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
σ2 is
γ is
Definition: Collection of operators and space
VKdV = (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 KDV 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 σ1 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
0
1
1
0
0
0
1
0
0
0
0
i
τ(x, t)_{xx}/ τ(x, t)
-τ(x, t)_x/ τ(x, t)
τ(x, t)_x/ τ(x, t)
i
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 B0,
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 VKdV 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) Hn (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 - ∑ Hn(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 Hn (x, t) = i δ/δx Hn+1 (x, t) + i δ/δx [H0 (x, t)] σ1 Hn (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 = {sn}. On this space define A = diag(i λn2), for an arbitrary fixed and bounded sequence of real numbers {λn} or more precisely A {sn} = { i λn2 sn}. In order to define the operator B(x) we fix an l2 sequence of non-zero entries {bn} and define:
B(x) = diag(bn) [cos (λn x - λn3 t) -i λn sin (λn x - λn3 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(xn), where the sequence {xn} is bounded from above AND from below (because X(0) must be invertible operator). Define next
┌ bn bm*[cos (λn x - λn3 t) λm sin (λm x – λm3 t) - λn sin (λn x - λn3 t) cos (λm x – λm3 t)]/(λn2 – λm2) if n≠m
X(x) = │
└ |bn|2 (2 λnx-4 λn3 t + sin (2(λn x - λn3 t)))/(4 λn) if n=m
Theorem. The collection
VKdV = (A, B(x, t), X(x, t); σ1, σ2, γ, γ*(x, t); H, Cp; 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 λn2 and at the closure of this set. This follows immediately from the realization formula (Realize).