On construction of solutions of evolutionary Non Linear Schrodinger equation

In this work we present an application of a theory of vessels to solution of the evolutionary Non Liner Schrodinger (NLS) equation. The classes of functions for which the initial value problem is solvable relies on the existence of an analogue of the inverse scattering theory for the usual NLS equation. This approach is similar to the classical approach of Zackarov-Shabbath for solving of evolutionary NLS equation, but has an advantage of simpler formulas and new techniques and notions to construct solutions of the evolutionary NLS equation.


Introduction and Background
Solution of the Non Lineat Schrödinger (NLS) evolutionary equation plays a special role in the theory of PDEs and in physics (optics and water waves). This equation can be defined as follows iyt + yxx + 2|y| 2 y = 0, y(x, 0) = β(x), where y = y(x, t) is a complex valued function of two real variables x, t and y(x, 0) = β(x) is the initial condition, defined on an interval I ⊆ R. Notice that constant 2 used in this work can be replaced with arbitrary number but scaling. This equation has so called integrability property, which enables to use the inverse scattering theory in order to solve it. It is usually done using Zakharov-Shabat system [ZABŠ74]. There are also numerous numerical solutions of this equation, among which we can mention split-step (Fourier) method. We are going to generalize the method of Zakharov-Shabat by introducing evolutionary NLS (regular) vessels. The setting we choose to work in involves bounded operators only and Hilbert space techniques. It is worth noticing that one can also generalize this theory to unbounded operators (as it was done in [Mela] for the Sturm-Liuoville differential equation). We assume from the beginning that the initial value β(x) is analytic on I ⊆ R function and moreover arises from a regular NLS vessel. It is on going research, whether every analytic on I function can be presented using a vessel and we tempt to believe that it is the case. The problem of construction of a vessel for a given β(x) is still open and complicated. On the other hand, we show examples of how to construct different β(x), which have a bounded spectrum, a spectrum on a bounded, continuous curve, or an infinite discrete spectrum. The problem of constructing of a vessel for a given β(x) is identical to the inverse scattering problem of the NLS equation. To emphasize this point let us briefly discuss how the construction of the solution of (1) is performed starting from the spectral data encoded in a 2 × 2 matrix-function S(λ) which is realized [BGR90] in the following manner S(λ) = I − B * 0 X −1 0 (λI − A) −1 B0. Here H is an auxiliary Hilbert space, on which there are defined bounded operators A, X0 and the operator B0 : C 2 → H. We also assume that x 0 B(y)σ2B * (y)dy we will obtain that the function is analytic (Theorem 2) on an interval, where X(x) is invertible. On the other hand, X(x) is invertible on an interval I including x0, because X0 is such. Indeed, at the definition of X(x) we add to X0 a bounded operator with bounded growth. Moreover, it turns out (Theorem 1) that the function maps solutions u(λ, x) of the trivial NLS equation (11) In order to obtain a solution of the evolutionary NLS (1), we evolve the operator B with respect to t as follows and redefine One of the most interesting results of this paper is Theorem 9, where we prove that the new satisfies (1) and coincides with β(x) for t = 0, proving the existence of solutions for (1) with initial value. The formulas presented here enable also to explicitly perform this construction for some basic and important cases. We show how to construct a vessel, for which the spectrum of A lies on a curve Γ (Section 3.2 for β(x) and 5.2 for β(x, t)), with a discrete set (Section 3.3 for β(x) and Section 5.3 for β(x, t)). We also discuss the general construction in Sections 3.1 and 5.1, Finally we present constructions of the Solitons in Section 5.4.

Definition of a regular NLS vessel and its properties
We define first parameters, which will be frequently used in the sequel. Other choice of these parameters generates solutions of the Sturm-Liuoville differential equation and the Kortweg-de-Vries equation (see [Mela, Melb] for details).
Definition 1 NLS vessel parameters are defined as follows: For each vessel NLS vessel there are exist three notions, which play a significant role in research.
Definition 3 Suppose that we are given an NLS vessel VNLS. Then its transfer function S(λ, x), the tau-functions τ (x) and the beta-function β(x) are defined as follows: The definition of β(x) may be considered as excessive, because actually the matrix-function γ * (x) turns to be using the self-adjointenss of X(x). Still, we will use these two notions extensively, so we have defined both of them. Notice that S(λ, x) is a 2 × 2 matrix-function, whose poles and singularities with respect to λ are determined by the operatorA only. Since all the involved operators, appearing at the definition of S(λ, x) are bounded, we can see that S(λ, x) is analytic in λ for all x ∈ I with value I (=identity 2 × 2 matrix) there. As a result, we can consider its Taylor series which is convergent at least for λ > A . Thus we define its (Markov) moments as follows Definition 4 n-th moment of the vessel VNLS is We will present in the next section basic properties of an NLS vessel VNLS by exploring all the objects (the transfer, the tau, the beta functions and the moments). We will also see that there is a standard technique for construction of such vessels.

The transfer and the tau function of an NLS vessel
The main reason to consider NLS vessel is the next Theorem 1, whose proof we present in full details, also it appeared in different settings in [Ls78,Ls01,Mel09,MVa,MVb,Mel11]. It turns out that we can see an NLS vessel as a Bäcklund transformation of the trivial NLS equation to a more complicated one. More precisely, the transfer function S(λ, x) of such a vessel maps solutions of the so called input LDE with the spectral parameter λ to solutions of the output LDE with the same spectral parameter As a result the following differential equation holds Theorem 1 (Vessel as a Bäcklund transformation) Suppose that VNLS is an NLS vessel, de- is a solution of the output LDE (12): Proof: we plug in the expression , then for all x ∈ I and λ ∈ spec(A) Let us differentiate the expression σ1G(λ, x)σ1 using formulas (3) and (5): Plugging this expression back into the equation (12) developed earlier and performing some obvious cancellations, we will obtain x)) = 0, using the linkage condition (6) and the differential equation (11).
Let us present next the significance (and well definedness) of the tau-function τ (x), defined in (9). Using vessel condition (5) X(x) has the formula and as a result Since σ2 has rank 2, this expression is of the form I + T , for a trace class operator T and since X0 is an invertible operator, there exists a non trivial interval (of length at least 1 (a, b) into the group G (the set of bounded invertible operators on H of the form I + T, for a trace-class operator T ) is said to be differentiable if F (x) − I is differentiable as a map into the trace-class operators. In our case, exists in trace-class norm. Israel Gohberg and Mark Krein [IG69, formula 1.14 on p. 163] proved that if A most important question, related to this theory is for what classes of β(x) are obtained. This question is answered in the next Theorem.
Theorem 2 Suppose that VNLS is an NLS vessel, defined in (2). Then the function β(x) is analytic on the interval I.
Proof: Notice that from the formula (3) it follows the operator B(x) is analytic in x. Since X(x) is invertible on I, the operator X −1 (x) is also analytic in x using the formula Thus β(x), defined by (10) is analytic on I.
Theorem 3 (Permanency conditions) Suppose that we are given an NLS regular vessel VNLS, then 1. if the Lyapunov equation (4) holds for a fixed x0 ∈ I, then it holds for all x ∈ I, for all x ∈ I, 3. det S(λ, x) = det S(λ, x0) for all x0, x ∈ I and all points of λ-analyticity of S(λ, x).
1 sp -stands for the trace in the infinite dimensional space.
Proof: Differentiating Lyapunov equation and using the vessel conditions (3), (5) we will obtain that from where the permanency of the Lyapunov equation follows. Similarly, differentiating S * (−λ, x)σ1S(λ, x) we will obtain zero and the permanency of (15) follows. For the last statement, using (13) we calculate for λ ∈ spec(A) We will not be using the last property in this statement, but we find it interesting by itself.

Moments
The following properties of the moments H ( x) of an NLS vessel are immediate from their definition as the coefficients of 1 λ n+1 at the Taylor series of S(λ, x).

Theorem 4 Let VNLS be an NLS vessel. Then its moments satisfy the following equations
Hn+1 Proof: Plugging the Taylor expansion formula into (13) and equating the coefficients of 1 λ n=1 we will obtain the first formula (17). The second formula (18) is obtained in the same manner from (18).
It turns out that using only the differential equations (17) As a result, we obtain that the formula for the tau-function (14) becomes For example, if we choose the initial parameters H 11 0 (0), H 22 0 (0) to be equal, we will obtain that under normalization τ (x0) = 1: 3 Examples of constructions of regular NLS vessels

Construction of an NLS vessel from a realized function
Construction of an NLS vessel from the scattering data (initial condition S(x0, λ)) can be performed as follows. Suppose that the function S(λ, x0) is realized [BGR90] as follows satisfying additionally AX0 + X0A * + B0B * 0 = 0 and X * 0 = X0. These two conditions are required by the permanency conditions (Theorem 3) and will hold for all x by the construction. then define B(x) as the unique solution of (3) with initial value B0: Then define and define γ * (x) (and hence β(x)) using (6). It is a straigtforward to check that all vessel conditions hold: Theorem 5 Suppose that we are given of a function where using an auxiliary Hilbert space H the operators act as follows: We show two special examples, arising from this construction for special cases of the choice of H.

Let us fix a bounded continuous curve Γ = {µ(t) | t ∈ [a, b]} (i.e. µ(t) is continuous) and define
We suppose without loss of generality that x0 = 0 and we construct a vessel, existing on an interval I including zero.
Let A = 2µ, i.e. it is a bounded operator acting within L 2 (Γ). Solving differential equation (3) we find that Notice that the adjoint B * (x) : H → C 2 is defined as follows It is a well-defined operator, because by the Cauchy-Schwartz inequality the integrals are finite. It turns out that the operator X(x) can be also explicitly defined as follows for any f ∈ H. Notice that X(x)f is a new function at H, for which we present its value at the point µ = µ(t) ∈ Γ: It almost immediate that such an operator satisfies the conditions (4), (5). For example for the Lyapunov equation, the expression (AX(x)f )(µ) + (X(x)A * f )(µ) is obtained by multiplying the expression under the integral in (X(x)f )(µ) by 2(µ+μ(s)). When it is canceled with the denominator, we obtain (−B(x)σ2B * (x)f )(µ). Similarly, differentiating (X(x)f )(µ) we cancel the denominator and switch the sign of the first term, so that (5) holds. there is only one problem, arising from the zero of the denominator µ(t) +μ(s) = 0. It can be overcome by requiring that either Γ ∩ (−Γ * ) = ∅, or b1, b2 are Hölder functions and b1(µ(t))b1(µ(s)) + b2(µ(t))b2(µ(s)) = 0, whenever µ(t) +μ(s) = 0 (so that the zero of the denominator is canceled by the numerator). We will make the following assumption on the curve, to simplify arguments: Then investigating the formula for X(x) we find that for each f ∈ H ab 2(µ(s))eμ (s)x µ +μ(s) f (µ(s))ds (24) Notice that this expression can be presented as where the functions are analytic functions of µ in C\(−Γ * ). Notice also that Γ ⊆ C\(−Γ * ) by the assumption on Γ. Thus the functions c1(µ), c2(µ) can have only isolated zeros on Γ or to be identically zero.
To simplify a proof that X(x) is an invertible operator, let us assume that b1 = b2 and that they are an analytic function of µ. Then the equality X(x)f = 0 is equivalent to where H = L 2 (Γ), A = 2µ, X(x) and B(x) are defined by (24) and by (23) for |b1(µ)| 2 +|b2(µ)| 2 = 0. Then the collection VNLS is an NLS regular vessel existing on I = R.
Proof: By the construction the operators are well defined and satisfy the vessel condition. Since the operator X(x) is invertible for all x ∈ R, we obtain that I = R.

Construction of a regular NLS vessel with a discrete spectrum
In this section we want to show how to construct a vessel, whose spectrum is a given set of numbers D = {2µn}. We define H = ℓ 2 , which the set of infinite sequences, summable in absolute value. Now we can imitate the construction of the vessel on a curve Γ using discretization as follows. we define first the operator A = diag(2µn) and for this operator to be bounded, we have to demand that the sequence D is bounded from below and from above, namely Definition 6 The sequence D is called bounded if there exist M > 0 such that |µn| < M for µn ∈ D. It is called separated from zero if there exist m > 0 such that 0 < m < |µn| for all µn ∈ D.
In the next definition we think of X(x) as an infinite matrix with the entry m, n denoted by [X(x)]n,m: we also assume that [X(x)]n,m = b1nb1m − b2nb2m 2 x, whenever µn +μm = 0. The fact that B(x) is a well defined operator is immediate from the definition. Indeed, since the sequence µn is bounded the term e ±µnx is uniformly bounded in absolute value by e M x . The fact that the operator X(x) is bounded also easily follows from the definitions and from the assumption on D: Under condition that X0 = X(0) is invertible, there exists a non trivial interval (of length at least 1 X −1 0 ) on which X(x) is invertible too. Thus we obtain the following Theorem.

Evolutionary regular NLS vessel
We present a construction of solutions of the equation (1) which has initial value β(x, 0) arising from a regular NLS vessel. For this we will insert dependence on the variable t into the vessel operators and postulate evolution of the operators B, X with respect to t. This is done in the next Definition.

Construction of a solution for evolutionary NLS vessel from a realized function
Suppose that β(x) was constructed from a realized function as in Section 3.1 Then the construction can proceed the following steps, each one requiring a solution of Linear Differential equation with initial value. Construct B(x) and X(x) by formulas (21)  All the vessel equations will be satisfied by the construction and can be easily verified.