Solitons of the vector KdV and Yamilov lattices
aa r X i v : . [ n li n . S I] O c t Solitons of the vector KdV and Yamilov lattices.
V.E. Vekslerchik
Usikov Institute for Radiophysics and Electronics12, Proskura st., Kharkov, 61085, UkraineE-mail: [email protected]
Abstract.
We study a vector generalizations of the lattice KdV equation and oneof the simplest Yamilov equations. We use algebraic properties of a certain class ofmatrices to derive the N -soliton solutions.AMS classification scheme numbers: 35Q51, 35C08, 11C20PACS numbers: 02.30.Ik, 05.45.Yv, 02.10.Yn Submitted to:
J. Phys. A: Math. Gen.
1. Introduction.
We study a generalization of two well-known equations, the lattice KdV equation,( u m +1 ,n +1 − u m,n ) ( u m +1 ,n − u m,n +1 ) = 1 , (1.1)and one of the equations from the Yamilov list, du n dt = 1 u n +1 − u n − . (1.2)Equation (1.1) has been introduced by Capel, Nijhoff and coauthors [1, 2, 3, 4] andnow is often referred to as equation H1 from the Adler–Bobenko–Suris list [5]. During itsmore than 30-year history it has attracted much attention and is one of the most-studieddiscrete integrable systems.Equation (1.2), sometimes referred to as Yamilov discretization of the Krichever–Novikov equation, is known since the work by Yamilov [6] who classified all integrablesemi-discrete equations of the form du n /dt = f ( u n − , u n , u n +1 ) using the generalizedsymmetry method (see also [7, 8]). Equation (1.2) is related to the the well-knownVolterra equation. It has been shown in [9] that it describes the simplest negative flowof the Volterra hierarchy.Despite their different appearance, equations (1.1) and (1.2) are known to be closelyrelated. For example, it has been demonstrated in [10] that generalized symmetries of(1.1) are described by (1.2). In other words, equation (1.1) can be viewed as describingthe B¨acklund transformations of equation (1.2). olitons of the vector KdV and Yamilov lattices. (cid:13)(cid:13) φ m +1 ,n +1 − φ m,n (cid:13)(cid:13) (cid:13)(cid:13) φ m +1 ,n − φ m,n +1 (cid:13)(cid:13) = 1 (1.3)and ddt φ n = φ n +1 − φ n − (cid:13)(cid:13) φ n +1 − φ n − (cid:13)(cid:13) . (1.4)Here and in what follows the vectors φ are 3-dimensional real vectors, φ =( φ , φ , φ ) T ∈ R , and k φ k denotes the standard Euclidean norm in R , k φ k = P i =1 φ i .Equations (1.3) and (1.4) can be viewed as ‘vectorizations’ of (1.1) and (1.2)alternative to ones discussed in [1] (compare equations (1.3) and (1.4) with equations(6.9) and (8.5) from [1]).In this paper, we do not discuss the questions related to the integrability ofequations (1.3) and (1.4) such as Lax representation, conservation laws, Hamiltonianstructures etc . We restrict ourselves with the problem of finding some particular solutions, namely the N -soliton ones.In the next section we introduce an auxiliary system which is closely related to theequations we want to solve. In section 3 we derive some solutions for this system usingthe straightforward calculations involving the soliton matrices discussed in [11]. Thesesolutions are used in section 4 to construct the N -soliton solutions for equations (1.3)and (1.4).
2. Auxiliary system.
To derive the soliton solutions we start from the bilinear difference vector equation T ξ φ − T η φ = ε ξη T ξη φ − φ k T ξη φ − φ k (2.1)where ε ξη is some skew-symmetric constant, ε ξη = − ε ηξ which we introduce to ensure theproper symmetry with respect to the interchange of ξ and η . The symbols T ξ stand for OK?the shifts, which can be viewed as a generalization of the translations φ ( x ) → φ ( x + δ ( ξ ))with some analytic function δ ( ξ ) and whose particular implementation in our case isspecified below (see (3.4)) while the double indices denote combined action of differentshifts, T ξη = T ξ T η .It is easy to show that each solution for (2.1) provide a solution for both (1.3) and(1.4). Indeed, taking the norm of both sides of (2.1) one immediately arrives at k T ξη φ − φ k k T ξ φ − T η φ k = | ε ξη | . (2.2)Thus, any solution for (2.1) solves at the same time the equation which is (up to aconstant in the right-hand side) nothing but the difference version of (1.3). This meansthat solutions for (2.1) can be converted, by fixing the values ξ and η , into ones for(1.3). olitons of the vector KdV and Yamilov lattices. T − η and taking the ξ → η limit one arrives at D η φ = T η φ − T − η φ (cid:13)(cid:13) T η φ − T − η φ (cid:13)(cid:13) (2.3)where D η is the differential operator defined as D η = lim ξ → η ε ξη (cid:0) T ξ T − η − (cid:1) (2.4)(note that the fact that ε ξη = − ε ηξ together wih the assumption of analytical dependenceof ε ξη on ξ and η yields ε ηη = 0).Of course, the correspondence between solutions of (1.3), (1.4) (or even theirdifference versions (2.2) and (2.3)) and (2.1) is not one-to-one. Each solution for (2.1)satisfies (2.2) but the reverse statement is not true. The similar situation is with (2.1)and (2.3). However, the fact that using (2.1) we actually make a reduction is not crucialfor our consideration because the aim of this work is to derive the soliton solutions, aset of particular solutions, and, as is shown in what follows, the soliton solutions standthis reduction.Comparison of the equations (2.2) and (2.3) with (1.1) and (1.2) suggests thefollowing way to derive solutions for the last two equations using the ones for (2.2)and (2.3): to identify the shits corresponding to some fixed parameter, say, µ and ν with the translations m → m + 1 and n → n + 1, and to introduce the t -dependencein such a way that the action of D ν defined in terms of the T -shifts leads to the sameresults as the differentiating with respect to t . Thus, we set T µ φ m,n = φ m +1 ,n , T ν φ m,n = φ m,n +1 (2.5)for equation (1.1) and T ν φ n = φ n +1 , D ν φ n = ∂∂t φ n (2.6)for equation (1.2).Rewriting (2.1) as a system A ξη ( T ξη φ − φ ) = f ξη ( T ξ φ − T η φ ) B ξη k T ξη φ − φ k = f ξη (2.7)where new constants A ξη and B ξη satisfy A ξη = − A ηξ , B ξη = B ηξ , A ξη B ηξ = ε ξη (2.8)one can note that the first equation of this system is nothing but the difference vectorMoutard equation which can be tackled in a standard way. Indeed, the substitutions φ = 1 τ ω , f ξη = ( T ξ τ )( T η τ ) τ ( T ξη τ ) (2.9)lead to the well-known bilinear equation A ξη ( τ ( T ξη ω ) − ( T ξη τ ) ω ) = ( T η τ )( T ξ ω ) − ( T ξ τ )( T η ω ) (2.10) olitons of the vector KdV and Yamilov lattices. ω , we arrive at a quadrilinear equation B ξη k τ T ξη ω − ( T ξη τ ) ω k = τ ( T ξ τ )( T η τ )( T ξη τ ) . (2.11)This means that we cannot use the standard direct methods like the Hirota approachand have to build solutions almost ‘from scratch’.
3. Soliton matrices.
In this section we construct solutions for the system (2.7) from the soliton matricesstudied in [11]. Partly, the calculations presented here are similar to ones of [11].However, this time we need more deep analysis of the properties of the soliton matrices:the results of [11] are not enough to tackle the quadrilinear restrictions discussed in theprevious section.
We define the soliton matrices by the so-called ‘rank one condition’ L A − A L = | ℓ ih a | L A − A L = | ℓ ih a | (3.1)where L and L are constant N × N diagonal matrices, | ℓ i and | ℓ i are constant N -columns while h a | and h a | are N -component rows that depend on the coordinatesdescribing the model.For our purposes it is helpful to rewrite this equation as an intertwining relation( L − | ℓ ih β | ) A = A L ( L − | ℓ ih β | ) A = A L (3.2)with constant N -rows h β , | which are defined as h a i | = h β i | A i , ( i = 1 , . (3.3)The shifts T are defined as the right multiplication T ζ A = A ( L + ζ ) ( L − ζ ) − T ζ A = A ( L − ζ ) ( L + ζ ) − (3.4)(we do not indicate the unit matrix explicitly and write L ± ζ instead of L ± ζ , etc). olitons of the vector KdV and Yamilov lattices. From (3.4) one can derive the action of the shifts T on the determinants ττ = det | A A | (3.5)and the inverse matrices G = (1 + A A ) − G = (1 + A A ) − . (3.6)The corresponding formulae can be written as T ζ ττ = 1 + 2 ζ K ζ h β ζ | A G A | ℓ ζ i (3.7)= 1 − ζ K ζ h β ζ | A G A | ℓ ζ i (3.8)and T ζ ττ ( T ζ − G = 2 ζ K ζ G A | ℓ ζ ih β ζ | G A (3.9) T ζ ττ ( T ζ − G = − ζ K ζ G A | ℓ ζ ih β ζ | G A (3.10)where constants K iζ are given by K iζ = 11 − h β i ζ || ℓ i i , ( i = 1 ,
2) (3.11)and h β ζ | = h β | ( L − ζ ) − h β ζ | = h β | ( L + ζ ) − | ℓ ζ i = ( L + ζ ) − | ℓ i| ℓ ζ i = ( L − ζ ) − | ℓ i . (3.12)Introducing the new functions p = 1 − h β | G | ℓ i ,s = 1 − h β | G | ℓ i , q = h β | G A | ℓ i ,r = h β | G A | ℓ i , (3.13)one can derive from (3.4) and (3.10) T ζ ττ ( T ζ − pqrs = 2 ζ K ζ − q ζ r ζ q ζ s ζ − p ζ r ζ q ζ r ζ (3.14)and T ζ ττ = K ζ ( p ζ s ζ + q ζ r ζ ) (3.15)where K ζ = K ζ K ζ (3.16)and p ζ = 1 − h β ζ | G | ℓ i s ζ = 1 − h β ζ | G | ℓ i q ζ = h β ζ | G A | ℓ i r ζ = h β ζ | G A | ℓ i . (3.17) olitons of the vector KdV and Yamilov lattices. By means of straightforward (although rather cumbersome) calculations based on (3.4)and (3.7)–(3.10) one can describe the ‘evolution’ of the functions p ζ , ..., s ξ , T η ττ ( T η − p ξ q ξ r ξ s ξ = 2 ηK η ξ − η p ξ q η r η − q ξ p η r η q ξ p η s η − p ξ q η s η r ξ p η s η − s ξ p η r η s ξ q η r η − r ξ q η s η , (3.18)and to obtain the following two-shift identity for the tau-functions: τ ( T ξη τ ) − ( T ξ τ )( T η τ ) = 4 ξηK ξ K η ( ξ − η ) ( p ξ q η − p η q ξ ) ( r η s ξ − r ξ s η ) τ . (3.19)Equations (3.14) together with (3.18) lead to T ξη p − p + ξ + η = ξ + ηξ − η f ξη ( T ξ p − T η p + ξ − η ) T ξη q − q = ξ + ηξ − η f ξη ( T ξ q − T η q ) T ξη r − r = ξ + ηξ − η f ξη ( T ξ r − T η r ) (3.20)where f ξη = ( T ξ τ )( T η τ ) τ ( T ξη τ ) . (3.21)We do not write similar expression for s because, as follows from (3.14), T ζ ( p + s ) = p + s , which means that p + s = constant .Introducing the new function w = p + χ (3.22)where χ is the ‘linear’ function defined by T ζ χ = χ + ζ (3.23)one can rewrite (3.20) as( T ξη − qrw = ξ + ηξ − η f ξη ( T ξ − T η ) qrw . (3.24)Finally, these equations together with (3.19), (3.14) and (3.15) yield( T ξη w − w ) − ( T ξη q − q ) ( T ξη r − r ) = ( ξ + η ) f ξη . (3.25)It is easy to note that the last two equations have the structure of system (2.7) with A ξη = ( ξ − η ) / ( ξ + η ), B ξη = 1 / ( ξ + η ) and hence ε ξη = ξ − η . The only differenceis that the quadratic form in (3.25) is not the Euclidean norm of the vector ( q, r, w ) T .Thus, the last problem we have to solve is to construct, of the functions q , r and w , thevectors φ with the appropriate norm. olitons of the vector KdV and Yamilov lattices. Till now, we have not specified whether the functions introduced in this section are realor complex. All formulae presented above are suitable for both cases. Here, we discussthe symmetry of the soliton matrices with respect to the comlex conjugation.It is easy to verify that the restrictions L = L , h β | = h β | , | ℓ i = | ℓ i , (3.26)where the overbar stands for the complex conjugation, lead to A = A . (3.27)It follows from (3.4) that to ensure the consistency of the action of the shifts T ζ withthe involution (3.27) we have to restrict ourselves with pure imaginary ζ ,Re ζ = 0 ⇒ T ζ A = T ζ A . (3.28)Hereafter, we use the ‘real’ shifts T R defined by T R λ = T iλ , (Im λ = 0) . (3.29)One can derive from (3.26), (3.27) and the definitions (3.13) the identities s = p, r = q (3.30)which are compatible with the action of the shifts T R λ , T R λ s = T R λ p, T R λ r = T R λ q. (3.31)We have already mentioned that p + s is constant with respect to the shifts. In thecontext of (3.30), this reads( T R λ − p = i ( T R λ −
1) Im p (3.32)which, together with the definition (3.23), implies( T R λ − w = i ( T R λ −
1) Im w. (3.33)Now, we can rewrite equation (3.25) in terms of q and w (cid:0) T R λµ Im w − Im w (cid:1) + (cid:12)(cid:12) T R λµ q − q (cid:12)(cid:12) = ( λ + µ ) f R λµ (3.34)where f R λµ = ( T R λ τ ) (cid:0) T R µ τ (cid:1) /τ (cid:0) T R λµ τ (cid:1) .Thus, we can formulate the main result of this section. Proposition 3.1
Vector φ defined as φ = (Re q, Im q, Im w ) T (3.35) with functions q , r and w defined in (3.13), (3.22) and (3.23) satisfies T R µ φ − T R ν φ = (cid:0) µ − ν (cid:1) T R µν φ − φ (cid:13)(cid:13) T R µν φ − φ (cid:13)(cid:13) (3.36) with arbitrary real µ and ν .olitons of the vector KdV and Yamilov lattices.
4. N-soliton solutions.
As follows from proposition 3.1, to obtain soliton solutions for (1.3) we have to maketwo simple steps. First, we introduce the dependence on m and n as φ m,n = (cid:0) T R µ (cid:1) m ( T R ν ) n φ . (4.1)Secondly, we have to rescale φ m,n in order to make the right-hand side of (1.3) equal tounity, φ m,n → (cid:12)(cid:12) µ − ν (cid:12)(cid:12) − / φ m,n . (4.2)After that, we can present the N -soliton solutions for (1.3) as follows. Proposition 4.1
The N -soliton solutions for the vector discrete KdV equation (1.3)can be presented as φ m,n = φ bg m,n + φ sol m,n (4.3) where the background part, φ bg m,n is the linear function of m and n , φ bg m,n = mµ + nν | µ − ν | / (4.4) and φ sol m,n = 1 | µ − ν | / Re h β | G m,n A m,n | i Im h β | G m,n A m,n | i− Im h β | G m,n | i . (4.5) Here A m,n = A H mµ H nν (4.6) with the constant matrices A and H µ,ν given by A = (cid:18) a k ¯ L j − L k (cid:19) j,k =1 ,...,N (4.7) H λ = diag (cid:18) L k + iλL k − iλ (cid:19) k =1 ,...,N , (4.8) and G m,n = (cid:0) A m,n A m,n (cid:1) − . (4.9) The constant N -row h β | is defined by h β | = ( β , ..., β N ) = ( a , ..., a N ) A − , the N -column | i is defined as | i = (1 , ..., and { a k , L k } k =1 ,...,N and µ, ν are arbitrary constants.olitons of the vector KdV and Yamilov lattices. −20 20 m −2020 n ϕ sol1 −20 20 m −2020 n ϕ sol2 −20 20 m −2020 n ϕ sol3 −202 −0.50.00.5 06 Figure 1.
The ( m, n )-dependence of the components of the two-soliton solution (4.5).
Note that we use L j for the elements of the diagonal matrix L , L = diag ( L , . . . , L N ) (4.10)and that we have eliminated some ‘redundant’ constants by replacing | ℓ , i with | i (thecomponents of the columns | ℓ , i can be ‘included’ in the arbitrary constants a k ).In the one soliton case ( N = 1) the matrix L becomes a scalar, L → L and we haveonly one a -parameter, a = a . The formulae from proposition 4.1 can be rewritten as φ sol m,n = ρ cosh h m,n (cid:0) cos ϕ m,n , sin ϕ m,n , e − h m,n (cid:1) T (4.11)where ρ = | Im L | / | µ − ν | / and h m,n and ϕ m,n are linear functions of m and n , h m,n = κ R ( µ ) m + κ R ( µ ) n + h ∗ (4.12) ϕ m,n = κ I ( µ ) m + κ I ( ν ) n + ϕ ∗ (4.13)where κ R ( λ ) = ln (cid:12)(cid:12)(cid:12)(cid:12) L + iλL − iλ (cid:12)(cid:12)(cid:12)(cid:12) , κ I ( λ ) = arg L + iλL − iλ , (4.14) h ∗ = ln (cid:12)(cid:12)(cid:12) a L (cid:12)(cid:12)(cid:12) , ϕ ∗ = arg a. (4.15)Calculating the norm of φ sol m,n , (cid:13)(cid:13) φ sol m,n (cid:13)(cid:13) = 4 ρ e h m,n → ( ρ as h m,n → −∞ h m,n → + ∞ (4.16)one can see that the obtained line soliton has a step- or kink-like structure: the (cid:13)(cid:13) φ sol m,n (cid:13)(cid:13) is bounded between 0 (which it attains in one asymptotic direction) and 2 ρ (which itattains in the opposite direction). However, the part of φ which is perpendicular to φ bg (the first two components in (4.11)) reveals typical soliton sech -behaviour.To illustrate the structure of the two-soliton solutions we calculate (4.5) for somefixed set of soliton parameters: L = 0 . i , L = 0 . i , a = 10, a = 9. To makethe plots more clear we present in figure 1 only the soliton part of the solution, φ sol . As olitons of the vector KdV and Yamilov lattices. φ (the part of φ whichis parallel to φ bg ) has the two-kink structure, while the first two (the part of φ whichis perpendicular to φ bg ) have the stucture of two solitons (with sign-alternation alongone of the directions). To obtain the solitons of equation (1.4) using the result of proposition 3.1 we have tointroduce the continuous variable t so that the differentiating d/dt reproduces the actionof the operator (2.4) or µ − ν (cid:0) T R µ ( T R ν ) − − (cid:1) . One can obtain from (3.4) that (cid:16) T R µ ( T R ν ) − − (cid:17) A = 2 i ( µ − ν ) A L ( L − iµ ) − ( L + iν ) − (4.17)which leads to ddt A ( t ) = i A ( t ) L (cid:0) L + ν (cid:1) − (4.18)or A ( t ) = A (0) exp ( i Ω t ) , Ω = L (cid:0) L + ν (cid:1) − (4.19)The n -dependence of the matrices A (and, hence, of φ ) is governed, as in the previoussection, by the matrix H = H ν from (4.8). Thus, we have all necessary to present thesolitons of (1.4). Proposition 4.2
The N -soliton solutions for the vector Yamilov equation (1.4) can bepresented as φ n ( t ) = φ bg n ( t ) + φ sol n ( t ) (4.20) where the background part, φ bg n ( t ) is the linear function of t and n , φ bg m,n = (cid:18) t ν + nν (cid:19) (4.21) and φ sol n ( t ) = Re h β | G n ( t ) A n ( t ) | i Im h β | G n ( t ) A n ( t ) | i− Im h β | G n ( t ) | i . (4.22) Here A n ( t ) = A H n exp ( i Ω t ) , (4.23) with the constant matrices A , H and Ω given by A = (cid:18) a k ¯ L j − L k (cid:19) j,k =1 ,...,N (4.24) H = diag (cid:18) L k + iνL k − iν (cid:19) k =1 ,...,N , (4.25) olitons of the vector KdV and Yamilov lattices. t −10 0 10 n −20 0 200 2 ϕ sol1 t −10 0 10 n −20 0 200 2 ϕ sol2 t −10 0 10 n −20 0 2005 ϕ sol3 Figure 2.
The ( t, n )-dependence of the components of the two-soliton solution (4.22). Ω = diag (cid:18) L k L k + ν (cid:19) k =1 ,...,N , (4.26) and G n ( t ) = (cid:16) A n ( t ) A n ( t ) (cid:17) − . (4.27) The constant N -row h β | is defined by h β | = ( β , ..., β N ) = ( a , ..., a N ) A − , the N -column | i is defined as | i = (1 , ..., and { a k , L k } k =1 ,...,N and ν are arbitrary constants. Clearly, the structure of the one soliton solution is the same as in the case of thevector discrete KdV equation, φ sol n ( t ) = ρ cosh h n ( t ) (cid:0) cos ϕ n ( t ) , sin ϕ n ( t ) , e − h n ( t ) (cid:1) T . (4.28)The differences are in that ρ = | Im L | and in the ‘dispersion laws’, h n ( t ) = − γt + κ R ( ν ) n + h ∗ , (4.29) ϕ n ( t ) = ωt + κ I ( ν ) n + ϕ ∗ (4.30)where ω = − Im a L , γ = Re a L (4.31)while the functions κ R,I ( ν ) and the constants h ∗ and ϕ ∗ are defined in (4.14) and (4.15).The two-soliton solution for L = i , L = 2 i , a = 2 + 2 i , a = 2 + 3 i and ν = − . φ which is perpendicular to φ bg has thestructure of two sech –solitons, while the part of φ which is parallel to φ bg reveals thetwo–kink behaviour.
5. Discussion.
To conclude, we would like to stress out once more the main difference between thecalculations of this work and other our works devoted to solitons of the vector latticemodels, for example, [13, 14]. In [13, 14], our starting point was some scalar identititiesfor the soliton matrices from [11]. These identies were enough to (i) derive the vector olitons of the vector KdV and Yamilov lattices. N -soliton solutions can be viewed as an indication of the integrability of the models(1.3) and (1.4). Thus, a natural continuation of this work is to study the correspondingrange problems mentioned in the Introduction (the Lax representation, conservationlaws, Hamiltonian structures etc ). However these questions are out of the scope of thispaper and may be considered in the following studies. References [1] Capel H W, Wiersma G L and Nijhoff F W 1986
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