Matrix KP: tropical limit and Yang-Baxter maps.

Transcription

1 Matrix KP: tropical limit and Yang-Baxter maps Aristophanes Dimakis a and Folkert Müller-Hoissen b a Dept of Financial and Management Engineering University of the Aegean Chios Greece dimakis@aegeangr b Max-Planck-Institute for Dynamics and Self-Organization Göttingen Germany folkertmueller-hoissen@dsmpgde arxiv:708069v [nlinsi] Feb 08 Abstract We study soliton solutions of matrix Kadomtsev-Petviashvili KP equations in a tropical limit in which their support at fixed time is a planar graph and polarizations are attached to its constituting lines There is a subclass of pure line soliton solutions for which we find that in this limit the distribution of polarizations is fully determined by a Yang-Baxter map For a vector KP equation this map is given by an R-matrix whereas it is a non-linear map in case of a more general matrix KP equation We also consider the corresponding Korteweg-deVries KdV reduction Furthermore exploiting the fine structure of soliton interactions in the tropical limit we obtain a new solution of the tetrahedron or Zamolodchikov equation Moreover a solution of the functional tetrahedron equation arises from the parameter-dependence of the vector KP R-matrix Introduction A line soliton solution of the scalar Kadomtsev-Petviashvili KP-II equation see eg [] is at fixed time t an exponentially localized wave on a plane The tropical limit in the sense of our work in [ ] also see [ 6] takes it to a piecewise linear structure a planar graph that represents the wave crest with values of the dependent variable attached to its edges In this work we consider the m n matrix potential KP equation φ xt φ xxxx φ yy 6φ x Kφ x x + 6φ x Kφ y φ y Kφ x = 0 where K is a constant n m matrix and φ an m n matrix depending on independent variables x y t and a subscript indicates a corresponding partial derivative We will refer to this equation as pkp K If φ is a solution of then φ R := φk and φ L := Kφ solve the ordinary m m respectively n n matrix potential KP equation We also note that if K = T K S with a constant m m matrix S and a constant n n matrix T then the m n matrix φ = SφT satisfies the pkp K equation as a consequence of In the vector case n = writing K = k k m and φ = φ φ m becomes the following system of coupled equations φ ixt φ ixxxx φ iyy 6 m j= k j φ ix φ jx x φ ix φ jy + φ iy φ jx = 0 i = m By choosing T = and any invertible m m matrix S that has K as its first row we have K = K S with K = 0 0 In terms of the new variable φ = Sφ the above system thus consists of one scalar pkp equation and m linear equations involving the dependent variable of the former

2 For u := φ x we obtain from the m n matrix KP equation u t u xxx uku x x u yy + uk u y dx u y dx Ku = 0 x The extension of the scalar KP equation to a matrix version achieves that solitons carry internal degrees of freedom polarization The Korteweg-deVries KdV reduction of is u t u xxx uku x = 0 which we will refer to as KdV K If K is the identity matrix this is the matrix KdV equation see eg [7] The -soliton solution of the latter yields a map from polarizations at t 0 to polarizations at t 0 It is known [8 9] that this yields a Yang-Baxter map ie a set-theoretical solution of the quantum Yang-Baxter equation also see [0] for the case of the vector Nonlinear Schrödinger equation Not surprisingly this is a feature preserved in the tropical limit The surprising new insight however is that this map governs the evolution of polarizations throughout the tropical limit graph of a soliton solution In case of a vector KdV equation ie KdV K with n = it is given by an R-matrix a linear map solution of the Yang-Baxter equation More generally we will explore in this work the tropical limit of pure see Section soliton solutions of the above K-modified matrix KP equation and demonstrate that a Yang-Baxter map governs their structure In case of the vector KP equation the expression for a pure soliton solution involves a function τ which is a τ-function of the scalar KP equation Its tropical limit at fixed t determines a planar graph and the vector KP soliton solution associates in this limit a constant vector polarization with each linear segment of the graph The polarization values are then related by a linear Yang- Baxter map represented by an R-matrix which does not depend on the independent variables x y t but only on the spectral parameters of the soliton solution Section summarizes a binary Darboux transformation for the pkp K equation and applies it to a trivial seed solution in order to obtain soliton solutions In Section we restrict out consideration to the subclass of pure soliton solutions This essentially disregards solutions with substructures of the form of Miles resonances Section addresses the tropical limit of pure soliton solutions The cases of two and three solitons are then treated in Sections and 6 Section 7 provides a general proof of the fact that in the vector case an R-matrix relates the polarizations at crossings The linearity of the Yang-Baxter map in the vector case is certainly related to the particularly simple structure of the vector pkp equation mentioned above In Section 8 we show how to construct a pure N-soliton solution of the vector KP equation from a pure N-soliton solution of the scalar KP equation N vector data and the aforementioned R-matrix Section 9 extends our exploration of the vector KP -soliton case and presents an apparently new solution of the tetrahedron Zamolodchikov equation see eg [] and references cited there In Section 0 we reveal the structure of the vector KP R-matrix which leads us to a more general two-parameter R-matrix Its parameter-dependence determines via a local Yang-Baxter equation [] also see [] a solution of the functional tetrahedron equation see eg [ ] ie the set-theoretical version of the tetrahedron equation Finally Section contains some concluding remarks Soliton solutions of the K-modified matrix KP equation The following describes a binary Darboux transformation for the pkp K equation This is a simple extension of what is presented in [] for example Let φ 0 be a solution of Let θ and

3 χ be m N respectively N n matrix solutions of the linear equations Then the system θ y = θ xx + φ 0x Kθ θ t = θ xxx + φ 0x Kθ x + φ 0y + φ 0xx Kθ χ y = χ xx χkφ 0x χ t = χ xxx + χ x Kφ 0x χkφ 0y φ 0xx Ω x = χkθ Ω y = χkθ x + χ x Kθ Ω t = χkθ xx + χ x Kθ x χ xx Kθ χkφ 0x Kθ is compatible and can thus be integrated to yield an N N matrix solution Ω If Ω is invertible then φ = φ 0 θ Ω χ is a new solution of For vanishing seed solution ie φ 0 = 0 soliton solutions are obtained as follows Let θ = A θ a e ϑpa χ = a= B e ϑqb χ b where P a Q b are constant N N matrices θ a χ b are constant m N respectively N n matrices and b= ϑp = x P + y P + t P If for all a b the matrices P a and Q b have no eigenvalue in common there are unique N N matrix solutions W ba of the Sylvester equations Then is solved by Q b W ba W ba P a = χ b Kθ a a = A b = B Ω = Ω 0 + A a= b= B e ϑqb W ba e ϑpa with a constant N N matrix Ω 0 and determines a soliton solution of and thus via u = φ x a solution of if Ω is everywhere invertible Remark Corresponding solutions of the pkp K hierarchy are obtained by replacing with ϑp = r= t r P r where t = x t = y t = t Pure soliton solutions In the following we restrict our considerations to the case where A = B = Then there remains only a single Sylvester equation Q W W P = χ Kθ More generally the following holds for any constant φ 0 But adding to φ a constant matrix is an obvious symmetry of the pkp K equation

4 Moreover we will restrict the matrices P and Q to be diagonal It is convenient to name the diagonal entries spectral parameters in two different ways P = diagp p N = diagp p N Q = diagp p N = diagq q N We further write θ = q p ξ q p ξ q N p N ξ N χ = η η N where ξ i are m-component column vectors and η i are n-component row vectors Then the solution of the above Sylvester equation is given by W = w ij w ij = q j p j q i p j η i K ξ j i j = N Furthermore we set Ω 0 = I N the N N identity matrix Hence Ω = Ω ij with Ω ij = δ ij + w ij e ϑp j ϑq i where δ ij is the Kronecker delta We call soliton solutions obtained from with the above restrictions pure solitons All what follows refers to them We introduce ϑ I := N ϑp iai if I = a a N { } N i= Instead of using a a N as a subscript or superscript we will simply write a a N in the following For example ϑ a a N = ϑ a a N From we find that the pure soliton solutions of the pkp K equation are given by with φ = F τ τ := e ϑ det Ω F := e ϑ θ e ϑp adjω e ϑq χ where adjω denotes the adjugate of the matrix Ω and := = Proposition τ and F have expansions τ = µ I e ϑi I {} N F = M I e ϑi 6 I {} N with constants µ I and constant m n matrices M I where µ = and M = 0

5 Proof From the definition of the determinant det Ω = ɛ i i N Ω i Ω NiN with the Levi-Civita symbol ɛ i i N and summation convention we know that det Ω consists of a sum of monomials of order N in the entries Ω ij Here the latter are given by If no diagonal term Ω ii appears in a monomial its phase factor is e ϑ ϑ If one diagonal entry Ω ii = + w ii e ϑp i ϑp i appears in a monomial the latter splits into two parts Only the part arising from the summand is different as now the phase factor is e ϑ ϑ ϑp i +ϑp i From monomials containing several diagonal entries of Ω we obtain summands with a phase factor of the form e ϑ ϑ ϑp i +ϑp i + ϑp ir +ϑp ir = e [ϑ ϑp i ϑp ir ]+ϑp i + +ϑp ir ϑ Finally from a monomial with N diagonal entries of Ω we also obtain a constant term namely Now our assertion follows since τ is det Ω multiplied by e ϑ Clearly µ = According to the Laplace cofactor expansion det Ω = N j= Ω ij adjω ji with respect to the i-th row the term Ω ij adjω ji consists of all summands in det Ω having Ω ij as a factor implies that a summand of e ϑ adjω ji then has a phase factor of the form e ϑ I ϑp j +ϑp i with some I { } N so that e ϑ e ϑp adjω e ϑq ji has the phase factor e ϑ I Hence 6 holds Furthermore no entry of e ϑp adjω e ϑq is constant hence M = 0 Remark The introduction of the redundant factor e ϑ in via the definitions and achieves that τ and F are linear combinations of exponentials e ϑ I I { } N in which case we have a very convenient labelling This is also so if we choose the factor e ϑ instead which leads to an expansion in terms of e ϑ I now with M = 0 Regularity of a pure soliton solution requires µ I 0 for all I { } N or equivalently µ I 0 for all I { } N and µ I 0 for at least one I If µ I = 0 for some I this means that the phase ϑ I is not present in the expression for τ In this case one has to arrange the data in such a way that M I = 0 in order to avoid unbounded exponential growth of the soliton solution in some phase region But we will disregard such cases and add the condition µ I > 0 I { } N to our definition of pure soliton solutions It follows that the corresponding solution of the KP equation is given by where u = τ IJ {} N p J p I µ I M J µ J M I e ϑi e ϑj p I = p a + + p NaN if I = a a N Using Jacobi s formula for the derivative of a determinant we obtain τ x = p τ + e ϑ tradjω Ω x = p τ e ϑ tradjω χkθ = p τ e ϑ trkθ adjω χ which implies and thus = p τ + trk F trkφ = ln τ x p 7 trku = ln τ xx Using in 7 and reading off the coefficient of e ϑ I we find trk M I = p I p µ I 8

6 Remark If n = 7 reads Kφ = ln τ x p and Ku = ln τ xx is a solution of the scalar KP equation If n > trku is not in general a solution of the scalar KP equation Remark Dropping the redundant factor e ϑ in and means that we have to multiply the above expressions and 6 for τ and F by e ϑ It is then evident that φ only depends on differences of phases of the form ϑp i ϑp i = p i p i x+p i p i y +p i p i t As a consequence setting p i = p i ie q i = p i eliminates the y-terms in all phases This means that under this condition for the parameters we could have started as well with ϑp = x P +t P hence without the y-term in In this way we make contact with the KdV K reduction of KP K Tropical limit of pure soliton solutions A crucial point is that we define the tropical limit of the matrix soliton solution via the tropical limit of the scalar function τ cf [ ] Let φ I := φ = M I ϑj J I µ I In a region where a phase ϑ I dominates all others in the sense that logµ I e ϑ I > logµ J e ϑ J for all participating J I the tropical limit of the potential φ is given by It should be noticed that these expressions do not depend on the coordinates x y t The boundary between the regions associated with the phases ϑ I and ϑ J is determined by the condition µ I e ϑ I = µ J e ϑ J Not all parts of such a boundary are visible at fixed time since some of them may lie in a region where a third phase dominates the two phases The tropical limit of a soliton solution at a fixed time t has support on the visible parts of the boundaries between the regions associated with phases appearing in τ On such a visible boundary segment the value of u is given by For I = a a N we set u IJ = p I p J φ I φ J I k a = a a k a a k+ a N At fixed time the set of line segments associated with the k-th soliton are obtained from with I = I k and J = I k for all possible I They satisfy p k p k x + p k p k y + p k p k t + ln µ I k µ Ik = 0 6

7 All these line segments have the same slope p k +p k in the xy-plane hence they are parallel The shifts between them are given by δ k IJ = ln µik µ Jk µ Ik µ Jk They give rise to the familiar asymptotic phase shifts of line solitons The tropical limit of u on a visible line segment of the k-th soliton is given by u Ik I k = p k p k φ Ik φ Ik The value of u at a visible triple phase coincidence is u IJL = 9 u IJ + u IL + u JL = 9 p I p J p L φ I + p J p I p L φ J + p L p I p J φ L Instead of the above expressions for the tropical values of u we will rather consider û IJ = φ I φ J p I p J which has the form of a discrete derivative Since 8 and imply the latter values are normalized in the sense that If I = a a N and i j let I ij a b = a a N ai aa j b The normalized tropical values of u satisfy trkφ I = p I p trkû IJ = 6 p i q i û Iij I ij + p j q j û Iij I ij = p i q i + p j q j û Iij I ij p i q i û Iij I ij + p j q j û Iij I ij = p i q i + p j q j û Iij I ij p i q i û Iij I ij + p j q j û Iij I ij = p i q i + q j p j û Iij I ij p i q i û Iij I ij + p j q j û Iij I ij = p i q i + q j p j û Iij I ij These identities are simply consequences of the definition of û IJ They linearly relate the normalized polarizations at points of the tropical limit graph where three lines meet Pure -soliton solutions Let N = Then we have where τ = e ϑ + κ e ϑ + κ e ϑ + α κ κ e ϑ κ ij = η i Kξ j α = q p q p κ κ q p q p κ κ 7

8 and κ F = q p q p ξ η + κ ξ η + κ ξ η p q p q q p + κ ξ η e ϑ + p q ξ η e ϑ + p q ξ η e ϑ q p The tropical values of the pkp K solution φ in the dominant phase regions are then given by φ = q p q p κ ξ η + κ ξ η α κ κ p q p q + κ ξ η + κ ξ η q p q p φ = p q ξ η κ φ = p q ξ η κ φ = 0 Remark The above values φ ab solve the following nonlinear equation + tr Kφ φ Kφ φ q p p q φ φ φ φ Kφ φ q p + φ φ Kφ φ p q φ φ φ φ = 0 Addressing more than two solitons non-zero counterparts of φ will show up as displayed in this equation For the tropical values of û along the phase region boundaries we obtain u in := û = α m q p ξ η ξ η K n q p K ξ η q p κ q p κ u in := û = ξ η κ u out := û = ξ η κ κ u out := û = α m q p ξ η q p κ ξ η K n q p K ξ η κ q p κ where m stands for the m m identity matrix For the in/out classification see Fig below All the matrices in have rank one which is not at all obvious from the form of φ ab We obtain the following nonlinear relation between incoming and outgoing polarizations where u out = α in u out = α in m q p u in K u in p p m q p u in K q q α in = p q p q p p q q tr Ku inku in n p q Ku in q q u in n p q Ku in p p We note that α α in = is a new nonlinear Yang-Baxter map with parameters p i q i i = Writing u ain = ξ ain η ain η ain Kξ ain u aout = ξ aout η aout η aout Kξ aout a = 8

9 determines ξ in/out and η in/out up to scalings We find and ξ out = α / in m p q ξ out = α / in K η in Kξ in ξ in η in p p m q p ξ in η in K q q η in Kξ in ξ in ξ in η out = α / in η in n q p K ξ in η in q q η in Kξ in η out = α / in η in n p q K ξ in η in p p η in Kξ in ξ in η in ; ξ in η in ξ out η out ; ξ out η out is another form of the above Yang-Baxter map Remark In we found that u in and u out have a simple elementary form They are the polarizations at the two boundary lines of the dominating phase region numbered by = see Fig We know from Proposition that it is special since M = 0 Considering an evolution in negative x-direction instead of y-direction thus offers a more direct derivation of the Yang-Baxter map Example Let m = and n = Choosing p = / p = / q = / q = / and η = 0 η = 0 ξ = / ξ = / K = we obtain the first contour plot at t = 0 shown in Fig Fig shows plots of the components of the transpose of u Choosing instead p close to q reveals an inner structure of crossing points see the second plot in Fig This is the phase shift mentioned in Section Remark We should stress that the relevant structures are actually three-dimensional and our figures only display a two-dimensional cross section Instead of displaying structures in the xy-plane at constant t we may as well look at those in the xt-plane at constant y The latter becomes relevant if we consider the KdV K reduction Remark Soliton solutions of KdV K are obtained from those of KP K by setting q i = p i i = N see Remark Then the above equations reduce to ξ out = α / in m p ξ out = α / in K ξ in η in Kξ in p p ξ in η in m p p p ξ in η in η in Kξ in K ξ in η out = α / in η in n p K ξ in η in p p η in Kξ in η out = α / in η in n p K ξ in η in p p η in Kξ in 9

10 Figure : The first is a contour plot of trku = ln τ xx for a -soliton solution of the matrix KP equation at t = 0 in the xy-plane using the data of Example Viewed as a process in y-direction the YB map takes the values of the KP variable on the lower two legs to those on the upper two A number ij indicates the respective dominating phase region In the second plot the value of p is replaced by / + 0 so that p is very close to q Here a boundary segment between phase regions and is visible The third plot presents an example where the parameters of the -soliton solution are now chosen such that the latter boundary is hidden and instead a boundary segment between phase regions and is visible Figure : Plot of the six components of the transpose of u at t = 0 for the solution of the matrix KP equation with the first data specified in Example The components are localized exactly where trku is localized cf Fig 0

11 with α in = + p p p p η in Kξ in η in Kξ in η in Kξ in η in Kξ in If K is the N N identity matrix this becomes the Yang-Baxter map first found by Veselov [8] also see [9 6] The factor α / in is missing in these publications but such a factor is necessary for the map to satisfy the Yang-Baxter equation One can avoid the square root at the price of having an asymmetric appearance of factors αin Pure column vector -soliton solutions We set n = Now the η i are scalars and drop out of the relevant formulas Introducing we have ˆξ i = ξ i Kξ i u in = p q p p ˆξ + q p p p ˆξ u in = ˆξ u out = ˆξ u out = p q p p ˆξ + q p p p ˆξ and thus u out u out = u in u in p p q p q p q p q q q p q p q Generalizing the matrix that appears on the right hand side to Rp i q i ; p j q j = pi p j p i q j p j q j p i q j p i q i p i q j q i q j p i q j and letting this act from the right on the i-th and j-th slot of a three-fold direct sum the Yang- Baxter equation holds This can be checked directly or inferred from a -soliton solution see Section 6 Remark 6 The reduction to vector KdV K via q i = p i see Remark leads to Rp i p j = pi p j p i +p j p j p i +p j p i p i +p j p j p i p i +p j This rules the evolution of initial polarizations at t 0 step by step along the tropical limit graph in two-dimensional space-time The R-matrix also describes the elastic collision of non-relativistic particles with masses p i in one dimension see [7] Pure row vector -soliton solutions Now we set m = Then the ξ i are scalars and drop out of the relevant formulas Introducing we have ˆη i = η i η i K u in = q p q q ˆη + p q q q ˆη u in = ˆη u out = ˆη u out = q p q q ˆη + p q q q ˆη

12 so that uout u out = q q q p q p q p p p q p q p q uin u in which determines a Yang-Baxter map Let Rp i q i ; p j q j := qj q i p j q i p i q i p j q i p j q j p j q i p j p i p j q i act on the i-th and j-th slot of a direct sum Then the Yang-Baxter equation holds We note that Rp i q i ; p j q j = Rq i p i ; q j p j 6 Pure -soliton solutions For N = we find where again κ ij = η i Kξ j and τ = κ κ κ β e ϑ + κ κ α e ϑ + κ κ α e ϑ +κ κ α e ϑ + κ e ϑ + κ e ϑ + κ e ϑ + e ϑ 6 α ij = p i q i p j q j κ ij κ ji p i q j p j q i κ ii κ jj β = + α + α + α + p q p q p q p q p q p q + p q p q p q κ κ κ p q p q p q κ κ κ Furthermore we obtain F = p q α κ κ ξ η + p q α κ κ ξ η κ κ κ κ κ κ +p q α κ κ ξ η + p q p q q p α κ κ ξ η + p q p q q p α κ κ ξ η + p q p q q p α κ κ ξ η + p q q p α κ κ ξ η + p q q p α κ κ ξ η p q p q + p q q p α κ κ ξ η e ϑ p q + p q κ ξ η + p q κ ξ η + p q p q κ ξ η q p p q p q κ ξ η e ϑ p q + p q κ ξ η + p q κ ξ η + p q p q κ ξ η q p p q p q κ ξ η e ϑ p q

13 + p q κ ξ η + p q κ ξ η + p q p q κ ξ η q p p q p q κ ξ η e ϑ p q +p q ξ η e ϑ + p q ξ η e ϑ + p q ξ η e ϑ 6 where α kij = q i p j q k p k κ ik κ kj q i p k q k p j κ ij κ kk Note that α ij = α ijj Recall that the coefficient of e ϑ abc in the expression for τ respectively F has been named µ abc respectively M abc The tropical value in the region where ϑ abc dominates all other phases is given by φ abc = M abc µ abc The corresponding values can be read off from 6 and 6 6 Pure vector KP -soliton solutions Now we restrict our considerations to the vector case n = Using we obtain ˆξ i = ξ i Kξ i û = ˆξ û = ˆξ û = ˆξ û = p q p q p p p p ˆξ p q p q p p p p ˆξ + p q p q p p p p ˆξ û = p q p q p p p p ˆξ p q p q p p p p ˆξ + p q p q p p p p ˆξ û = p q p q p p p p ˆξ p q p q p p p p ˆξ + p q p q p p p p ˆξ û = p q p p ˆξ p q p p ˆξ û = p q p p ˆξ p q p p ˆξ û = p q p p ˆξ p q p p ˆξ û = p q p p ˆξ p q p p ˆξ û = p q p p ˆξ p q p p ˆξ û = p q p p ˆξ p q p p ˆξ The contour plots in Fig show the structure at fixed t with t < 0 and t > 0 respectively The lines extending to the bottom are numbered by from left to right displayed as blue red green respectively Thinking of three particles undergoing a scattering process in y-direction they carry polarizations that change at crossing points As y increases we have û û û û û û û û û û û û

14 Figure : Yang-Baxter relation in terms of vector KP line solitons These are contour plots in the xy-plane horizontal x- and vertical y-axis of a -soliton solution at negative respectively positive t A number abc indicates the respective dominating phase region for t < 0 where the i-th entry contains the polarization of the i-th particle and û û û û û û û û û û û û for t > 0 The numbers ij assigned to the steps refer to the particles involved In both cases we start and end with the same vectors and this implies the Yang-Baxter equation for the associated transformations Let V a a a b b b be the column vector formed by the coefficients of û a a a b b b with respect to ˆξ ˆξ ˆξ The following matrices are composed of these column vectors U = V V V U = V V V U = V V V U = V V V U = V V V U = V V V They represent the triplets of polarizations constituting the above chains Next we define matrices r = U U = U U r = U U = U U r = U U = U U which turn out to be given in terms of the R-matrix For example The Yang-Baxter equation reads r = p p p q 0 p q p q 0 0 p q p q 0 q q p q r r r = r r r Fig shows plots of Ku for a choice of the parameters 6 Vector KdV -soliton solutions We impose the KdV reduction see Remark and replace by ϑp = x P + t P + s P

15 Figure : Plots of the scalar Ku for a Yang-Baxter line soliton configuration of a vector KP equation at times t < 0 t = 0 and t > Figure : Tropical limit graph and dominating phase regions of a vector KdV solution in twodimensional space-time x horizontal t vertical at s = 0 s = 0 and s = 0 See Section 6 Numbers in red attached to lines identify appearances of the respective soliton Here bounded lines are formally associated with a pair of a virtual anti-soliton indicated by a bar over the respective number and a virtual soliton At s = 0 a composite of three virtual solitons shows up

16 The additional last term introduces the next evolution variable s of the KdV hierarchy also see Remark Let us consider for simplicity the KdV K equation with m = and K = and the special solution with parameters θ = I χ = The tropical limit graph is displayed in Fig for p = / p = / p = and different values of s We have the matrix p +p p +p U = p p p p 0 0 û û û = p p +p p +p p p p p p p 0 p p +p p p p p p p p of initial polarizations The next values û abcdef are then obtained by application of the R-matrix from the right and so forth following either the left or the right graph in Fig in upwards ie t- direction Since the initial and the final polarizations are the same the Yang-Baxter equation holds Here the R-matrix describes the time evolution of polarizations in the tropical limit Fig suggests to think of R-matrices as being associated with bounded lines which may be thought of as representing virtual solitons Interaction of two solitons then means exchange of a virtual soliton At s = 0 see the plot in the middle of Fig something peculiar occurs namely a sort of three-particle interaction This is a degenerate special case to which the Yang-Baxter description does not apply But this is not relevant for our conclusions 7 Tropical limit of pure vector solitons and the R-matrix We set n = vector case The following results describe what happens at a crossing of two solitons numbered by i and j depicted as a contour plot in Fig 6 Lemma 7 p i p j φ Iij + q i q j φ Iij = p i q j φ Iij + q i p j φ Iij 7 Proof This is quickly verified for the -soliton solution N = But at a crossing a general solution φ is equivalent to a -soliton solution since there the four elementary phases ϑp ia ϑp ja a = dominate all others hence the exponential of any other phase vanishes in the tropical limit Remark 7 7 can be regarded as a vector version of a scalar linear quadrilaterial equation satisfying consistency on a cube see in [8] Such a linear relation does not hold in the matrix case where m n > But with φ ab replaced by φ Iij ab in which case we have φ Iij 0 in general is a nonlinear counterpart of 7 The latter can be deduced from it for n = by using Theorem 7 ûiiji ij û IijI ij Rpi q i ; p j q j = û IijI ij û IijI ij 7 with Rp i q i ; p j q j = pi p j p i q j p j q j p i q j p i q i p i q j q i q j p i q j 6

17 I ij I ij I ij I ij Figure 6: A crossing of solitons with numbers i and j at fixed time in the xy-plane and the numbers of the four phases that are involved Figure 7: Three-soliton configuration for t < 0 and t > 0 respectively in the xy-plane Numbers specify dominating phase regions Proof Using we can directly verify that the following relations hold as a consequence of 7 In matrix form this is 7 p i p j û p i q Iij I ij + p j q j û j p i q Iij I ij = û Iij I ij j p i q i û p i q Iij I ij + q i q j û j p i q Iij I ij = û Iij I ij j We already know that Rp i q i ; p j q j satisfies the Yang-Baxter equation 8 Construction of pure vector KP soliton solutions from a scalar KP solution and the R-matrix Given a τ-function for a pure N-soliton solution of the scalar KP equation the Yang-Baxter R- matrix found above can be used to construct a pure N-soliton solution of the vector KP equation We will explain this for the case N = The τ-function of the pure -soliton solution of the scalar KP-II equation is given by τ = abc= abc e ϑ abc abc = p b p a p c p a p c p b as obtained by the Wronskian method see eg [9] Comparison with shows that µ I = I Starting at the bottom of both graphs in Fig 7 we associate a column vector ξ i with the i-th 7

18 soliton counted from left to right and normalize it such that Kξ i = Accordingly we set û = ξ û = ξ û = ξ By consecutive application of the R-matrix we find the polarizations on the further line segments proceeding in y-direction There are different ways to proceed but they are consistent since R satisfies the Yang-Baxter equation For example û û = û û Rp q ; p q This leads to û = p p p q ξ + p q p q ξ û = p q p q ξ + q q p q ξ û = p p p q ξ + p q p q ξ û = p q p q ξ + q q p q ξ û = p p p p p q p q ξ + p p p q p q p q ξ + p q p q ξ û = p p p q p q p q ξ + p q p q p q p q ξ + q q p q ξ û = p q p q ξ + p q q q p q p q ξ + q q q q p q p q ξ û = p p p q ξ + p q p q p q p q ξ + p q q q p q p q ξ û = p p p q p q p q ξ + p p q q p q + p q p q p q p q p q p q + p q q q p q p q ξ By integration of we find φ abc up to a single constant Setting φ = 0 we obtain φ = ξ p q + ξ p q + ξ p q φ = p p p q q q p q p q ξ + + p q ξ p q p q q p q q q q p q + + p q ξ p q q p φ = p p p q ξ + p q p q ξ + p q ξ p q p q φ = p q ξ + p q ξ φ = p p p p p q ξ + p p p q p q ξ + p q p q ξ p q p q p q p q p q φ = p q ξ φ = p p p q ξ + p q p q ξ p q p q From we can read off M abc and thus obtain via and the solution φ = τ abc= M abc e ϑ abc of the vector pkp equation This procedure can easily be applied to a larger number of solitons 8 ξ

19 Figure 8: Passing to the tropical limit and zooming into the second interaction point in the tropical limit of Fig shows the tropical origin of phase shifts Again the left figure refers to t < 0 the right to t > 0 9 A solution of the tetrahedron equation In this section we address again the case of three pure solitons see Section 6 Because of the occurrence of phase shifts in the tropical limit the crossing points of solitons are not really points Fig 8 shows this for the second interaction in the vertical y-direction in Fig or Fig 7 We may think of associating with the additional edge in the left plot t < 0 the polarization of the boundary between the phase regions numbered by = and and in the right plot t > 0 that of the boundary between the phases and But this does not lead us to something meaningful Instead we make an educated guess and associate a mean value of vectors with an additional edge and V = V + V V = V + V V = V + V ˆV = V + V ˆV = V + V ˆV = V + V The inner boundary lines shown in the two plots in Fig 8 are now associated with V respectively ˆV In the first plot the boundary between phase regions and is hidden but there is also a polarization associated with it namely V V is the mean value of the latter and V which belongs to the visible inner boundary segment Now we have Si j k = V V V ˆV ˆV ˆV p p i p p j p p k q q i q q j q q k = δ ijk γ ijk where γ ijk = γ ijk rs with γ ijk = ρ k ρ i p k + q k q i + ρ j p k + q k p j γ ijk = ρ j ρ i p j + q j q i ρ k p j + q j q k γ ijk = ρ iρ j p i + q i p j + ρ k p i + q i q k γ ijk = ρ ρ i ρ j k ρ ρ i k ρ i q i + p k + q k γ ijk ρ = ρ ρ i ρ j k k ρ j ρ k γ ijk = ρ ρ i ρ k j ρ ρ i j ρ i p j + q j q i γ ijk ρ = ρ ρ i ρ k j j ρ j ρ k ρj p k + q k p j + ρ k ρ j ρ k p j + q j q k 9

20 Figure 9: KP line solitons remain parallel while moving The two chains of line configurations here shifts are disregarded show the two different ways in which a -soliton solution can evolve from the same initial to the same final configuration The two chains represent the higher Bruhat order B cf [] This implies the tetrahedron equation Here red numbers attached to lines enumerate the four solitons Also the crossings of pairs of them are enumerated by numbers in black Time evolution proceeds by inversion of triangles In the first chain the first step is the inversion of the triangle formed by the crossings numbered The second step is the inversion of the triangle formed by the crossings The latter involves the solitons with numbers and = ρ ρ j ρ k i ρ i ρj p i + q i p j + ρ ρ i γ ijk = ρ ρ j ρ k i ρ j ρ i i ρ k p i + q i q k ρ k γ ijk Here we set δ ijk = p j q i ρ i ρ j + q k q i ρ i ρ k + q k p j ρ j ρ k ρ i = p i q i Let S αβγ i j k be the 6 6 matrix which acts via Si j k on the positions α β γ of a 6-component column vector and as the identity on the others Let R = S R = S R 6 = S 6 R 6 = S 6 We verified that this constitutes a to our knowledge new solution of the tetrahedron Zamolodchikov equation see eg [] and references cited there R R R 6 R 6 = R 6 R 6 R R 9 An explanation for the choices of numbers in the above definition of R αβγ can be found in Fig 9 Also see [] Remark 9 The KdV K reduction q i = p i yields S KdV i j k = p i+p k p j p k p i p k p j +p k p j p i +p k p i +p j p j +p k p i p j p k p i +p j p i p k p i p j p k p i p k p j +p k p i p j p k p i p k p j +p k p i p j p j p k p j p i +p k p i +p j p j +p k p i +p j p j +p k p i p j p k p i +p j p i p k p i p j p i +p k p i +p j p i p k which determines a simpler solution of the tetrahedron equation 0

21 0 A generalization of the vector KP R-matrix and a solution of the functional tetrahedron equation The vector KdV R-matrix is obtained from the one-parameter R-matrix see eg [ ] for a similar R-matrix x + x Rx = x x by setting x = p p /p + p The local Yang-Baxter equation R x R y R z = R Z R Y R X where indices αβ indicate on which components of a three-fold direct sum R acts determines the map x y z X Y Z given by X = xy x + z xyz Y = x + z xyz Z = yz x + z xyz A similar map appeared in [ ] A general argument cf eg [] and references cited there implies that Rx y z := X Y Z solves the functional tetrahedron equation 9 where a product of R s now has to be interpreted as composition of maps This tetrahedron map is involutive Setting x = p p /p +p y = p p /p + p and z = p p /p + p it becomes the identity Correspondingly the vector KP R-matrix is obtained from the more general two-parameter R-matrix x y Rx y = x y by setting x = p p /p q and y = p q /p q The local Yang-Baxter equation R x y R z u R v w = R V W R Z U R X Y determines the map x y; z u; v w X Y ; Z U; V W where with X = z C Y = z A x C Z = x C U = B V = vz x y A W = u w B A = uvx ux vy + xz B = uwx ux wy + AB A u wx Bv x y C = AB A u w Bvz x y Then Rx y; z u; v w := X Y ; Z U; V W solves the functional ie set-theoretical tetrahedron equation

22 Conclusions In this work we explored pure soliton solutions of matrix KP equations in a tropical limit In case of the reduction to matrix KdV this consists of a planar graph in two-dimensional space-time with polarizations assigned to its edges Given initial polarizations the evolution of them along the graph is ruled by a Yang-Baxter map For the vector KdV equation this is a linear map hence an R matrix The classical scattering process of matrix KdV solitons resembles in the tropical limit the scattering of point particles in a -dimensional integrable quantum field theory which is characterized by a scattering matrix that solves the quantum Yang-Baxter equation We have shown that all this holds more generally for KP K where the tropical limit at a fixed time t is given by a graph in the xy-plane with polarizations attached to the soliton lines Moreover the vector KP case provides us with a realization of the classical straight-string model considered in [] It should be noticed however that KP line solitons in the tropical limit are not in general straight because of the appearance of phase shifts As a side product of our explorations of the tropical limit of pure vector KP solitons we derived apparently new solutions of the tetrahedron Zamolodchikov equation Whether these solutions are relevant eg for the construction of solvable models of statistical mechanics in three dimensions has still to be seen Another subclass of soliton solutions of the vector KP equation consists of those for which the support at fixed time is a rooted and generically binary tree in the tropical limit For the scalar KP equation this has been extensively explored in [ ] Instead of the Yang-Baxter equation the pentagon equation see [] and references therein now plays a role in governing corresponding vector solitons This will be treated in a separate work Acknowledgment AD thanks V Papageorgiou for a very helpful discussion References [] Kodama Y 07 KP Solitons and the Grassmannians SpringerBriefs in Mathematical Physics vol Singapore: Springer [] Dimakis A and Müller-Hoissen F 0 KP line solitons and Tamari lattices J Phys A: Math Theor 00 [] Dimakis A and Müller-Hoissen F 0 Associahedra Tamari Lattices and Related Structures Progress in Mathematics vol 99 ed Müller-Hoissen F Pallo J and Stasheff J Basel: Birkhäuser pp 9 [] Dimakis A and Müller-Hoissen F 0 KdV soliton interactions: a tropical view J Phys Conf Ser [] Biondini G and Chakravarty S 006 Soliton solutions of the Kadomtsev-Petviashvili II equation J Math Phys [6] Chakravarty S and Kodama Y 008 Classification of the line-soliton solutions of KPII J Phys A: Math Theor 709 [7] Goncharenko V 00 Multisoliton solutions of the matrix KdV equation Theor Math Phys [8] Veselov A 00 Yang-Baxter maps and integrable dynamics Phys Lett A [9] Goncharenko V and Veselov A 00 New Trends in Integrability and Partial Solvability NATO Science Series II: Math Phys Chem vol ed Shabat A et al Dordrecht: Kluwer pp 9 97 [0] Tsuchida T 00 N-soliton collision in the Manakov model Progr Theor Phys 8 [] Dimakis A and Müller-Hoissen F 0 Simplex and polygon equations SIGMA 0 [] Maillet J and Nijhoff F 989 Integrability for multidimensional lattice models Phys Lett B 89 96

23 [] Sergeev S 998 Solutions of the functional tetrahedron equation connected with the local Yang-Baxter equation for the ferro-electric condition Lett Math Phys 9 [] Kashaev R Korepanov I and Sergeev S 998 Functional tetrahedron equation Theor Math Phys [] Chvartatskyi O Dimakis A and Müller-Hoissen F 06 Self-consistent sources for integrable equations via deformations of binary Darboux transformations Lett Math Phys [6] Suris Y and Veselov A 00 Lax matrices for Yang-Baxter maps J Nonl Math Phys 0 0 [7] Kouloukas T 07 Relativistic collisions as Yang-Baxter maps Phys Lett A 8 9 [8] Atkinson J 009 Linear quadrilateral lattice equations and multidimensional consistency J Phys A: Math Theor 00 [9] Hirota R 00 The Direct Method in Soliton Theory Cambridge Tracts in Mathematics vol Cambridge: Cambridge University Press