Integrability of auto-Bäcklund transformations,and solutions of a torqued ABS equation
aa r X i v : . [ n li n . S I] F e b Integrability of auto-B¨acklund transformations,and solutions of a torqued ABS equation
Xueli Wei , Peter H. van der Kamp , Da-jun Zhang ∗ Department of Mathematics, Shanghai University, Shanghai 200444, P.R. China Department of Mathematics and Statistics, La Trobe University, Victoria 3086, Australia
February 25, 2021
Abstract
An auto-B¨acklund transformation for the quad equation Q1 is considered as a dis-crete equation, called H2 a , which is a so called torqued version of H2. The equationsH2 a and Q1 compose a consistent cube, from which a auto-B¨acklund transforma-tion and a Lax pair for H2 a are obtained. More generally it is shown that auto-B¨acklund transformations admit auto-B¨acklund transformations. Using the auto-B¨acklund transformation for H2 a we derive a seed solution and a one-soliton solution.From this solution it is seen that H2 a is a semi-autonomous lattice equation, as thespacing parameter q depends on m but it disappears from the plain wave factor. Key Words: auto-B¨acklund transformation, consistency, Lax pair, soliton solution,torqued ABS equation, semi-autonomous.
The subtle concept of integrability touches on global existence and regularity of solutions,exact solvability, as well as compatibility and consistency (cf. [1]). In the past two decades,the study of discrete integrable system has achieved a truly significant development, whichmainly relies on the effective use of the property of multidimensional consistency (MDC).In the two dimensional case, MDC means the equation is Consistent Around the Cube(CAC) and this implies it can be embedded consistently into lattices of dimension 3 andhigher [2–4]. In 2003, Adler, Bobenko and Suris (ABS) classified scalar quadrilateral equa-tions that are CAC (with extra restrictions: affine linear, D4 symmetry and tetrahedronproperty) [5]. The complete list contains 9 equations.In this paper, our discussion will focus on two of them, namelyQ1 δ ( u, e u, b u, be u ; p, q ) = p ( u − b u )( e u − be u ) − q ( u − e u )( b u − be u ) + δpq ( p − q ) = 0 (1.1)and H2( u, e u, b u, be u ; p, q ) = ( u − be u )( e u − b u ) + ( q − p )( u + e u + b u + be u ) + q − p = 0 . (1.2) ∗ Corresponding author. Email: djzhang@staff.shu.edu.cn u = u ( n, m ) is a function on Z , p and q are spacing parameters in the n and m direction respectively, δ is an arbitrary constant which we set equal to 1 in the sequel, andconventionally, tilde and hat denote shifts u = u ( n, m ) , e u = u ( n + 1 , m ) , b u = u ( n, m + 1) , be u = u ( n + 1 , m + 1) . (1.3)H2 is a new equation due to the ABS classification, while Q1 δ extends the well known cross-ratio equation, or lattice Schwarzian Korteweg-de Vries equation Q1 δ =0 . Note that spacingparameters p and q can depend on n and m respectively, which leads to nonautonomousequations.For a quadrilateral equation that is CAC, the equation itself defines its own (natural)auto-B¨acklund transformation (auto-BT), cf. [5]. For example, the systemQ1 δ ( u, e u, u, e u ; p, r ) = 0 , Q1 δ ( u, b u, u, b u ; q, r ) = 0 , where r acts as a wave number, composes an auto-BT between Q1 δ ( u, e u, b u, be u ; p, q ) = 0 andQ1 δ ( u, e u, b u, be u ; p, q ) = 0. Such a property has been employed in solving CAC equations,see e.g. [6–10].Some CAC equations allow auto-BTs of other forms. For example, in [11] it was shownthat the coupled system A : ( u − e u )( e u − u ) − p ( u + e u + u + e u + p + 2 r ) = 0 , (1.4a) B : ( u − b u )( b u − u ) − q ( u + b u + u + b u + q + 2 r ) = 0 (1.4b)provides an auto-BT between Q : Q1 ( u, e u, b u, be u ; p, q ) = 0 (1.5)and Q : Q1 ( u, e u, b u, be u ; p, q ) = 0, and, that H2 acts as a nonlinear superposition principlefor the BT (1.4). One can think of the auto-BT as equations posed on the side faces of aconsistent cube with Q and Q respectively on the bottom and the top face, as in Figure 1.Here one interprets u = u ( n, m, l + 1), and r serves as a spacing parameter for the thirddirection l . The superposition principle can be understood as consistency of a 4D cube,see [12, 13]. tt t tt tt t ✧✧✧✧✧ ✧✧✧✧✧✧✧✧✧✧ A b AQQ e B B e u be uu e u b uu be u b u p qr Figure 1: Consistent cube for
A, B and Q .2n [14] the auto-BT (1.4) and its superposition principle have been derived from thenatural auto-BT for H2, employing a transformation of the variables and the parameters.The equationH2 a ( u, e u, b u, be u ; p, q ) = H2( u, be u, b u, e u ; p + q, q )= ( u − e u )( be u − b u ) − p ( u + e u + b u + be u + p + 2 q ) = 0 (1.6)was identified as a torqued version of the equation H2. The superscript a refers to the a dditive transformation of the spacing parameter. In [11] equation (1.6) appeared as partof an auto-BT for Q1 . The corresponding consistent cube is a special case of [15, Eq.(3.9)]. In [14] equation (1.6) was shown to be an integrable equation in its own right, withan asymmetric auto-BT given by A = H2 a = 0 and B = H2 = 0. Here we provide analternative auto-BT for equation (1.6) to the one that was provided in [14].In section 2, we establish a simple but quite general result, namely that if a systemof equations A = B = 0 comprises an auto-BT then both equations A = 0 and B = 0admit an auto-BT themselves. In particular, the equation H2 a given by (1.6) is CAC,with H2 a and Q1 providing its an auto-BT. We construct a Lax pair for H2 a , whichis asymmetric. In section 3, we employ the auto-BT for H2 a to derive a seed-solutionand the corresponding one-soliton solution. In the seed-solution the spacing parameter q depends explicitly on m , which makes H2 a inherent semi-autonomous. Some conclusionsare presented in section 4. H2 a To have a consistent cube with H2 a and Q1 on the side faces, providing an auto-BT forH2 a , we assign equations to six faces as follows: Q : H2 a ( u, e u, b u, be u ; p, q ) = 0 , Q : H2 a ( u, e u, b u, be u ; p, q ) = 0 , (2.1a) A : Q1 ( u, e u, u, e u ; p, r ) = 0 , b A : Q1 ( b u, be u, b u, be u ; p, r ) = 0 , (2.1b) B : H2 a ( u, u, b u, b u ; r, q ) = 0 , e B : H2 a ( e u, e u, be u, be u ; r, q ) = 0 . (2.1c)Then, given initial values u, e u, b u, u , by direct calculation, one can find that the value be u isuniquely determined. Thus, the cube in Figure 1 with (2.1) is a consistent cube.By means of such a consistency, the side equations A and B , i.e. A : p ( u − u )( e u − e u ) − r ( u − e u )( u − e u ) + pr ( p − r ) = 0 , (2.2a) B : ( u − u )( b u − u ) − r ( u + u + b u + b u + r + 2 q ) = 0 , (2.2b)compose an auto-BT for the H2 a equation (1.6). Here r acts as the B¨acklund parameter.We note that the order of the variables in the equations (2.1) is quite particular. Sinceequation (1.6) is not D4 symmetric, i.e. we haveH2 a ( u, u, b u, b u ; r, q ) = H2 a ( u, b u, u, b u ; q, r ) , one has to be careful. The above result is explained by the following useful result, cf. [16,Section 2.1] where the same idea was used to reduce the number of triplets of equationsto consider for the classification of consistent cubes.3 emma 2.1. Let A ( u, e u, u, e u ; p, r ) = 0 , B ( u, b u, u, b u ; q, r ) = 0 (2.3) be an auto-BT for Q ( u, e u, b u, be u ; p, q ) = 0 . (2.4) Then we have (i) Q ( u, e u, u, e u ; p, r ) = 0 , B ( u, u, b u, be u ; r, q ) = 0 (2.5) is an auto-BT for A ( u, e u, b u, be u ; p, q ) = 0; (2.6) and (ii) Q ( u, u, e u, e u ; r, p ) = 0 , A ( u, u, b u, b u ; r, q ) = 0 (2.7) is an auto-BT for B ( u, e u, b u, be u ; p, q ) = 0 . (2.8) Proof. If A = B = 0 is an auto-BT of Q = 0, then they compose a consistent cube asin Figure 1. We prove the result by relabelling the fields at the vertices, cf. [13, Lemma2.1]. For (i) we interchange b u ↔ u and q ↔ r , and for (ii) we perform the cyclic shifts b u → e u → u → b u and q → p → r → q .Applying (i) to the consistent cube with (1.4) and (1.5) we obtain (2.1). Applying (ii)yields the same, as Q1 has D4 symmetry.3D consistency can be used to construct Lax pairs for quadrilateral equations (cf.[3, 5, 17]). To achieve a Lax pair for H2 a , we rewrite (2.2) as e u = u ( p e u − ru ) + ( p − r )( pr − e uu )( p − r ) u + r e u − pu , (2.9a) b u = − r + b u − r ( q + b u + u ) r − u + u . (2.9b)Then, introducing u = G/F and ϕ = ( G, F ) T , from (2.9) we have e ϕ = Lϕ, b ϕ = M ϕ, (2.10)where L = γ − ur − ( p − r ) e u pu e u + ( p − r ) pr − p ( p − r ) u + r e u ! ,M = γ ′ b u − r ( − r + b u )( r − u ) − r ( q + u + b u )1 r − u ! , with γ = √ p − ( u − e u ) , γ ′ = √ q + u + b u . The linear system (2.10) is compatible for solutionsof (1.6) in the sense that H2 a is a divisor of ( b LM ) − ( f M L ) , where the square can betaken either as matrix multiplication, or as component-wise multiplication.4 Seed and one-soliton solution
Our idea of constructing solutions for (1.6) is to use its auto-BT (2.2). First, we need tohave a simple solution as a “seed”. To find such a solution, we take u = u in the BT (2.2),i.e. ( u − e u ) = p ( p − r ) , u + b u = − q − r . (3.1)Such a treatment is called fixed point idea , which has proved effective in finding seedsolutions [6, 8]. Proposition 3.1.
Parametrising p = αa , α = − aca − , q = ( − m β − c , (3.2) and setting the seed BT parameter equal to r = c , the equations (3.1) allow the solution u = ( − m ( αn + βm + c ) (3.3) where c is a constant.Proof. By direct calculation; with the given parametrisations the equations (3.1) read( u − e u ) = α , u + b u = ( − m β. It can be verified directly that (3.3) also provides a solution to (1.6). Next, we derive theone-soliton solution for (1.6), from the auto-BT (2.2) with u = u as a seed solution. Proposition 3.2.
The equation (1.6) , with lattice parameters (3.2) , admits the one-solitonsolution u = ( − m (cid:18) αn + βm + c + ck − k − ρ n,m ρ n,m (cid:19) , (3.4) where ρ n,m = ρ , (cid:18) a + ka − k (cid:19) n m − Y i =0 ( − i − k ( − i + k (3.5) with constant ρ , , is the plain wave factor.Proof. Let u = u + ( − m ( κ + ν ) , (3.6)where κ = kr . With (3.2) and parametrising the first BT parameter by r = c − k , (3.7)then substitution of u = u and u = u into the auto-BT (2.2) yields e ν = νE + ν + E − , b ν = νF + ( m ) ν + F − ( m ) , (3.8)where E ± = − r ( a ± k ) , F ± ( m ) = r (( − m ∓ k ) . (3.9)5he difference system (3.8) can be linearized using ν = fg and Φ = ( f, g ) T , which leads toΦ( n + 1 , m ) = M Φ( n, m ) , Φ( n, m + 1) = N ( m )Φ( n, m ) , (3.10)where M = E + E − ! , N ( m ) = F + F − ! , (3.11)By “integrating” (3.10) we haveΦ( n, m ) = M ( n )Φ(0 , m ) , Φ( n, m ) = N ( m )Φ( n, , (3.12)where M ( n ) = E n + E n − − E n + κ E n − , N ( m ) = m − Q i =0 F + ( i ) 01 − ( − m m − Q i =0 F + ( i ) m − Q i =0 F − ( i ) . Thus, we get a solution to (3.12):Φ( n, m ) = M ( n ) N ( m )Φ(0 , , (3.13)from which ν = f /g is obtained as ν = E n + m − Q i =0 F + ( i ) · ν , E n − m − Q i =0 F − ( i ) + (cid:16) E n − m − Q i =0 F − ( i ) − E n + m − Q i =0 F + ( i ) (cid:17) ν , κ , (3.14)where ν , = f , g , . Introducing the plain wave factor ρ n,m = ρ , (cid:18) E + E − (cid:19) n m − Y i =0 F + ( i ) F − ( i ) (3.15)with constant ρ , , the above ν is written as ν = − κρ n,m ρ n,m (3.16)where some constants are absorbed into ρ , = − ν , κ + ν , . Substituting (3.16) into (3.6)yields the one-soliton solution (3.4), which solves (1.6) with (3.2) and (3.7). Note that inthe plain wave factor (3.15) n, m ∈ Z , and when m ≤ Q m − i =0 ( · ) is consideredas Q i = m − ( · ).It is interesting that the solution has an oscillatory factor ( − m in m -direction andin the plain wave factor ρ n,m the spacing parameter q for m -direction does not appear.Considering the parametrization (3.2) where p is constant while q depends on m , we cansay that the H2 a equation (1.6) is semi-autonomous.6 Conclusions
In this paper, we have shown that equations which constitute an auto-BT for a quadequation admit auto-BTs themselves. We have focussed on one such equation, the torquedH2 equation denoted H2 a (1.6), which forms an auto-BT for Q1 . This equation is notpart of the ABS list of CAC quad equations, as it is not symmetric with respect to( n, p ) ↔ ( m, q ). The integrability of this equation is guaranteed as it is part of a consistentcube, cf. [14]. The equations H2 a and Q1 comprise an auto-BT from which a Lax pair wasobtained. Using this auto-BT we have derived a seed solution and a one-soliton solution.The parametrisation of these solutions show that H2 a is a semi-autonomous equation. Wehope to be able to construct higher order soliton solutions in a future paper. Acknowledgments
This work was supported by a La Trobe University China studies seed-funding researchgrant, and by the NSF of China [grant numbers 11875040 and 11631007].
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