Neumann problem for the Korteweg-de Vries equation

We consider Neumann initial-boundary value problem for the Korteweg-de Vries equation on a half-line{A formula is presented} We prove that if the initial data u₀ ∈ H₁ ^{0, frac(21,4)} ∩ H₂^{1, frac(7, 2)} and the norm {norm of matrix} u₀ {norm of matrix}_H¹0, _{frac(21, 4)} + {norm of matrix} u₀ {norm of matrix}_H²1, _{frac(7, 2)} {less-than or slanted equal to} ε, where ε > 0 is small enough {Mathematical expression}, 〈 x 〉 = sqrt(1 + x²) and λ ∫₀^∞ x u₀ ( x ) d x = λ θ < 0. Then there exists a unique solution u ∈ C ( [ 0, ∞ ), H₂^{1, frac(7, 2)} ) ∩ L² ( 0, ∞ ; H₂^{2, 3} ) of the initial-boundary value problem (0.1). Moreover there exists a constant C such that the solution has the following asymptotics{A formula is presented} for t → ∞ uniformly with respect to x > 0, where η = - 9 θ λ ∫₀^∞ A i^{′ 2} ( z ) d z and A i ( q ) is the Airy function{A formula is presented}.