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Register a new userRogue waves on the double-periodic background in the focusing nonlinear Schrodinger equation
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The double-periodic solutions of the focusing nonlinear Schrodinger equation have been previously obtained by the method of separation of variables. We construct these solutions by using an algebraic method with two eigenvalues. Furthermore, we characterize the Lax spectrum for the double-periodic solutions and analyze rogue waves arising on their background. Magnification of the rogue waves is studied numerically.
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The extreme events are investigated for an $n$-component nonlinear Schrodinger ($n$-NLS) system in the focusing Kerr-like nonlinear media, which appears in many physical fields. We report and discuss the novel multi-parametric families of vector rational rogue wave (RW) solutions featuring the parity-time (PT) symmetry, which are characterized by non-identical boundary conditions for the components, and consistent with the degeneracy of $n$ branches of Benjamin-Feir instability. Explicit examples of PT-symmetric vector RWs are presented. Some parameter constraints can make some components generate the RWs with high amplitudes due to many-body resonant interactions.Effect of a non-integrable deformation of the model on the excitation of vector RWs is also discussed. These results will be useful to design the RW experiments in multi-component physical systems.
We apply the method of nonlinear steepest descent to compute the long-time asymptotics of the Korteweg-de Vries equation with steplike initial data leading to a rarefaction wave. In addition to the leading asymptotic we also compute the next term in the asymptotic expansion of the rarefaction wave, which was not known before.
We propose a novel, analytically tractable, scenario of the rogue wave formation in the framework of the small-dispersion focusing nonlinear Schrodinger (NLS) equation with the initial condition in the form of a rectangular barrier (a box). We use the Whitham modulation theory combined with the nonlinear steepest descent for the semi-classical inverse scattering transform, to describe the evolution and interaction of two counter-propagating nonlinear wave trains --- the dispersive dam break flows --- generated in the NLS box problem. We show that the interaction dynamics results in the emergence of modulated large-amplitude quasi-periodic breather lattices whose amplitude profiles are closely approximated by the Akhmediev and Peregrine breathers within certain space-time domain. Our semi-classical analytical results are shown to be in excellent agreement with the results of direct numerical simulations of the small-dispersion focusing NLS equation.
We study the spectral (linear) stability and orbital (nonlinear) stability of the elliptic solutions for the focusing modified Korteweg-de Vries (mKdV) equation with respect to subharmonic perturbations and construct the corresponding breather solutions to exhibit the unstable or stable dynamic behavior. The elliptic function solutions of mKdV equation and the fundamental solutions of Lax pair are exactly represented by using the theta function. Based on the `modified squared wavefunction (MSW) method, we construct all linear independent solutions of the linearized KdV equation, and then provide a necessary and sufficient condition of the spectral stability for the elliptic function solutions with respect to subharmonic perturbations. In the case of spectrum stable, the orbital stability of the elliptic function solutions with respect to subharmonic perturbations is established under a suitable Hilbert space. Using Darboux-Backlund transformation, we construct the breather solutions to exhibit the unstable or stable dynamic behavior. Through analyzing the asymptotical behavior, we find the breather solution under the $mathrm{cn}$-background is equivalent to the elliptic function solution adding a small perturbation as $ttopminfty$.
A prototypical example of a rogue wave structure in a two-dimensional model is presented in the context of the Davey-Stewartson~II (DS~II) equation arising in water waves. The analytical methodology involves a Taylor expansion of an eigenfunctionof the models Lax pair which is used to form a hierarchy of infinitely many new eigenfunctions. These are used for the construction of two-dimensional (2D) rogue waves (RWs) of the DS~II equation by the even-fold Darboux transformation (DT). The obtained 2D RWs, which are localized in both space and time, can be viewed as a 2D analogue of the Peregrine soliton and are thus natural candidates to describe oceanic RW phenomena,as well as ones in 2D fluid systems and water tanks.
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