1. In this section, we again consider Dirac system (1.1) on the semi-axis [0, ∞) with a bounded potential v:
kv(x)k ≤M forx∈R+. (4.1)
According to Definition 2.5, the Weyl function of system (1.1) such that (4.1) holds is given by the right-hand side of (2.5). From Proposition 2.2 we see that the Weyl function ϕ is unique. Moreover, it follows from the proof of Corollary 3.12 that
ϕ(z) = 2iz Z ∞
0
e2ixzΦ(x)dx, e−2(M+ε)xΦ(x)∈L2m2×m1, (4.2)
and for any 0< l <∞ we have
Φ(x)≡Φ1(x) (0< x < l). (4.3) Since Φ1 ≡ Φ, we get that Φ1(x) does not depend on l for l > x. Compare this with the proof of Proposition 4.1 in [15], where the fact thatE(x, t) (and so Φ1) does not depend onl follows from the uniqueness of the factorizations of operatorsSl−1. See also Section 3 in [4] on the uniqueness of the accelerant.
Using (4.2), (4.3) and Theorem 3.14 we get our next result.
Theorem 4.1 The unique Weyl function of Dirac system (1.1) satisfying (4.1) admits representation
ϕ(z) = 2iz Z ∞
0
e2ixzΦ1(x)dx, e−2ηxΦ1(x)∈L2m2×m1 (η > M). (4.4) The procedure to recover Φ1 and v from ϕ is given in Theorem 3.14.
2. The well-known “focusing” nonlinear Schr¨odinger (fNLS) equation 2vt+ i(vxx+ 2vv∗v) = 0
vt := ∂
∂tv
, (4.5)
where v(x, t) is an m1 ×m2 matrix function, is interesting from a mathe-matical point of view and plays an essential role (in its scalar, matrix and multicomponent forms) in numerous applications (see, e.g., [2,44]). The fNLS equation (4.5) is equivalent (see [11, 48] and references therein) to the zero curvature equation
Gt−Fx+ [G, F] = 0, [G, F] :=GF −F G, (4.6) where the m×m (m =m1+m2) matrix functions G(x, t, z) and F(x, t, z) are given by the formulas
G= izj +jV, F = i z2j−izjV − Vx+jV2 /2
, (4.7)
and j and V are defined in (1.2). The zero curvature representation (4.6) of the integrable nonlinear equations appeared soon after the seminal Lax pairs
(see a historical discussion in [11] and original papers [1,25,47]). Furthermore, condition (4.6) is the compatibility condition for the auxiliary linear systems
yx =Gy, yt=F y, (4.8)
and in the case of the fNLS equation the first of those systems is (in view of (4.7)) the skew-self-adjoint Dirac system (1.1). The compatibility condi-tion was studied in a more rigorous way and the corresponding important factorization formula for fundamental solutions was introduced in [38, 40].
It was proved in greater detail and under weaker conditions in [34]. More specifically, we have the following proposition.
Proposition 4.2 [34]. Let m ×m matrix functions G and F and their derivatives Gt and Fx exist on the semi-strip
D={(x, t) : 0≤x <∞, 0≤t < a}, (4.9) let G, Gt and F be continuous with respect to x and t on D, and let (4.6) hold. Then we have the equality
u(x, t, z)R(t, z) =R(x, t, z)u(x,0, z), R(t, z) := R(0, t, z), (4.10) where u(x, t, z) and R(x, t, z) are normalized fundamental solutions given, respectively, by:
ux =Gu, u(0, t, z) =Im; Rt =F R, R(x,0, z) =Im. (4.11) The equality (4.10) means that the matrix function
y =u(x, t, z)R(t, z) =R(x, t, z)u(x,0, z)
satisfies both systems (4.8) in D. Moreover, the fundamental solution u ad-mits the factorization
u(x, t, z) =R(x, t, z)u(x,0, z)R(t, z)−1. (4.12) To derive the evolution ϕ(t, z) of the Weyl functions of the skew-self-adjoint Dirac systems yx(x, t, z) =G(x, t, z)y(x, t, z), we rewrite (4.12) in the form
u(x, t, z)−1 =R(t, z)u(x,0, z)−1R(x, t, z)−1. (4.13)
Note that an additional parameter t appears now in the functions discussed in Sections 2 and 3 (see, e.g, u(x, t, z) andϕ(t, z)). We partitionR(t, z) into blocks R={Rij}2i,j=1 (and R11 is an m1 ×m1 block here).
Theorem 4.3 Let an m1×m2 matrix function v(x, t) in D be continuously differentiable together with its first derivatives and let vxx exist. Assume that v satisfies the fNLS equation (4.5) as well as the following inequalities:
sup
(x,t)∈Dkv(x, t)k ≤M, sup
x∈R+kvx(x, t)k<∞ for each 0≤t < a. (4.14) Then the evolution ϕ(t, z) of the Weyl functions of Dirac systems yx = Gy, where G has the form (4.7), is given (for ℑ(z)> M) by the equality
ϕ(t, z) = R21(t, z) +R22(t, z)ϕ(0, z)
R11(t, z) +R12(t, z)ϕ(0, z)−1
. (4.15) P r o o f. SinceV =V∗andjV =−V j, it follows from (4.7) thatF(x, t, z)∗ =
−F(x, t, z). Hence, using (4.7) and (4.11) we get
∂
∂t R(x, t, z)∗jR(x, t, z)
=R(x, t, z)∗(jF(x, t, z)−F(x, t, z)j)R(x, t, z)
=R(x, t, z)∗ i(z2−z2)Im+ (z+z)V(x, t) + iVx(x, t)j
R(x, t, z). (4.16) In view of (4.14) and (4.16), for each 0 ≤ t < a there are numbers M1,2(t) (M1 > M >0, M2 >0) and the corresponding quarterplane
D1 ={z : ℑz > M1, ℜz > M2} ⊂C, (4.17) such that the inequality
∂
∂t R(x, t, z)∗jR(x, t, z)
<0 (4.18)
holds inD1. Inequality (4.18) and the initial condition R(x,0, z) =Im imply R(x, t, z)∗jR(x, t, z)≤j, or, equivalently,
R(x, t, z)∗−1
jR(x, t, z)−1 ≥j, z∈ D1. (4.19) Let P(z) satisfy (2.3) and thus satisfy (2.1). Let Pe(x, t, z) (sometimes we will also write Pe(z), omitting x and t) be determined by the equality
Pe(x, t, z) :=R(x, t, z)−1P(z). (4.20)
In view of (4.19) the matrix function Pe satisfies (2.1) in D1.
Because of the smoothness conditions onv, we see thatGandF given by (4.7) satisfy the requirements of Proposition 4.2, that is, (4.13) holds. Using (4.13) and (4.20), we see that
u(x, t, z)−1P(z) =R(t, z) According to (4.21) and (4.23) we have (in the quarterplane D1) the equality
0 Im2 Taking into account Definition 2.1, we see that the left-hand side of (4.24) belongs N(x, t, z) and φ(x, t, z) ∈ N(x,0, z), where φ is defined in (4.22).
Therefore, (4.15) follows from (2.5) and (4.24) (see also (2.24)), when x tends to infinity. Although, we first derived (4.15) only for D1, we see that it holds everywhere in CM via analyticity.
Remark 4.4 Theorems 4.1 and 4.3 can be applied to recover solutions of the fNLS. Theorems on the evolution of the Weyl functions also make up the first step in proofs of uniqueness and existence of the solutions of nonlinear equations via the ISpT method (see, for instance, [35]).
Acknowledgement. The work of I.Ya. Roitberg was supported by the German Research Foundation (DFG) under grant no. KI 760/3-1 and the work of A.L. Sakhnovich was supported by the Austrian Science Fund (FWF) under Grant no. Y330.
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B. Fritzsche, Fakult¨at f¨ur Mathematik und Informatik, Mathematisches Institut, Universit¨at Leipzig, Johannisgasse 26, D-04103 Leipzig, Germany, e-mail: Bernd.Fritzsche@math.uni-leipzig.de
B. Kirstein, Fakult¨at f¨ur Mathematik und Informatik, Mathematisches Institut, Universit¨at Leipzig, Johannisgasse 26, D-04103 Leipzig, Germany,
e-mail: Bernd.Kirstein@math.uni-leipzig.de
I. Roitberg, Fakult¨at f¨ur Mathematik und Informatik, Mathematisches Institut, Universit¨at Leipzig, Johannisgasse 26, D-04103 Leipzig, Germany, e-mail: Inna.Roitberg@math.uni-leipzig.de
A.L. Sakhnovich, Fakult¨at f¨ur Mathematik, Universit¨at Wien, Nordbergstrasse 15, A-1090 Wien, Austria, e-mail: al−sakhnov@yahoo.com