Abstract
Abstract. We construct a multi-parametric family of quasi-rational solutions to the focusing NLS equation, presenting a profile of multiple rogue waves. These solutions have also been used by us to construct a large family of smooth, real localized rational solutions of the KP-I equation quite different from the multi-lumps solutions first constructed in Bordag et al. (1977). The physical relevance of both equations is very large. From the point of view of geosciences,the focusing NLS equation is relevant to the description of surface waves in deep water, and the KP-I equation occurs in the description of capillary gravitational waves on a liquid surface, but also when one considers magneto-acoustic waves in plasma (Zhdanov, 1984) etc. In addition, there are plenty of equations of physical importance, having their origin in fiber optics, hydrodynamics, plasma physics and many other areas, which are gauge equivalent to the NLS equation or to the KP-I equation. Therefore our results can be easily extended to a large number of systems of physical interest to be discussed in separate publications.
Highlights
IntroductionKharif et al (2009). Since the notion of rogue waves has appeared in several other fields such as nonlinear optics where a Peregrine breather has been observed recently for the first time (Kibler et al, 2010)
In oceanography a rogue wave is a unexpectedly high wave strongly localized in space-time recently, certain authors have speculated about “long-life” rogue waves
Even if testimonies about such freak phenomena have been available for a long time, the study of rogue waves has been booming for a couple of decades, following the first scientific recording of an appearance of a rogue wave in the ocean
Summary
Kharif et al (2009). Since the notion of rogue waves has appeared in several other fields such as nonlinear optics where a Peregrine breather has been observed recently for the first time (Kibler et al, 2010). We discuss an important class of solutions of (1), representing the rational 2n-parametric modulation of the plane wave solution of fixed amplitude B. This class of solutions was first constructed in 1986 in the article by Eleonskii et al (1986). We conjecture that for higher values of n “in general position” the number of these maxima is equal to n(n + 1)/2 This conjecture is supported by the tested solutions corresponding to n = 3 and n = 4. In some exceptional cases corresponding to the “higher order” Peregrine breathers constructed in (Akhmediev et al, 2009a,b,c, 2010), the number of local maximums for n = 2 is equal 5 and one of them is much higher than others. Their qualitative analysis is still far from being completed but already we have a reason to consider some of the related solutions as two-dimensional rogue waves
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