The decay of the continuous spectrum for solutions of the Korteweg-deVries equation

Abstract

The asymptotic behavior of the solution u(x, t) of the Korteweg‐deVries equation u_t + uu_x + u_xxx = 0 is investigated for the class of problems where the initial data does not give rise to an associated discrete spectrum. It is shown that the behavior is different in the three regions (i) ${\mathrm{x\gg t}}^{\text{1/3}}$ , (ii) x = 0(t^1/3), (iii) ${\mathrm{x\ll -\hspace{0.17em}t}}^{\text{1/3}}$ . Asymptotic solutions in each of these regions are found which match at their respective boundaries. One of the Berezin‐Karpman similarity solutions is the asymptotic state in (ii) and u(x, t) decays like 1/t^2/3 in this region. For region (i) there is exponential decay, whereas in region (iii) the structure of u(x, t) is highly oscillatory and the amplitudes of the oscillations decay as 1/t^1/2 when − x/t is of order unity. For x/t large and negative these amplitudes decay at least as 1/(− x/t)^1/4t^1/2.