Regular symmetric NLS Vessels

We will see that construction of a regular NLS vessel gives rise to a solution of the evolutionary Non Linear Shrödinger equation (eNLS). The results presented here can be found in Onconstruction of solutions of evolutionary Non Linear Schrodinger equation (Arxiv).

NOTICE that this construction as actually the same construction as for the regular KdV vessels, with vessel parameters changed.

Definition: NLS vessel parameters are defined as follows:

σ₁ is I (the identity matrix)

σ₂ is

γ is the zero matrix

Definition: Collection of operators and space

V_NLS = (A, B(x, t), X(x, t); σ₁, σ₂, γ, γ_*(x, t); H, C²; I = [a, b]x[c, d]),

where A, X(x, t)= X*(x, t): H →H , B(x, t): C² →H are differentiable bounded operators, σ₁, σ₂, γ are SL parameters and γ_* (x, t) = - (γ*(x, t))* is a 2x2 matrix, is called a regular NLS vessel if the following conditions hold (we suppress the dependence on x, t for the corresponding operators):

(DB) δ/δ x [B σ₁] = - A B­ σ₂ - B γ ,

(DX) δ/δ x X = B σ₂ B*,

(DBt) δ/δ t [B σ₁] = i A δ/δ x B

(DXt) δ/δ t X = i A B σ₂ B* – i B σ₂ B* A* + i B γ B*

(Linkage) γ_* = γ + σ₂ B* X^-1B σ₁- σ₁ B* X^-1B σ₂ Linkage condition

(Lyapunov) A X + X A*+ B σ₁ B* = 0 Lyapunov equation

And where the operator X is invertible on a rectangle I.

The notion of tau function τ(x, t) and the transfer function S(λ, x, t) remain the same as for regular vessels but there is added a dependence on the variable t. The matrix-function γ_*(x, t) is as follows

where |β (x, t)|² = (τ_x(x, t)/ τ(x, t))_x. The following theorem explains why we call such vessels as NLS.

Theorem (NLS equation): the function β(x, t) satisfies the following evolutionary NLS equation

(eNLS) i β_t (x, t) + β(x, t)_{xx} + 2 | β(x, t)|² β(x, t) = 0

Standard construction of an NLS vessel

Let A, X₀=X₀^*: H →H, B₀: C² →H be operators satisfying

· Lyapunov equation: A X₀ + X₀ A*+ B₀σ₁ B₀*=0,

· X₀ is invertible

Then

_1. Define B(x) as the unique solution of (DB) with the initial value B_0,

2. Define X(x) as the unique solution of (DX) with the initial value X₀. It will be invertible on an interval I, since it is invertible at 0 and is continuous,

3. Define B(x, t) as the solution of the “wave” equation (DBt) with initial B(x) from above,

4. Define X(x, t) by integrating the right hand side of (DXt) and using the initial condition X(x) from above,

5. Define γ_*(x) on I, where X(x, t) is invertible using (Linkage).

Transfer function

Definition: Transfer function of the vessel V_NLS is defined as follows

(Realize) S(λ, x, t) = I - B*(x, t) X^-1(x, t)( λ I - A)^-1 B(x, t) σ₁

defined for λ out the spectrum of A and for x,t in I.

This function is symmetric, analytic in x and in t inside of the rectangle I, and multiplication by it maps solutions of LDEs. See details in regular vessels. But it has now a nice evolution formula with respect to t, which is easy to explain using moments.

Moments and evolutionary equation of the transfer function

Similarly to the regular vessels, we define moment as follows

(DefH) H_n (x, t) = B*(x, t) X^-1(x, t) Aⁿ B(x, t), n = 0, 1, 2, 3,…

Coming from the coefficients of the Taylor series of

(SMomnts) S(λ, x, t) = I - ∑ H_n(x, t) σ₁ / λⁿ⁺¹

In order to derive the KdV equation one can study the evolution of moments, of the transfer function and finally of the potential. The details can be found in On completely integrable polynomial PDEs arising from Sturm-Liouville differential equation using evolutionary vessels. KdV Hierarchy (Arxiv).

Theorem: The moments satisfy the following evolutionary formula

(DHt) δ/δt H_n (x, t) = i δ/δx H_n+1 (x, t) + i δ/δx [H₀ (x, t)] σ₁ H_n (x, t)

Proof: using differential equations of the vessel and algebraic equations to combine/cancel terms. □

Corollary: The transfer functions satisfies the following differential equation

(DSt) δ/δt S (λ, x, t) = i λ δ/δx S(λ, x, t) + i δ/δx [H₀ (x, t)] σ₁ S(λ, x, t)

Proof: immediate from (SMomnts) and (DHt). □

Corollary: the matrix-function satisfies

(DGt) δ/δt γ_*(x, t) = - i γ_*(x, t) δ/δx [H₀ (x, t)] σ₁+ i σ₁ δ²/δx² [H₀ (x, t)] σ₁ + i σ₁ δ/δx [H₀ (x, t)] γ_*(x, t)

Proof: immediate immediate from the considering the coefficient of λ^-1 at the expression

σ₂ δ/δt S (λ, x, t) - δ/δt S (λ, x, t) σ₁^-1 σ₂

using (Linkage) and then using (DSt). □

Proof (of Theorem (NLS equation)): Plugging here the formula for γ_*(x, t) as a matrix, depending on β (x, t) and considering the 12 entry we will obtain the evolutionary NLS equation(eNLS) for β (x, t). □