General rogue wave solutions to the Sasa-Satsuma equation

Wu, Chengfa; Zhang, Guanxiong; Changyan, Shi; Feng, Bao-Feng

doi:10.48550/arxiv.2206.02210

“…Soliton solutions on the zero background in this equation were derived by Sasa and Satsuma in their original paper [9]. Later, rational solutions on a constant background, including rogue waves, were also derived [8,[11][12][13][14][15][16][17][18][19]. The solutions that will be the starting point of this paper are a certain class of rational solutions which, in the language of Darboux transformation, are associated with a scattering matrix admitting a triple eigenvalue.…”

Section: A a Class Of Rational Solutionsmentioning

confidence: 99%

“…It has been shown in [17] that the Sasa-Satsuma equation (1) under boundary conditions (2) admits the following solutions…”

Section: Appendixmentioning

confidence: 99%

“…where C is some constant. The above result can be made even more general by allowing each of p and q to take different values in different blocks of the determinant (A.66) [17,21]. But that generalization is not necessary for our purpose.…”

Section: Appendixmentioning

confidence: 99%

“…Different ways to satisfy the dimension reduction condition (A.73) will lead to different types of solutions to the Sasa-Satsuma equation. One type of such solutions -rogue waves, were derived in [17]. To derive rational solutions in Theorem 1, a different dimension reduction is needed.…”

Section: Appendixmentioning

confidence: 99%

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Partial-Rogue Waves that Come from Nowhere But Leave with a Trace in the Sasa-Satsuma Equation

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“…Since roots of the Yablonskii-Vorob'ev polynomial hierarchy on the complex plane come in shapes such as triangles, pentagons and heptagons, previous numerical reports on this subject can then be analytically explained. Further significant progress in this direction was made in [54], where NLS rogue patterns associated with the Yablonskii-Vorob'ev hierarchy were shown to be universal in integrable systems, as long as rogue wave solutions of the integrable systems can be expressed by τ functions whose matrix elements are Schur polynomials with index jumps of two, as in the generalized derivative NLS equations, the Boussinesq equation, the Manakov system, and many others [54][55][56][57].A natural next question is, are there other shapes of rogue wave patterns in integrable systems? If so, what special polynomials would be associated with such rogue patterns?In this article, we show that there are indeed other shapes of rogue patterns in integrable systems.…”

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Rogue wave patterns associated with Okamoto polynomial hierarchies

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Yang²

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Preprint

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We show that new types of rogue wave patterns exist in integrable systems, and these rogue patterns are described by root structures of Okamoto polynomial hierarchies. These rogue patterns arise when the τ functions of rogue wave solutions are determinants of Schur polynomials with index jumps of three, and an internal free parameter in these rogue waves gets large. We demonstrate these new rogue patterns in the Manakov system and the three-wave resonant interaction system. For each system, we derive asymptotic predictions of its rogue patterns under a large internal parameter through Okamoto polynomial hierarchies. Unlike the previously reported rogue patterns associated with the Yablonskii-Vorob'ev hierarchy, a new feature in the present rogue patterns is that, the mapping from the root structure of Okamoto-hierarchy polynomials to the shape of the rogue pattern is linear only to the leading order, but becomes nonlinear to the next order. As a consequence, the current rogue patterns are often deformed, sometimes strongly deformed, from Okamoto root structures, unless the underlying free parameter is very large. Our analytical predictions of rogue patterns are compared to true solutions, and excellent agreement is observed, even when rogue patterns are strongly deformed from Okamoto root structures. I. INTRODUCTIONRogue waves are large and spontaneous nonlinear wave excitations that "come from nowhere and disappear with no trace" [1]. Such waves were first studied in oceanography, since they posed a threat even to large ships [2,3]. Later, these waves were also investigated in optics and other physical fields due to their peculiar nature [4,5]. From a theoretical point of view, an important fact is that, many integrable nonlinear wave equations admit explicit rational solutions that correspond to rogue waves. This fact was first reported by Peregrine [6], who presented a simple rogue wave solution for the nonlinear Schrödinger (NLS) equation. Peregrine's solution was later generalized, and more intricate rogue wave solutions in the NLS equation were derived [7][8][9][10][11]. Since the NLS equation governs nonlinear wave packet evolution in a wide range of physical systems [12,13], these theoretical rogue wave solutions of the NLS equation then motivated a lot of rogue-wave experiments, ranging from water waves to optical waves to acoustic waves [14][15][16][17][18][19][20]. These combined theoretical and experimental studies significantly deepened our understanding of physical rogue wave events. Due to this success, rogue wave solutions have also been derived in many other physical integrable equations, such as the derivative NLS equations for circularly polarized nonlinear Alfvén waves in plasmas and shortpulse propagation in a frequency-doubling crystal [21-26], the Manakov equations for light transmission in randomly birefringent fibers and interaction between two incoherent light beams in crystals [27-34], the three-wave resonant interaction equations [13,[35][36][37][38][39], and many others. Such theoreti...

show abstract

Rogue wave patterns associated with Okamoto polynomial hierarchies

Yang

¹

,

Yang

²

2023

Stud Appl Math

4

0

View full text Add to dashboard Cite

We show that new types of rogue wave patterns exist in integrable systems, and these rogue patterns are described by root structures of Okamoto polynomial hierarchies. These rogue patterns arise when the τ functions of rogue wave solutions are determinants of Schur polynomials with index jumps of three, and an internal free parameter in these rogue waves gets large. We demonstrate these new rogue patterns in the Manakov system and the three‐wave resonant interaction system. For each system, we derive asymptotic predictions of its rogue patterns under a large internal parameter through Okamoto polynomial hierarchies. Unlike the previously reported rogue patterns associated with the Yablonskii–Vorob'ev hierarchy, a new feature in the present rogue patterns is that the mapping from the root structure of Okamoto‐hierarchy polynomials to the shape of the rogue pattern is linear only to the leading order, but becomes nonlinear to the next order. As a consequence, the current rogue patterns are often deformed, sometimes strongly deformed, from Okamoto‐hierarchy root structures, unless the underlying internal parameter is very large. Our analytical predictions of rogue patterns are compared to true solutions, and excellent agreement is observed, even when rogue patterns are strongly deformed from Okamoto‐hierarchy root structures.

show abstract

General rogue wave solutions to the Sasa-Satsuma equation

Cited by 3 publications

References 47 publications

Partial-Rogue Waves that Come from Nowhere But Leave with a Trace in the Sasa-Satsuma Equation

Partial-Rogue Waves that Come from Nowhere But Leave with a Trace in the Sasa-Satsuma Equation

Rogue wave patterns associated with Okamoto polynomial hierarchies

Rogue wave patterns associated with Okamoto polynomial hierarchies

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