On certain properties of the compact Zakharov equation

Long-time evolution of a weakly perturbed wavetrain near the modulational instability (MI) threshold is examined within the framework of the compact Zakharov equation for unidirectional deep-water waves (Dyachenko and Zakharov, JETP Lett., vol. 93, 2011, pp. 701–705). Multiple-scale solutions reveal that a perturbation to a slightly unstable uniform wavetrain of steepness μ slowly evolves according to a nonlinear Schrodinger equation (NLS). In particular, for small carrier wave steepness μ <μ ₁≈ 0.27 the perturbation dynamics is of focusing type and the long-time behaviour is characterized by the Fermi–Pasta–Ulam recurrence, the signature of breather interactions. However, the amplitude of breathers and their likelihood of occurrence tend to diminish as μ increases while the Benjamin–Feir index (BFI) decreases and becomes nil at μ ₁ . This indicates that homoclinic orbits persist only for small values of wave steepness μ μ ₁ , in agreement with recent experimental and numerical observations of breathers. When the compact Zakharov equation is beyond its nominal range of validity, i.e. for μ >μ ₁ , predictions seem to foreshadow a dynamical trend to wave breaking. In particular, the perturbation dynamics becomes of defocusing type, and nonlinearities tend to stabilize a linearly unstable wavetrain as the Fermi–Pasta–Ulam recurrence is suppressed. At μ =μ c≈ 0.577 , subharmonic perturbations restabilize and superharmonic instability appears, possibly indicating that wave dynamical behaviour changes at large steepness, in qualitative agreement with the numerical simulations of Longuet-Higgins and Cokelet (Proc. R. Soc. Lond. A, vol. 364, 1978, pp. 1–28) for steep waves. Indeed, for μ >μ c a multiple-scale perturbation analysis reveals that a weak narrowband perturbation to a uniform wavetrain evolves in accord with a modified Korteweg–de Vries/Camassa–Holm type equation, again implying a possible mechanism conducive to wave breaking.