The large-time development of the solution to an initial-value problem for the Korteweg–de Vries equation: III. Pure soliton solutions

In this paper, we consider an initial-value problem for the Korteweg–de Vries equation. The normalized Korteweg–de Vries equation considered is given by uτ+uux+uxxx=0,−∞0, where xx and ττ represent dimensionless distance and time respectively. In particular, we consider the case when the initial data is given by u(x,0)=6N(N+1)k2sech2(kx), where k>0k>0 and NN is a positive integer. The method of matched asymptotic coordinate expansions is used to obtain the large-ττ asymptotic structure of the solution to this problem, which exhibits the formation of a NN soliton solution structure in x>0x>0. Further, this solution is a pure soliton solution with no propagating oscillatory behavior in x<0x<0. For N>1N>1 we determine the correction to the propagation speed of each of the NN solitons as τ→∞τ→∞ and the rate of convergence to each of the NN solitons as τ→∞τ→∞.