Neumann problem for the Korteweg–de Vries equation

We consider Neumann initial-boundary value problem for the Korteweg–de Vries equation on a half-lineequation(0.1){ut+λuux+uxxx=0,t>0,x>0,u(x,0)=u0(x),x>0,ux(0,t)=0,t>0. We prove that if the initial data u0∈H10,214∩H21,72 and the norm ‖u0‖H10,214+‖u0‖H21,72⩽ε, where ε>0ε>0 is small enough Hps,k={f∈L2;‖f‖Hps,k=‖〈x〉k〈i∂x〉sf‖Lp<∞}, 〈x〉=1+x2 and λ∫0∞xu0(x)dx=λθ<0. Then there exists a unique solution u∈C([0,∞),H21,72)∩L2(0,∞;H22,3) of the initial-boundary value problem (0.1). Moreover there exists a constant C such that the solution has the following asymptoticsu(x,t)=Cθ(1+ηlogt)−1t−23Ai′(xt3)+O(ε2t−23(1+ηlogt)−65) for t→∞t→∞ uniformly with respect to x>0x>0, where η=−9θλ∫0∞Ai′2(z)dz and Ai(q)Ai(q) is the Airy functionAi(q)=12πi∫−i∞i∞e−z3+zqdz=1πRe∫0∞e−iξ3+iξqdξ.