ABSTRACT
Dynamics and properties of breathers for the modified Korteweg-de Vries equations with negative cubic nonlinearities are studied. While breathers and rogue waves are absent in a single component waveguide for the negative nonlinearity case, coupling can induce regimes of modulation instabilities. Such instabilities are correlated with the existence of rogue waves and breathers. Similar scenarios have been demonstrated previously for coupled systems of nonlinear Schrödinger and Hirota equations. Both real- and complex-valued modified Korteweg-de Vries equations will be treated, which are applicable to stratified fluids and optical waveguides, respectively. One special family of breathers for coupled, complex-valued equations is derived analytically. Robustness and stability of breathers are studied computationally. Knowledge of the growth rates of modulation instability of plane waves provides an instructive prelude on the robustness of breathers to deterministic perturbations. A theoretical formulation of the linear instability of breathers will involve differential equations with periodic coefficient, i.e., a Floquet analysis. Breathers associated with larger eigenvalues of the monodromy matrix tend to suffer greater instability and increased tendency of distortion. Predictions based on modulation instability and Floquet analysis show excellent agreements. The same trend is obtained for simulations conducted with random noise disturbances. Linear approaches like modulation instabilities and Floquet analysis, thus, generate a very illuminating picture of the nonlinear dynamics.
ABSTRACT
The nonlinear Schrödinger equation possesses doubly periodic solutions expressible in terms of the Jacobi elliptic functions. Such solutions can be realized through doubly periodic patterns observed in experiments in fluid mechanics and optics. Stability and robustness of these doubly periodic wave profiles in the focusing regime are studied computationally by using two approaches. First, linear stability is considered by Floquet theory. Growth will occur if the eigenvalues of the monodromy matrix are of a modulus larger than unity. This is verified by numerical simulations with input patterns of different periods. Initial patterns associated with larger eigenvalues will disintegrate faster due to instability. Second, formation of these doubly periodic patterns from a tranquil background is scrutinized. Doubly periodic profiles are generated by perturbing a continuous wave with one Fourier mode, with or without the additional presence of random noise. Effects of varying phase difference, perturbation amplitude, and randomness are studied. Varying the phase angle has a dramatic influence. Periodic patterns will only emerge if the perturbation amplitude is not too weak. The growth of higher-order harmonics, as well as the formation of breathers and repeating patterns, serve as a manifestation of the classical problem of Fermi-Pasta-Ulam-Tsingou recurrence.
ABSTRACT
Three-wave (triad) resonance in a uniformly stratified fluid is investigated as a case study of energy transfer among oscillatory modes. The existence of a degenerate triad is demonstrated explicitly, where two components have identical group velocity. An illuminating example is a resonance involving waves from modes 1, 3, 5 families, but many other combinations are possible. The physical applications and nonlinear dynamics of rogue waves derived analytically in the literature are examined. Exact solutions with four free parameters (two related to the amplitudes of the background plane waves, two related to the frequencies of slowly varying envelopes) describe motions localized in both space and time. The differences between rogue waves of the degenerate versus the nondegenerate cases are highlighted. The phase and profile of the degenerate case rogue waves are correlated. The volume or energy of the rogue wave (defined as the total extent or energy contents of the fluid set in motion for the duration of the rogue wave) may change drastically, if the wave envelope parameters vary. Pulsating modes (breathers) have been studied previously by layered-fluid and modified Korteweg-de Vries models. Here we extend the consideration to stratified fluids but for the simpler case of nondegenerate triads. Instabilities of fission and fusion of breathers are confirmed computationally with Floquet analysis. This knowledge should prove useful for energy transfer processes in the oceans.