Inverse scattering up to smooth functions for the dirac-zs-akns system

We consider the Dirac-ZS-AKNS system ψ_1x = -ikψ₁ + q₁(x)ψ₂, ψ_2x = ikψ₂ + q₂ (x)ψ₁, (1) where q₁(x),q₂(x) ∈ W^n,1(ℝ)( the space of functions with n derivatives in L¹), n ∈ ℕ ∪ 0. (2) We consider for (1) the transition matrix T(k) = (a(k)/b(k) c(k)/d(k)), k ∈ ℝ, and, in addition, for the case of the Dirac system (i.e. for the selfadjoint case q₂(x) = q₁(x)̄)) the scattering matrix S(k)-(s₁₁(k)/s₂₁(k) s₁₂(k)/s₂₂(k)) = 1/-c(k) b(k)/1), k ∈ ℝ. We can divide main results of the present work into three parts. I. We show that the inverse scattering transform and the inverse Fourier transform give the same solution, up to smooth functions, of the inverse scattering problem for (1). More precisely, we show that, under condition (2) with n ∈ ℕ, the following formulas are valid: q₁(x) = 2č(2x) in W^n,1 (ℝ)/W^n+1,1(ℝ), (3) q₂(x) = 2b̌(-2x) in W^n,1 (ℝ)/W^n+1,1 (ℝ), and, in addition, for the case of the Dirac system q₁(x) = -2š2₁(2x) in W^n,1 (ℝ)/W^n+1,1 (ℝ), (4) q₂(x) = 2š₁₂(-2x) in W^n,1 (ℝ)/W^n+1,1 (ℝ), where φ̌(x) = (2π)^-1 ∫_ℝ e^-ikx φ(k)dk, W^n,1 (ℝ)/W^n+1,1 (ℝ) denotes the factor space. II. Using (3), (4), we give the characterization of the transition matrix and the scattering matrix for the case of the Dirac system under condition (2) with n ∈ ℕ. III. As applications of the results mentioned above, we show that 1) for any real-valued initial data θ(0, x) ∈ W^n,1 (ℝ), n ∈ ℕ, the Cauchy problem for the sh-Gordon equation θ_xt = shθ has a unique solution θ(t, x) such that θ_x(t, x) ∈ C([0, ∞[, W^n-1,1 (ℝ)) and θ(t, x) → 0 as |x| → ∞ for any t ≥ 0, 2) in addition, for n ∈ ℕ, for such a solution the following formula is valid: θ(t, x) = θ(0, x) in W_loc^n,1 (ℝ)/W_loc^n+1,1(ℝ), where W_loc^n,1 (ℝ) denotes the space of functions locally integrable with n derivatives. We give also a review of preceding results.