Abstract

We show that the limit infimum, as time t goes to infinity, of any uniformly bounded in time H3/2+ ∩ L1 solution to the intermediate long wave (ILW) equation converges to zero locally in an increasing-in-time region of space of order t/log(t). Also, for solutions with a mild L1norm growth in time it is established that its lim inf converges to zero, as time goes to infinity. This confirms the nonexistence of breathers and other solutions for the ILW model moving with a speed “slower” than a soliton. We also prove that in the far field linearly dominated region, the L2-norm of the solution also converges to zero as time approaches infinity. In addition, we deduced several scenarios for which the initial value problem associated to the generalized Benjamin–Ono and the generalized ILW equations cannot possess time periodic solutions (breathers). Finally, as previously demonstrated in solutions of the KdV and Benjamin–Ono equations, we establish the following propagation of regularity result: if the datum u0 ∈ H3/2+(R) ∩ Hm((x0, ∞)), for some x0 ∈ R, m ∈ Z+, m ≥ 2, then the corresponding solution u(·, t) of the ILW equation belongs to Hm(β, ∞) for any t > 0 and β ∈ R.