Non-Symmetric Boussinesq Vessels

Most of the material presented in this text can be found here. Mathematica implementations of solitons see in Soliton examples below.

Definition: Collection of operators and spaces

V_Bouss = (C(x,t), A_ζ, X(x,t), A, B(x,t); σ₁, σ₂, γ; σ_1t, σ_2t, γ_t; γ_*(x,t); K, C³; Ω),

where K is a Krein space; C(x,t): K → C^3, A, A_ζ, X(x,t): K →K , B(x,t): C³ →K are linear operators, σ₁ = σ₁*, σ₂ = σ₂*, σ_1t = σ_1t *, σ_2t = σ_2t *, γ = - γ*, γ_t = -γ_t*, γ_* (x,t) = - (γ_*(x,t))*are 3x3 matrices, Ω is an open subspace of R² is called a non-symmetric vessel if the following conditions hold:

· (AReg) the operator A has a dense domain D (A) with a symmetric spectrum with respect to the imaginary axis. A is the generator of a C₀ semi-group on H, R(λ) = (λ I - A)^-1is the Resolvent.

· B(x,t) satisfies regularity assumptions (for all x in I and all λ out of the spectrum of A)

(Bσ₂) the image of R (λ) B(x,t) σ₂ is in D (A),

(Bγ) the image of R (λ) B(x,t) γ is in H.

Flow with respect to x:

1. B(x,t) also satisfies

(ResDB) 0 = δ/δx [R(λ) B(x,t)] σ₁ + A R(λ) B(x,t) σ₂ + R(λ) B(x,t) γ.

2. (DC) σ₁ δ/δx C(x,t) u = (γ C - σ₂ C A_ζ ) u,

where u is in D(A),

3. X(x,t) is a bounded, self-adjoint, invertible on some interval I and satisfies

(DX) δ/δx X(x,t) = B(x,t) σ₂ C(x,t).

Flow with respect to t:

4. (DBt) δ/δt B(x,t) σ_1t = - (A B(x,t) σ_2t + B γ_t),

5. (DCt) σ_1t δ/δt C(x,t) u = (γ_t C - σ_2t C A_ζ ) u ,

6. (DXt) δ/δt X (x,t) = B(x,t) σ_2t C(x,t).

Algebraic relations

7. The Lyapunov equation holds for all (x,t) in Ω:

(Lyapunov) A X(x,t) + X(x,t) A_ζ + B(x,t) σ₁ C(x,t) = 0.

8. γ_*(x,t) satisfies the linkage condition on Ω

(Linkage) γ_*(x,t) = γ+ σ₂ C(x,t) X^-1(x,t) B(x,t) σ₁- σ₁ C(x,t) X^-1(x,t) B(x,t) σ_2.

Theorem Suppose that V_Bouss is a Boussinesq vessel and τ(x,t) = det(X(x_0,t_0)^-1 X (x,t)) is its tau function, defined for an arbitrary point (x_0,t_0) in Ω.

Then the coefficient q(x,t)=-3/2 δ²/δx² ln τ (x,t) satisfies the Boussinesq equation

(Boussinesq) q_tt (x,t) = δ²/δx² [3q_xx(x,t)- 12 q²(x,t)]

on Ω.

Proof can be found here (see Theorem 17)

Soliton examples

The classical soliton

is implemented in Mathematica (software) here. Another example of a soliton

is presented here. Changing these two examples and using higher dimensional inner spaces K, one can obtain solitons of a more complicated nature.