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Abstract Up to the third-order rogue wave solutions of the Sasa–Satsuma (SS) equation are derived based on the Hirota’s bilinear method and Kadomtsev–Petviashvili hierarchy reduction method. They are expressed explicitly by rational functions with both the numerator and denominator being the determinants of even order. Four types of intrinsic structures are recognized according to the number of zero-amplitude points. The first- and second-order rogue wave solutions agree with the solutions obtained so far by the Darboux transformation. In spite of the very complicated solution form compared with the ones of many other integrable equations, the third-order rogue waves exhibit two configurations: either a triangle or a distorted pentagon. Both the types and configurations of the third-order rogue waves are determined by different choices of free parameters. As the nonlinear Schrödinger equation is a limiting case of the SS equation, it is shown that the degeneration of the first-order rogue wave of the SS equation converges to the Peregrine soliton.
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General rogue waves in the three-wave resonant interaction systems
Abstract General rogue waves in (1+1)-dimensional three-wave resonant interaction systems are derived by the bilinear method. These solutions are divided into three families, which correspond to a simple root, two simple roots and a double root of a certain quartic equation arising from the dimension reduction, respectively. It is shown that while the first family of solutions associated with a simple root exists for all signs of the nonlinear coefficients in the three-wave interaction equations, the other two families of solutions associated with two simple roots and a double root can only exist in the so-called soliton-exchange case, where the nonlinear coefficients have certain signs. Many of these rogue wave solutions, such as those associated with two simple roots, the ones generated by a $$2\times 2$$ block determinant in the double-root case, and higher-order solutions associated with a simple root, are new solutions which have not been reported before. Technically, our bilinear derivation of rogue waves for the double-root case is achieved by a generalization to the previous dimension reduction procedure in the bilinear method, and this generalized procedure allows us to treat roots of arbitrary multiplicities. Dynamics of the derived rogue waves is also examined, and new rogue wave patterns are presented. Connection between these bilinear rogue waves and those derived earlier by Darboux transformation is also explained.
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- Award ID(s):
- 1910282
- PAR ID:
- 10253053
- Date Published:
- Journal Name:
- IMA Journal of Applied Mathematics
- Volume:
- 86
- Issue:
- 2
- ISSN:
- 0272-4960
- Page Range / eLocation ID:
- 378 to 425
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1093/imamat/hxab005
