General higher—order breathers and rogue waves in the two‐component long‐wave–short‐wave resonance interaction model

Rao, J. V.; Malomed, Boris A.; Mihalache, Dumitru; He, Jingsong

doi:10.1111/sapm.12523

“…Many other integrable systems possess such rogue waves, such as the coupled Hirota equations 75 and the two-component longwave-short-wave resonant interaction system. 76 Thus, rogue patterns we reported in this article will arise in all such systems and are universal as well.…”

Section: Discussionmentioning

confidence: 58%

“…In the Darboux transformation framework, rogue waves in the form of determinants of Schur polynomials with index jumps of 3 would arise when the underlying scattering matrix admits a triple eigenvalue. Many other integrable systems possess such rogue waves, such as the coupled Hirota equations 75 and the two‐component long‐wave‐short‐wave resonant interaction system 76 . Thus, rogue patterns we reported in this article will arise in all such systems and are universal as well.…”

Section: Discussionmentioning

confidence: 61%

Rogue wave patterns associated with Okamoto polynomial hierarchies

Yang

¹

,

Yang

²

2023

Stud Appl Math

4

0

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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‐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.

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“…In the Darboux transformation framework, it means that these rogue waves should come from the underlying scattering matrix having a triple eigenvalue. Many other integrable systems admit such rogue waves, such as the coupled Hirota equations [72] and the two-component long-wave-short-wave resonant interaction system [73]. Thus, rogue patterns we reported in this article will arise in all such systems and are universal as well.…”

Section: Discussionmentioning

confidence: 64%

Rogue wave patterns associated with Okamoto polynomial hierarchies

Yang¹,

Yang²

2022

Preprint

0

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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...

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