4 Generalized discrete Heisenberg magnet model and its auxiliary linear systems
4.2 Discrete Dirac systems depending on an additional continuous time parameter
Let Σ = {α, S0, ϑ1, ϑ2} be a strongly admissible quadruple, and assume that i6∈ σ(α). We shall associate with Σ a family of strongly admissible quadru-ples depending on two parameters t and k, where t ∈R+ and k ∈N0. This will be done in two steps.
Step 1: construction of Σt. In this first step we just require Σ to be admissible, not necessarily strongly admissible. From item (i) in Lemma A.1 we know that −i 6∈ σ(α). Together with the assumption that i 6∈ σ(α) this implies that both α−iIn and α+iIn are invertible. This allows us to define Λ(t) = Λ0(t) and S(t) = S0(t) using (4.18) (and (4.17)) and (4.16), respectively.
Lemma 4.4. LetΣ ={α, S0, ϑ1, ϑ2}be an admissible quadruple, and assume that i 6∈ σ(α). Let ϑj(t), j = 1,2, and S(t) be defined by (4.18) and (4.16), respectively. Then on some interval interval −ε1 < t < ε2 the quadruple
Σt={α, S(t), ϑ1(t), ϑ2(t)} (4.21) is admissible too. Moreover, if Σ is strongly admissible, then the same holds true for the quadruple Σt for each t ∈R.
Proof. Since S0 is positive definite, the same holds true forS(t) provided t is sufficiently small. Put
Υ(t) :=αS(t)−S(t)α∗−iΛ(t)Λ(t)∗. (4.22)
Here Λ(t) is defined by (4.17). Since {α, S0, ϑ1, ϑ2}is an admissible quadru-ple, it follows that Υ(0) = 0. After some easy transformations, using (4.17) and (4.16) and differentiating the right-hand side of (4.22), we see that Υ satisfies the linear differential equation
d
dtΥ(t) =−(α+iIn)−1 αΥ(t) + Υ(t)α∗
(α∗−iIn)−1+
−(α−iIn)−1 αΥ(t) + Υ(t)α∗
(α∗+iIn)−1. (4.23) Since the initial value Υ(0) is zero, it follows that Υ(t)≡0, that is,
αS(t)−S(t)α∗ =iΛ(t)Λ(t)∗. (4.24) Using (4.18) and (4.17), we conclude that {α, S(t), ϑ1(t), ϑ2(t)} is an admis-sible quadruple.
Next, assume Σ is strongly admissible. The identity (4.24) was already proved above. By Definition 3.3, the fact that Σ is strongly admissible means that the pair α and ϑ1 is controllable. But then the first equality in (4.18) tells us that
Span
n[−1
ν=0
Imανϑ1(t) =e−2t(α−iIn)−1 Span
n[−1
ν=0
Im ανϑ1(0)
=Cn, (4.25) that is, the pair α and ϑ1(t) is also controllable. Furthermore, since Σ is strongly admissible item (ii) of Lemma A.1 shows that σ(α) ⊂ C+. From (4.24) and σ(α)⊂C+ follows representation (A.7) ofS0(t) =S(t). Since the pair α and ϑ1(t) is controllable representation (A.7) implies that S(t) > 0.
Hence, again using Definition 3.3, the quadruple Σt is strongly admissible.
Step 2: construction of Σt,k. Let Σ = {α, S, ϑ1, ϑ2} be a strongly admissible quadruple, and assume that i 6∈ σ(α). Let Σt be the quadrupple defined by (4.21). Hence, according to the previous lemma, the quadruple Σt
is also strongly admissible for each t∈R. But then we can apply the results of Section 3 to associate with each Σt a family of quadruples Σt,k, k ∈N0.
Since Σt is strongly admissible, we know from Lemma 3.2 that α is in-vertible. In fact, by item (ii) in Lemma A.1, all eigenvalues of α are in C+. Following the constructions given in Section 3, put
ϑ1(t, k) = (In+iα−1)kϑ1(t), ϑ2,k(t) = (In−iα−1)kϑ2(t), (4.26) Λ(t, k) =
ϑ1(t, k) ϑ2(t, k)
. (4.27)
Furthermore, define S(t, k) by
S(t, k+ 1) =S(t, k) +α−1S(t, k)α−∗+
+α−1S(t, k)α−∗Λ(t, k)jΛ(t, k)∗, k= 0,1,2, . . . , S(t,0) =S(t).
Using these matrices we set
Σt,k ={α, S(t, k),(In+iα−1)kϑ1(t),(In−iα−1)kϑ2(t)}. (4.28) By Lemma 3.2, the quadruple Σt,k is also strongly admissible.
Let Σ ={α, S, ϑ1, ϑ2} be a strongly admissible quadruple, and let Σt be the strongly admissible quadruple defined by (4.21). Following (3.12) put
Ck(t) =j+ Λ(t, k)∗S(t, k)−1Λ(t, k)+
−Λ(t, k+ 1)∗S(t, k+ 1)−1Λ(t, k+ 1), k ∈N0. Then, by Definition 3.3, the sequence {Ck(t)}k∈N0 is the pseudo-expontential potential generated by Σt. The corresponding SkDDS is given by
yk+1(t, z) =
Im+iz−1Ck(t)
yk(t, z) (k∈N0).
Furthermore, we know (see Propositon 3.6) thatCk(t) admits a factorization of the form:
Ck(t) = Uk(t)jUk(t)∗, Uk(t)Uk(t)∗ =Im, k ∈N0. (4.29) We also note that (4.9) and (4.10) are immediate consequences of the iden-tities (3.38) and (3.39). Indeed, since Σt ={α, S(t), ϑ1(t), ϑ2(t)} is strongly admissible, we just apply Lemma 3.10 with Σt= Σt,0 in place of Σ0. Thus it remains to prove the identity (4.8) in order to prove Theorem 4.2. First, we formulate a proposition, which will be proved later.
Proposition 4.5. LetΣ ={α, S0, ϑ1, ϑ2}be a strongly admissible quadruple, assume thati6∈σ(α), and letΣt,kbe the strongly admissible quadruple defined by (4.28), where k ∈N0 and t∈R. Then the function Yk(t, z) defined by
Yk(t, z) =WΣt,k(−z)
Im+ i zj
k
exp
−2t P1
z+i+ P2 z−i
, (4.30) where z 6= ±i and z 6∈ σ(−α), is a solution of equation (4.2) as well as of equation (4.3).
Proof of Theorem 4.2. As soon as Proposition 4.5 is proved we can prove Theorem 4.2 by using Proposition 4.1. Indeed, assume that the function Y defined by (4.30) satisfies (4.2) and (4.3). Then
d
But then the above identity yields the compatibility equation (4.1), and we can apply Proposition 4.1 to get the identity (4.8).
In order to prove Proposition 4.5 we need to compute the derivatives d
dtΛ(t, k), d
dtS(t, k), d
dtΛ(t, k)∗S(t, k)−1, d
dtWΣt,k. (4.31) This will be done in a couple of steps.
The first derivative in (4.31). Recall that Λ(t, k) is given by (4.26) and (4.27). It follows that But then, using (4.18) and (4.17), we see that
d
Here P1 and P2 are the projctions defined in (4.19) and (4.20), respectively.
The second derivative in (4.31). In order to compute the second deriva-tive in (4.31) we present an alternaderiva-tive way to obtain the quadruple Σt,k
given by (4.28). As before, let Σ = {α, S0, ϑ1, ϑ2} be a strongly admissible quadruple, and assume that i 6∈ σ(α). We know that σ(α) ⊂ C+. In a particular α is non-singular. As in Section 3, let
Σk ={α, Sk,(In+iα−1)kϑ1,(In−iα−1)kϑ2}.
From Lemma 3.2 we know that Σk is again a strongly admissible quadruple.
The assumption i6∈σ(α) allows us to apply Lemma 4.4 to Σk in place of Σ.
Put
ϑ˜1(k, t) =e−2t(α−iIn)−1(In+iα−1)kϑ1, (4.33) ϑ˜2(k, t) =e−2t(α+iIn)−1(In−iα−1)kϑ1, (4.34)
Λ(k, t) =˜ ϑ˜1(k, t) ˜ϑ2(k, t)
, (4.35)
and let ˜S(k, t) be given by d
dtS(k, t) =˜
=−h
(α−iIn)−1S(k, t) + (α˜ +iIn)−1S(k, t)+˜
+ ˜S(k, t)(α∗+iIn)−1+ ˜S(k, t)(α∗−iIn)−1+ (4.36) + 2(α2+In)−1
αΛ(k, t)j˜ Λ(k, t)˜ ∗+ ˜Λ(k, t)jΛ(k, t)˜ ∗α∗
(α∗)2+In
−1i , S(k,˜ 0) = Sk.
Note that (4.36) is the analogue of (4.16) with Σk in place of Σ. Thus, by Lemma 4.4, the quadruple
Σk,t ={α,S(k, t),˜ ϑ˜1(k, t),ϑ˜2(k, t)}
is strongly admissible.
Lemma 4.6. For each k ∈N0 and t∈R we have Σt,k= Σk,t. In particular, d
dtS(t, k) =
=−h
(α−iIn)−1S(t, k) + (α+iIn)−1S(t, k)+
+S(t, k)(α∗+iIn)−1 +S(t, k)(α∗−iIn)−1+ (4.37) + 2(α2+In)−1
αΛ(t, k)jΛ(t, k)∗+ Λ(t, k)jΛ(t, k)∗α∗
((α∗)2+In)−1i . Proof. From (4.18) and (4.26) and the identities (4.33) and (4.34) we see that
ϑ1(t, k) = ˜ϑ1(k, t) =e−2t(α−iIn)−1(In+iα−1)kϑ1, ϑ2(t, k) = ˜ϑ2(k, t) =e−2t(α+iIn)−1(In−iα−1)kϑ2.
It follows that Λ(t, k) = ˜Λ(k, t). Recall that Σt,k and Σk,t are both strongly admissible. Then item (ii) in Lemma A.1 tells us that S(t, k and ˜S(k, t) are uniquely determined by Λ(t, k) and ˜Λ(k, t). Thus S(t, k and ˜S(k, t) coincide, and hence Σt,k = Σk,t. Thus Σt,k= Σk,t.
Finally, using Λ(t, k) = ˜Λ(k, t) and S(t, k) = ˜S(k, t), we see that (4.37)
follows from (4.36).
The third derivative in (4.31). From (4.32) and (4.37) we shall derive the identity
d
dt Λ(t, k)∗S(t, k)−1
=Hk+(t)Λ(t, k)∗S(t, k)−1(α−iIn)−1+
+Hk−(t)Λ(t, k)∗S(t, k)−1(α+iIn)−1. (4.38) Here Hk+ and Hk− are the functions given by (4.19) and (4.20), respectively.
To do this, note that according to (4.32) we have d
dt Λ(t, k)∗S(t, k)−1
=−2P1Λ(t, k)∗(α∗+iIn)−1+
−2P2Λ(t, k)∗(α∗−iIn)−1+
−Λ(t, k)∗S(t, k)−1d
dtS(t, k)
S(t, k)−1. (4.39) Identity (3.7) implies that
(α∗±iIn)−1S(t, k)−1 =S(t, k)−1(α±iIn)−1+
+i(α∗±iIn)−1S(t, k)−1Λ(t, k)Λ(t, k)∗S(t, k)−1(α±iIn)−1. (4.40)
By using (4.37), (4.40) and the equalities
2(α2+In)−1 =i (α+iIn)−1−(α−iIn)−1 ,
2(α2+In)−1α= (α−iIn)−1+ (α+iIn)−1, (4.41) after some calculations, we see that (4.39) can be rewritten in the form:
d
dt Λ(t, k)∗S(t, k)−1
=
=Vk+(t)Λ(t, k)∗S(t, k)−1(α−iIn)−1+Vk−(t)Λ(t, k)∗S(t, k)−1(α+iIn)−1. Here
Vk+(t) =Im+j−iΛ(t, k)∗S(t, k)−1(α−iIn)−1Λ(t, k)j+
+ijΛ(t, k)∗(α∗−iIn)−1S(t, k)−1Λ(t, k) + Λ(t, k)∗S(t, k)−1×
×(α−iIn)−1Λ(t, k)jΛ(t, k)∗(α∗−iIn)−1S(t, k)−1Λ(t, k), (4.42) and
Vk−(t) =Im−j+iΛ(t, k)∗S(t, k)−1(α+iIn)−1Λ(t, k)j+
−ijΛ(t, k)∗(α∗+iIn)−1S(t, k)−1Λ(t, k)−Λ(t, k)∗S(t, k)−1×
×(α+iIn)−1Λ(t, k)jΛ(t, k)∗(α∗+iIn)−1S(t, k)−1Λ(t, k). (4.43) It remains to show thatVk+(t) =Hk+(t) andVk−(t) =Hk−(t). To do this note that (3.51) and (3.52) tell us that Hk+(t) andHk−(t) are also given by
Hk+(t) =Im+WΣt,k(i)jWΣt,k(−i)∗, (4.44) Hk−(t) =Im−WΣt,k(−i)jWΣt,k(i)∗. (4.45) Now use (4.47) and compute the products. This shows that Hk+(t) is equal to the right-hand side of (4.42) and Hk−(t) is equal to the right-hand side of (4.43). In other words,Vk+(t) =Hk+(t) andVk−(t) =Hk−(t), and hence (4.38) is proved.
The fourth derivative in (4.31). In this part we shall show that d
dtWΣt,k
(λ) =
Hk+(t)
λ−i +Hk−(t) λ+i
WΣt,k(λ)+
−2WΣt,k(λ) P1
λ−i + P2
λ+i
. (4.46)
First, recall that WΣt,k is given by
WΣt,k(z) =Im+iΛ(t, k)∗S(t, k)−1(zIn−α)−1Λ(t, k). (4.47) Using (4.32) and (4.38), we obtain
d dtWΣt,k
(λ) =i
Hk+(t)Λ(t, k)∗S(t, k)−1(α−iIn)−1(λIn−α)−1Λ(t, k)+
+Hk−(t)Λ(t, k)∗S(t, k)−1(α+iIn)−1(λIn−α)−1Λ(t, k)+
−2Λ(t, k)∗S(t, k)−1(α−iIn)−1(λIn−α)−1Λ(t, k)P1+
−2Λ(t, k)∗S(t, k)−1(α+iIn)−1(λIn−α)−1Λ(t, k)P2
. (4.48)
Clearly we have
(α−iIn)−1(λIn−α)−1 = (λ−i)−1 (α−iIn)−1+ (λIn−α)−1
, (4.49) (α+iIn)−1(λIn−α)−1 = (λ+i)−1 (α+iIn)−1+ (λIn−α)−1
. (4.50) Taking into account (4.47), (4.49) and (4.50), we rewrite (4.48) as
d dtWΣt,k
(λ) = (λ−i)−1Hk+(t) WΣt,k(λ)−WΣt,k(i) + + (λ+i)−1Hk−(t) WΣt,k(λ)−WΣt,k(−i)
+
−2(λ−i)−1 WΣt,k(λ)−WΣt,k(i) P1
−2(λ+i)−1 WΣt,k(λ)−WΣt,k(−i)
P2. (4.51) Applying (3.46) and (3.47) with Σt,k in place of Σk we see that
Hk+(t)WΣt,k(i) = 2WΣt,k(i)P1, Hk−(t)WΣt,k(−i) = 2WΣt,k(−i)P2. (4.52) In view of (4.52), formula (4.51) can be rewritten as (4.46).
Proof of Proposition 4.5. LetY be the matrix fucntion defined by (4.30).
The fact that Yk(t, z) is a solution of equation (4.2) follows by applying Theorem 3.5 with Σt in place of Σ. Similarly, using (4.46) and the fact that the matrices P1, P2, and j are mutually commutative, we see that Yk(t, z) is
a solution of the system (4.3). Indeed,
Thus Y has the desired properties.
Remark 4.7. The equalityHk− = 2Im−(Hk+)∗ is immediate from (4.44)and (4.45).
A remark on the continuous space variable x analogue. The transfer matrix function of the form (2.10) depending on the continuous parameter t and discrete parameter k, which we use in this section, appeared in [27], whereas the transfer matrix function (of the form (2.10)) depending on con-tinuous parameters x and t appeared first in [36, 37] (see also [22] and [23]).
To explain the case of two continuous parameters in more detail we begin with the following analogue of Lemma 4.4.
Lemma 4.8. Let Σ ={α, S0, ϑ1, ϑ2}be a strongly admissible quadruple, and let k be a positive integer. Define
ϑ1(t) =e−itαkϑ1, ϑ2(t) =eitαkϑ2, Λ(t) =
The proof of the above lemma follows the same line of reasoning as used in the proof of Lemma 4.4. We omit the details.
Next, using the above lemma, we construct a family of pseudo-exponential potentials depending continuously on an additional variable t ∈ R. The starting point is the pseudo-exponential potential v in (2.1) defined by the strongly admissible quadruple Σ = {α, S0, ϑ1, ϑ2}. Fix t ∈ R, and let Σ(t) be the strongly admissible quadruple defined in Lemma 4.8, i.e., Σ(t) = {α, S(t), ϑ1(t), ϑ2(t)}. In particular, S(t) is given by (4.53). We apply Lemma 2.1 with Σ(t) in place of Σ. Put
ϑ1(x, t) =e−ixαϑ1(t) =e−i(xα+itαk)ϑ1, ϑ2(x, t) =eixαϑ2(t) =ei(xα+itαk)ϑ2,
Λ(x, t) =
ϑ1(x, t) ϑ2(x, t) , S(x, t) =S(t) +
Z x 0
Λ(s, t)jΛ(s, t)∗ds, x≥0. (4.54) Then, by Lemma 2.1, the quadruple
Σ(x, t) ={α, S(x, t), ϑ1(x, t), ϑ2(x, t)} (4.55) is admissible for each x ≥ 0. Furthermore, the corresponding pseudo-exponential is given by
v(x, t) = 2ϑ∗1ei(xα∗+it(α∗)k)S(x, t)−1ei(xα+itαk)ϑ2, x≥0. (4.56) In this way we obtain a family of pseudo-exponential potentials depending on the additional parameter t.
Fork= 2,3 the potentialv in (4.56) satisfies important integrable nonlin-ear equations; see, e.g., [36, Section 2] and [23, Section 4]. Namely, for k = 2 the function v is a solution of the matrix nonlinear Schr¨odinger equation
2∂v
∂t +i∂2v
∂x2 + 2ivv∗v = 0, (4.57) and for k = 3 it satisfies the matrix modified Korteweg-de Vries equation
4∂v
∂t + ∂3v
∂x3 + 3∂v
∂xv∗v +vv∗∂v
∂x
= 0. (4.58)
The proofs of the above results can be obtained by direct computa-tions as in [36, Section 2] and [23, Section 4] or by applying the generalized B¨acklund-Darboux transformation [38] to auxiliary linear systems and using
compatibility condition (zero curvature equation) Gt−Fx+GF −F G= 0, which is a continuous case equivalent of the compatibility condition (4.1).
For instance, equation (4.57) is equivalent to the compatibility condition Gt−Fx+GF −F G= 0, where G=izj+V(x, t),
F =i z2j −izjV(x, t)− Vx(x, t) +jV(x, t)2)/2 , and V has the form (1.2).