c i,j is give by equatio (4.19).

13 Resoace ad web structure i discrete solito systems: the 2D Toda lattice (a) 15 (b) (c) 15 (d) Figure 2. Resoat solutios of the fully discrete two-dimesioal Toda lattice: (a) (2, 1)-solito solutio (i.e., a Y-juctio) at l =, with p 1 = 1/1,p 2 = 1/2,p 3 = 1, (b) (2, 2)-solito solutio at l = 4, with p 1 = 1/1,p 2 = 1/2,p 3 = 2,p 4 = 15; (c) (3, 2)-solito solutio at l = 8, with p 1 = 1/2,p 2 = 1/2,p 3 = 2,p 4 = 1,p 5 = 6; (d ) (3, 3)-solito solutio at l = 8, with p 1 = 1/2,p 2 = 1/2,p 3 = 2,p 4 = 1,p 5 = 2,p 6 = 12. I all cases δ = κ = 1/4; the horizotal axis is ad the vertical axis is ad each figure is a plot of Q l,m, i logarithmic greyscale. Note that the values of both m ad i the horizotal ad vertical axes are discrete. Proof. Oce the obvious modificatios are made, the proof proceeds exactly like i the semi-cotiuous case, amely theorem 3.3. Like its couterpart i the semi-cotiuous case, theorem 5.3 determies the complete structure of asymptotic patters of the solutios Q l,m, give by (4.11). Ideed, theorem 5.3 ca be summarized as follows: as m, the fuctio w l,m, has N jumps, movig with velocities c j,n+ +j for j = 1,...,N ;asm,w l,m, has N + jumps, movig with velocities c i,n +i for i = 1,...,N +. Sice each jump represets a lie solito of Q l,m,, the whole solutio therefore represets a (N,N + )-solito. The velocity of each of the asymptotic lie solitos i the (N,N + )-solito is determied from the c η graph of the levels of itersectios. Note that, give a set of M phases (as determied by the parameters p i for i = 1,...,M), the same graph ca be used for ay (N,N + )-solito with N + N + = M. As a example, i figure 2 we show a (2, 1)-solito solutio, a (2, 2)-solito solutio, a (2, 3)-solito solutio ad a (3, 3)-solito solutio. I particular, if M = 2N, wehaven + = N = N, ad theorem 5.3 implies that the velocities of the N icomig solitos are equal to those of the N outgoig solitos. I the

14 11832 K-i Maruo ad G Biodii case of the ordiary multi-solito solutios of the discrete 2DTL equatio, the τ fuctio (4.6) does ot cotai all the possible combiatios of phases ad therefore theorem 5.3 should be modified. However, as i the semi-cotiuous case, the idea of usig the levels of itersectio is still applicable, ad oe ca fid that the asymptotic velocities for the ordiary N-solitos geerated by the Casorati determiat (4.6) with f i = e θ (2i 1) +e θ (2i) (i = 1,...,N) ad p 1 <p 2 < <p 2N are c 2i 1,2i = ( log ( 1+κp 1 ) ( 2i log 1+κp 2i 1))/ (log p2i log p 2i 1 ). Note that, like i the semi-cotiuous case, these velocities are differet from those of the resoat N-solito solutios. The resoat solutios of the fully discrete 2DTL provide the basis for the costructio of the resoat solutio of the ultra-discrete 2DTL, as is show i the ext two sectios. 6. The ultra-discrete two-dimesioal Toda lattice We ow tur our attetio to a ultra-discrete aalogue of the 2DTL equatio. Usig equatios (4.2) ad (4.3), we first write the 2DTL equatio (4.4) i biliear form as (1 δκ)τ l+1,m, τ l,m+1, τ l+1,m+1, τ l,m, + δκτ l,m,+1 τ l+1,m+1, 1 =. (6.1) We defie the differece operator as = e ( + )( + l + + ) m, (6.2) where from here o the symbols + l, + m ad + will be used to deote the differece operators + l = e l 1, + m = e m 1, + = e 1, (6.3) ad where the shift operators e l, e m ad e are defied by e f l,m, = f l,m,+1, etc. That is, f l,m, = f l+1,m+1, 1 + f l,m,+1 f l+1,m, f l,m+1,. (6.4) Usig equatios (6.3) ad (6.2), we ca rewrite equatio (6.1)as (1 δκ) + δκ exp[ log τ l,m, ] = exp [ + l + m log τ l,m,], (6.5) which becomes, takig a logarithm ad applyig (assumig δκ 1), [ ] δκ log 1+ 1 δκ exp( log τ l,m, ) = + l + m log τ l,m,. (6.6) We ow take a ultra-discrete limit of equatio (6.6) followig equatio (2.1) [9, 1]. This is accomplished by choosig the lattice itervals as δ ɛ = e r/ɛ, κ ɛ = e s/ɛ, (6.7) where r, s Z are some predetermied iteger costats, ad by defiig vl,m, ɛ = ɛ log τl,m, ɛ. (6.8) Takig the limit ɛ + i equatio (6.6) ad otig that lim ɛ + ɛ log(1+e X/ɛ ) = max(,x), we the obtai + l + m v l,m, = max(,v l,m, r s), (6.9) where v l,m, = lim ɛ +vl,m, ɛ. That is, usig equatio (6.8), v l,m, = lim ɛ ɛτɛ + l,m,. (6.1) Equatio (6.9) is the ultra-discrete aalogue of the 2DTL equatio ad ca be cosidered a cellular automato i the sese that v l,m, takes o iteger values.

15 Resoace ad web structure i discrete solito systems: the 2D Toda lattice Let us briefly discuss ordiary solito solutios of the ultra-discrete 2DTL equatio (6.9). As show i [9, 1], solito solutios for the ultra-discrete 2DTL equatio (6.9) are obtaied by a ultra-discretizatio of the solito solutio of the discrete 2DTL equatio (4.4). For example, a oe-solito solutio for equatio (4.4) is give by τ l,m, = 1+η 1, (6.11) with φ(p i ) η i = α i φ(q i ), (6.12) ad where φ(p) = p (1+δp) l (1+κp 1 ) m (6.13) as before. We itroduce a ew depedet variable ρl,m, ɛ = ɛ log τ l,m,, (6.14) ad ew parameters P 1,Q 1,A 1 Z as e P1/ɛ = p 1, e Q1/ɛ = q 1, e A1/ɛ = α 1. (6.15) Takig the limit ɛ +, we the obtai ρ l,m, = max(, 1 ), (6.16) where i = A i + (P i Q i ) + l{max(,p i r) max(,q i r)} + m{max(, Q i s) max(, P i s)}, (6.17) with e r/ɛ = δ ad e s/ɛ = κ as before, ad where ρ l,m, = lim ɛ + ρl,m, ɛ. Accordig to equatios (6.1) ad (6.14), the oe-solito solutio for equatio (6.9) is the give by v l,m, = ρ l+1,m+1, 1 + ρ l,m,+1 ρ l+1,m, ρ l,m+1,. (6.18) Usig a similar procedure we ca costruct a two-solito solutio. Equatio (4.4) admits a two-solito solutio give by τ l,m, = 1+η 1 + η 2 + θ 12 η 1 η 2, (6.19) with θ 12 = (p 2 p 1 )(q 1 q 2 ) (q 1 p 2 )(q 2 p 1 ), (6.2) ad where η i = α i φ(p i )/φ(q i ) as before (i = 1, 2). I order to take the ultra-discrete limit of the above solutio, we suppose without loss of geerality that the solito parameters satisfy the iequality <p 1 <p 2 <q 2 <q 1. (6.21) Itroducig agai the depedet variable ρl,m, ɛ = ɛ log τ l,m,, as well as iteger parameters P i,q i ad A i as e Pi/ɛ = p i, e Qi/ɛ = q i, e Ai/ɛ = α i (6.22) (i = 1, 2), ad takig the limit of small ɛ, we obtai ρ l,m, = max(, 1, 2, P 2 Q 2 ), (6.23) where i (i = 1, 2) was defied i equatio (6.17), with ρ l,m, = lim ɛ + ρl,m, ɛ agai, ad where v l,m, is obtaied from ρ l,m, usig equatio (6.18). Note that P 1 <P 2 <Q 2 <Q 1.

16 11834 K-i Maruo ad G Biodii More i geeral, startig from equatios (4.6) ad (4.9) (with <p 1 <p 2 < < p M <q M <q N 1 < <q 1 ) ad repeatig the same costructio, oe obtais the N-solito solutio of the ultra-discrete 2DTL equatio (6.9) as[1] ρ l,m, = max µ i i + µ i µ i (P i Q i ) (6.24) µ=,1 1 i M 1 i<i M where max µ=,1 idicates maximizatio over all possible combiatios of the itegers µ i =, 1, with i = 1,...,M.Agai,v l,m, is obtaied from ρ l,m, via equatio (6.18). Ordiary solito solutios correspodig to the above choices were preseted i [9, 1]. I the ext sectio we show how this basic costructio ca be geeralized to obtai resoat solito solutios. 7. Resoace ad web structure i the ultra-discrete 2D Toda lattice equatio Followig [1], we ow costruct more geeral solutios of the ultra-discrete 2DTL equatio (6.9) which display solito resoace ad web structure. We first cosider the case of a (2, 1)-solito for equatio (4.4), which is give by τ l,m, = ξ 1 + ξ 2 + ξ 3, (7.1) where ξ i = α i φ(p i ) (7.2) (i = 1, 2, 3), with φ(p) = p (1+δp) l (1+κp 1 ) m (7.3) as before, ad where agai we take <p 1 <p 2 <p 3. As i the previous sectio, we itroduce the ew depedet variable ρl,m, ɛ = ɛ log τ l,m,, (7.4) ad ew parameters as e Pi/ɛ = p i, e Ai/ɛ = α i (7.5) (i = 1, 2, 3), with e r/ɛ = δ ad e s/ɛ = κ as before. Takig the limit ɛ +, we the obtai ρ l,m, = max(r 1,R 2,R 3 ) (7.6) where ρ l,m, = lim ɛ + ρl,m, ɛ as before, but where ow R i = A i + P i + l max(,p i r) m max(, P i s) (7.7) (i = 1, 2, 3). Figure 3 shows that this solutio, which agai ca be called a (2, 1)-solito, is a Y-shape solutio. Note however that the (2, 1)-solito i figure 3 looks like a (1, 2)-solito, i the sese that there are two solitos for large positive m ad oly oe for large egative m. I geeral, a (N,N + )-solito of the discrete 2DTL equatio (4.1) leads to a (N +,N )-solito of equatio (6.9) whe takig the ultra-discrete limit. We also ote that, iterestigly, a L-shape solutio ca be obtaied istead of a Y-shape solutio for differet solutio parameters. A example of such a L-shape solito is show i figure 4. No aalogue of this solutio exists i the 2DTL ad its fully discrete versio. Next, we cosider the case of a (2, 2)-solito for equatio (4.4) followig [1]. Let us cosider the followig τ fuctio, τ l,m, = f l,m, f l,m,+1, (7.8) f l,m,+1 f l,m,+2

17 Resoace ad web structure i discrete solito systems: the 2D Toda lattice m m m m Figure 3. A (2, 1)-resoat solito solutio (i.e., a Y-juctio) for the ultradiscrete 2DTL equatio (6.9), with P 1 = 5,P 2 = 1,P 3 = 4,r = 3,s = 1,A 1 A 3 = 1,A 2 A 3 = 1. From left to right, the four plots represet the solutio respectively at l =,l = 2,l = 4ad l = 6. The dots idicate zero values of v l,m,. m m m m Figure 4. A L-shape (2, 1)-resoat solito solutio for equatio (6.9), with P 1 = 5,P 2 = 1,P 3 = 4,r = 3,s = 1,A 1 A 3 = 1,A 2 A 3 = 1. From left to right, the four plots represet the solutio respectively at l =,l = 2,l = 4 ad l = 6. where f l,m, = ξ 1 + ξ 2 + ξ 3 + ξ 4, (7.9) where ξ i (i = 1,...,4) is agai defied as i equatio (7.3), ad where <p 1 <p 2 < p 3 < p 4 holds. We itroduce agai the ew parameters e P k/ɛ = p k ad e A k/ɛ = α k (k = 1,...,4) ad the ew depedet variable ρ ɛ l,m, = ɛ log τ l,m,. Takig the limit ɛ +, we the obtai ρ l,m, = max (K ij +2P j ), (7.1) 1 i<j 4 where ρ l,m, = lim ɛ + ρl,m, ɛ, as before, ad K ij = R i + R j, (7.11) ad with R j give by equatio (7.7) as before. Figure 5 shows the temporal evolutio of a (2, 2)-solito solutio. Note the appearace of a hole i figure 5. Like i the 2DTL (2.1) ad its fully discrete versio (4.1), we ow cosider more geeral resoat solutios for the ultra-discrete 2DTL (6.9). We start from the geeral τ fuctio

18 11836 K-i Maruo ad G Biodii (a) (b) (c) (d) Figure 5. Sapshots illustratig the temporal evolutio of a (2, 2)-resoat solito solutio for equatio (6.9), with P 1 = 7,P 2 = 5,P 3 = 1,P 4 = 3,r = 4,s = 2, A 1 = 1,A 2 = 6,A 3 =,A 4 = 2: (a) l = 12; (b) l = 7; (c) l = 4; (d) l = 7. As i figures 3 ad 4, the horizotal axis is ad the vertical axis is m. Sice the iteractio exteds over a wider rage of values of m ad, the solutio is ow plotted i greyscale, i a similar way as i figures 1 ad 2; the values of v l,m, however are still discrete, as i figures 3 ad 4. defied i equatio (4.11), ad itroduce agai the parameters e Pk/ɛ = p k ad e Ak/ɛ = α k (k = 1, 2,...,M) ad the variable ρl,m, ɛ = ɛ log τ l,m,, together with e r/ɛ = δ ad e s/ɛ = κ. Takig the limit ɛ +, we the obtai the followig solutio of the ultra-discrete 2DTL (6.9): [ N N ρ l,m, = max R ij +2 (j 1)P ij ], (7.12) 1 i 1 < <i N M where agai lim ɛ + ρl,m, ɛ = ρ l,m,, with the maximum beig take amog all possible combiatios of the idices i j (j = 1,...,N), ad where oce more we have R i = A i + P i + l max(,p i r) m max(, P i s). (7.13) Equatio (7.12) produces complicated solito solutios displayig resoace ad web structure. As a example, i figures 6 ad 7 we show some sapshots of the time evolutio of a (3, 3)-resoat solito solutio ad a (4, 4)-resoat solito solutio. Ideed, we cojecture that, similar to its couterparts for the 2DTL ad i its fully discrete aalogue, equatio (7.12) yields the (N,N + )-solito solutio of the ultra-discrete 2DTL equatio (6.9), with N + = N j=2

19 Resoace ad web structure i discrete solito systems: the 2D Toda lattice (a) (b) (c) (d) Figure 6. Sapshots illustratig the temporal evolutio of a (3, 3)-resoat solito solutio for equatio (6.9), with P 1 = 1,P 2 = 7,P 3 = 5,P 4 = 1,P 5 = 4,P 6 = 5, r = 7,s = 4,A 1 = 8,A 2 = 6,A 3 =,A 4 = 2,A 5 = 4,A 6 = 7: (a) l = 15, (b) l = 1, (c) l =, (d) l = 1. ad N = M N. Ulike the semi-cotiuous ad fully discrete cases, however, we were uable to prove this cojecture usig the techiques itroduced i [1]. I this respect, it should be oted that solutios of the ultra-discrete 2DTL arise as a result of the properties of the maximum fuctio, ad therefore their study might require the use of techiques from tropical algebraic geometry, which is a subject of curret research [13 15]. It should also be oted that the iteractio patters i the ultra-discrete system differ somewhat from their aalogues i the semi-cotiuous ad fully discrete cases. I particular, low-amplitude iteractio arms may disappear whe cosiderig the ultra-discrete limit. Furthermore, the specific iteractio patters i the ultra-discrete limit deped o the value of the parameters r ad s ad differet kids of solutios may appear for differet values of r ad s. I particular, large values of r ad s ted to result i the productio of several vertical solitos, as show i figures 6 ad 7. I order to preserve the solito cout i these cases, all the outgoig vertical solitos should be couted as oe, as should the icomig oes. I this sese, a set of outgoig or icomig vertical lies ca be cosidered as a boud state of several solitos. A full characterizatio of these pheomea ad their parameter depedece is however outside the scope of this work.

20 11838 K-i Maruo ad G Biodii (a) (b) (c) (d) Figure 7. Sapshots illustratig the temporal evolutio of a (4, 4)-resoat solito solutio for equatio (6.9), with P 1 = 15,P 2 = 12,P 3 = 9,P 4 = 3,P 5 = 1,P 6 = 1,P 7 = 4,P 8 = 7,r = 7,s = 4,A 1 = 8,A 2 = 6,A 3 =,A 4 = 2,A 5 = 4,A 6 = 7,A 7 = 8,A 8 = 1: (a) l = 1, (b) l =, (c) l = 1, (d) l = Coclusios We have demostrated the existece of solito resoace ad web structure i discrete solito systems by presetig a class of solutios of the two-dimesioal Toda lattice (2DTL) equatio, its fully discrete aalogue ad their ultra-discrete limit. Solito resoace ad web structure had bee previously foud for oliear partial differetial equatios such as the KP ad ckp systems. Note that the 2DTL is a differetial-differece equatio, its fully discrete versio is a differece equatio ad its ultra-discrete limit is a cellular automato; therefore our fidigs show that resoace ad web structure pheomea are rather geeral features of two-dimesioal itegrable systems whose solutios are expressed i determiat form. A full characterizatio of the solutios, icludig the study of asymptotic amplitudes ad velocities ad the resoace coditio, was provided both i the semi-cotiuous ad i the fully discrete case. Their aalogue i the ultra-discrete case, together with a aalysis of the itermediate patters of iteractios, is out of the scope of this work, ad remais as a problem for further research. Of particular iterest is the ultra-discrete 2DTL, where ew types of solutios such as the L-shape solito show i figure 4 appear.

21 Resoace ad web structure i discrete solito systems: the 2D Toda lattice Fially, we ote that the class of solutios preseted i this work is just oe of the possible choices that yield resoace ad web structure. Just like with the KP ad ckp equatios, the class of solito solutios of each of the systems we have cosidered (amely, the 2DTL ad its fully discrete ad its ultra-discrete aalogues) is much wider, ad icludes also partially resoat solutios. The solutios described i this work represet the extreme case i which all of the iteractios amog the various solitos are resoat, whereas ordiary solito solutios represet the opposite case where oe of the iteractios amog the solitos are resoat. Ibetwee these two situatios, a umber of itermediate cases exist i which oly some of the iteractios are resoat. As i the case of the KP equatio, the study of these partially resoat solutios remais a ope problem. Ackowledgmets We thak M J Ablowitz, S Chakravarty, Y Kodama, J Matsukidaira ad A Nagai for may helpful discussios. KM ackowledges support from the Rotary Foudatio ad the 21st Cetury COE program Developmet of Dyamic Mathematics with High Fuctioality at Faculty of Mathematics, Kyushu Uiversity. GB was partially supported by the Natioal Sciece Foudatio, uder grat umber DMS Refereces [1] Biodii G ad Kodama Y 23 O a family of solutios of the Kadomtsev Petviashvili equatio which also satisfy the Toda lattice hierarchy J. Phys. A: Math. Ge [2] Hirota R, Ito M ad Kako F 1988 Two-Dimesioal Toda Lattice Equatios Prog. Theor. Phys. Suppl [3] Hirota R, Ohta Y ad Satsuma J 1988 Wroskia structures of solitos for solito equatios Prog. Theor. Phys. Suppl [4] Isojima S, Willox R ad Satsuma J 22 O various solutio of the coupled KP equatio J. Phys. A: Math. Ge [5] Isojima S, Willox R ad Satsuma J 23 Spider-web solutio of the coupled KP equatio J. Phys. A: Math. Ge [6] Matsukidaira J, Satsuma J, Takahashi D, Tokihiro T ad Torii M 1997 Toda-type cellular automato ad its N-solito solutio Phys. Lett. A [7] Media E 22 A N solito resoace solutio for the KP equatio: iteractio with chage of form ad velocity Lett. Math. Phys [8] Miles J W 1977 Diffractio of solitary waves J. Fluid Mech [9] Moriwaki S, Nagai A, Satsuma J, Tokihiro T, Torii M, Takahashi D ad Matsukidaira J Dimesioal Solito Cellular Automato (Lodo Math. Soc. Lecture Notes Series vol 255) (Cambridge: Cambridge Uiversity Press) [1] Nagai A, Tokihiro T, Satsuma J, Willox R ad Kajiwara K 1997 Two-dimesioal solito cellular automato of deautoomized Toda type Phys. Lett. A [11] Newell A C ad Redekopp L G 1977 Breakdow of Zakharov Shabat theory ad solito creatio Phys. Rev. Lett [12] Park J H H, Steiglitz K ad Thursto W P 1986 Solito-like behaviour i automata Physica D [13] Richter-Gebert J, Sturmfels B ad Theobald T 23 First steps i tropical geometry Preprit math. AG/ [14] Speyer D ad Sturmfels B 23 The tropical Grassmaia Preprit math. AG/34218 [15] Speyer D ad Williams L K 23 The tropical totally positive Grassmaia Preprit math. CO/ [16] Takahashi D ad Matsukidaira J 1995 O discrete solito equatios related to cellular automata Phys. Lett. A [17] Takahashi D ad Satsuma J 199 A solito cellular automato J. Phys. Soc. Japa [18] Tokihiro T, Takahashi D, Matsukidaira J ad Satsuma J 1996 From solito equatios to itegrable cellular automata through a limitig procedure Phys. Rev. Lett