Rogue wave modes for a derivative nonlinear Schrödinger model

Rogue waves in fluid dynamics and optical waveguides are unexpectedly large displacements from a background state, and occur in the nonlinear Schrödinger equation with positive linear dispersion in the regime of positive cubic nonlinearity. Rogue waves of a derivative nonlinear Schrödinger equation are calculated in this work as a long-wave limit of a breather (a pulsating mode), and can occur in the regime of negative cubic nonlinearity if a sufficiently strong self-steepening nonlinearity is also present. This critical magnitude is shown to be precisely the threshold for the onset of modulation instabilities of the background plane wave, providing a strong piece of evidence regarding the connection between a rogue wave and modulation instability. The maximum amplitude of the rogue wave is three times that of the background plane wave, a result identical to that of the Peregrine breather in the classical nonlinear Schrödinger equation model. This amplification ratio and the resulting spectral broadening arising from modulation instability correlate with recent experimental results of water waves. Numerical simulations in the regime of marginal stability are described.

Figure 1

Plot of the amplitude of the rogue wave mode, |$A$| of Eqs. (12) and (13), versus $x$ and $t$ for σ = 1, γ = 4, and ρ₀ = 1: (a) three-dimensional view and (b) two-dimensional contour plot. In (b) the top and bottom ovals represent displacement equal to 0.5, the middle oval represents displacement equal to 2, and the two hyperbolic curves represent displacement equal to 1.

Figure 2

Plot of the amplitude of the rogue wave mode, |$A$| of Eqs. (12) and (13), versus $x$ and $t$ for σ = 3, γ = 4, and ρ₀ = 1: (a) three-dimensional view and (b) two-dimensional contour plot. In (b) the top and bottom ovals represent displacement equal to 0.5, the middle oval represents displacement equal to 2, and the two hyperbolic curves represent displacement equal to 1.

Figure 3

Plot of the amplitude of the rogue wave mode, |$A$| of Eqs. (12) and (13), versus $x$ and $t$ for σ = 1, γ = 6, and ρ₀ = 1: (a) three-dimensional view and (b) two-dimensional contour plot. In (b) the top and bottom ovals represent displacement equal to 0.5, the middle oval represents displacement equal to 2, and the two hyperbolic curves represent displacement equal to 1.

Figure 4

Growth phase of a rogue wave mode in a parameter regime with finite modulation instability for σ = ρ₀ = 1, γ = 4, and Eq. (16): (a) plot of the exact solution (12) and (13) and (b) numerical simulation of Eq. (2) with an exact solution perturbed by a 10% noise as the initial condition.

Figure 5

Growth phase of a rogue wave mode in a parameter regime with marginal modulation instability for σ = ρ₀ = 1, γ = 2.1, and Eq. (16): (a) plot of an exact solution (12) and (13) and (b) numerical simulation of Eq. (2) with an exact solution perturbed by a 10% noise as the initial condition.