Asymptotics for \(t\to\infty\) of solutions to the generalized Korteweg-de Vries equation (Q1363257)

We find the asymptotics for \(t\to\infty\) of solutions \(u(x,t)\) to the Cauchy problem for the generalized KdV equation \[ u_t+ {\partial \over\partial x} (u^n)+ {1\over 3} u_{xxx} =0, \quad u|_{t=0} =\overline u(x), \tag{1} \] with the integral power \(n>3\) of nonlinearity. Note that equation (1) is conservative, and the decay of its solutions in time is caused by dispersion effects on account of a redistribution of energy between higher harmonics, rather than by dissipative effects.