On certain properties of the compact Zakharov equation
AbstractLong-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 $\mu $ slowly evolves according to a nonlinear Schrodinger equation (NLS). In particular, for small carrier wave steepness $\mu <\mu _{1}\approx 0.27$ 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 $\mu $ increases while the Benjamin–Feir index (BFI) decreases and becomes nil at $\mu _{1}$. This indicates that homoclinic orbits persist only for small values of wave steepness $\mu \ll \mu _{1}$, 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 $\mu >\mu _{1}$, 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 $\mu =\mu _{c}\approx 0.577$, 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 $\mu >\mu _{c}$ 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.