Regular KdV vessels

We will see that construction of a regular KdV vessel gives rise to a solution of the Korteweg-de-Vries (KdV) equation. The results presented here can be found in Solution of the KdVequation using evolutionary vessels (Arxiv).

Definition: SL (Sturm Liouville) vessel parameters are defined as follows

σ₁ is

The following theorem explains why we call such vessels as KdV.

Theorem (KdV equation): the function q(x, t) = -2[ln τ(x, t)]_{xx} satisfies the KdV equation

(KdV) q_t = - 3/2 q q_x + 1/4 q_{xxx}

Proof: “Brute force” can be found here. □

Standard construction of a KdV 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).

Examples of solitons

Constructed from symmetric vessels are as follows: two dimensional case, three dimensional (rational) solution case.

Transfer function

Definition: Transfer function of the vessel V_KdV 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 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). □

Finally, some extra effort required to show that the equation (DGt) and (KdV) are equivalent. It is quite simple and done in the article, cited above.

Theorem: The formulas (DGt) and (KdV) are equivalent.

Proof: Using the formula for γ_*(x, t) via tau function from above. Some extra work is required to understand the structure of the first moment H₀ (x, t). It turns out that the moment equations (DH), presented in regular vessels impose conditions on this moment so that its derivative is actually also expressed via the tau function. Plugging all these formulas we will obtain the equivalence. □

Proof2 (of Theorem (KdV equation)): the last Theorem actually gives an alternative proof this very important theorem. □

Standard model for the continuous spectrum on a bounded/symmetric curve

Standard model for the discrete spectrum (on the imaginary axis)

Let H be the space l² = {s_n}. On this space define A = diag(i λ_n²)_, for an arbitrary fixed and bounded sequence of real numbers {λ_n} or more precisely A {s_n} = { i λ_n² s_n}. In order to define the operator B(x) we fix an l² sequence of non-zero entries {b_n} and define:

B(x) = diag(b_n) [cos (λ_n x - λ_n³ t) -i λ_n sin (λ_n x - λ_n³ t)]

It is an infinite sequence of rows of two entries. Notice that B*(0) σ₁ B(0) = 0 and it is necessary to choose X(0) so that (Lyapunov) holds or that A X(0) + X(0) A* = 0. Since A is diagonal, we choose X(0) = diag(x_n), where the sequence {x_n} is bounded from above AND from below (because X(0) must be invertible operator). Define next

┌ b_n b_m*[cos (λ_n x - λ_n³ t) λ_m sin (λ_m x – λ_m³ t) - λ_n sin (λ_n x - λ_n³ t) cos (λ_m x – λ_m³ t)]/(λ_n² – λ_m²) if n≠m

X(x) = │

└ |b_n|² (2 λ_nx-4 λ_n³ t + sin (2(λ_n x - λ_n³ t)))/(4 λ_n) if n=m

Theorem. The collection

V_KdV = (A, B(x, t), X(x, t); σ₁, σ₂, γ, γ_*(x, t); H, C^p; I),

For the operators defined above is a regular vessel.

Proof: Equations (DB), (Lyapunov), (DX), (DBt), (DXt) follow immediately. Equation (Linkage) serves to define γ_*(x, t) □

The transfer function of this vessel has poles at λ = i λ_n² and at the closure of this set. This follows immediately from the realization formula (Realize).