A connection between the maximum displacements of rogue waves and the dynamics of poles in the complex plane

Liu, T. Y.; Chiu, T. L.; Clarkson, Peter A.; Chow, K. W.

doi:10.1063/1.5001007

Cited by 9 publications

(5 citation statements)

References 24 publications

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“…To substantiate our earlier works [8,10], we study further examples of Schrödinger equations here, namely, those with coherent coupling [14,16] and quintic (fifth-order) nonlinearities [15]. The conjecture is again verified for the first and second-order rogue waves of these models.…”

Section: Discussionsupporting

confidence: 61%

“…We proposed a conjecture (Section 2) which will hopefully provide an important step in this direction [8][9][10]. If the transverse variable of the Schrödinger equation is allowed to be complex, the real parts of the pole trajectories at the turning points will be identical to or will closely correlate with the locations of maximum heights of the rogue wave in physical space.…”

Section: Discussionmentioning

confidence: 99%

“…The parameters ρ 1 and ρ 2 can be obtained from the derivations outlined in earlier references [15]. They may also be readily determined by examining the far field condition [|x|, |t| → ∞] of Equations (10) and (11), i.e., they are the roots of the algebraic equation:…”

Section: A Schrödinger Equation With Quintic Non-linearitymentioning

confidence: 99%

“…Experimentally, the occurrence of rogue waves is realized through wave channels in hydrodynamics and fiber laser setting in optics [7]. Our goal is to provide still another perspective, namely, utilizing the dynamics of pole trajectories in the complex plane to elucidate the properties of rogue waves [8][9][10].…”

Section: Introductionmentioning

confidence: 99%

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The Dynamics of Pole Trajectories in the Complex Plane and Peregrine Solitons for Higher-Order Nonlinear Schrödinger Equations: Coherent Coupling and Quintic Nonlinearity

2020

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The Peregrine soliton is an exact, rational, and localized solution of the nonlinear Schrödinger equation and is commonly employed as a model for rogue waves in physical sciences. If the transverse variable is allowed to be complex by analytic continuation while the propagation variable remains real, the poles of the Peregrine soliton travel down and up the imaginary axis in the complex plane. At the turning point of the pole trajectory, the real part of the complex variable coincides with the location of maximum height of the rogue wave in physical space. This feature is conjectured to hold for at least a few other members of the hierarchy of Schrödinger equations. In particular, evolution systems with coherent coupling or quintic (fifth-order) nonlinearity will be studied. Analytical and numerical results confirm the validity of this conjecture for the first-and second-order rogue waves.

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Section: Discussionsupporting

confidence: 61%

Section: Discussionmentioning

confidence: 99%

Section: A Schrödinger Equation With Quintic Non-linearitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The Dynamics of Pole Trajectories in the Complex Plane and Peregrine Solitons for Higher-Order Nonlinear Schrödinger Equations: Coherent Coupling and Quintic Nonlinearity

2020

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“…To gain further insight, we shall use pole analysis [51] in the complex plane to study the locations of maximum displacements of these transient pulses. The underlying conjecture is that the maximum displacement of these rogue modes in the physical plane will coincide with the turning points of the trajectories in the complex plane, if the spatial variable x is allowed to be complex (while time t remains real).…”

Section: Second-order Rational Soliton Solutionmentioning

confidence: 99%

Rational solitons for non-local Hirota equations: Robustness and cascading instability

Pan

Yin²,

Chow³

2023

Front. Phys.

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The Hirota equation is a higher-order non-linear Schrödinger equation by incorporating third-order dispersion. Two pairs of non-local Hirota equations are studied. One is a parity transformed conjugate pair, and the other is a conjugate PT-symmetric pair. For the first pair, rational solitons are derived by the Darboux transformation, and are shown computationally to exhibit robust propagation properties. These rational solitons can exhibit both elastic and inelastic interactions. One particular case of an elastic collision between dark and “anti-dark” solitons is demonstrated. For the second pair, a “cascading mechanism” illustrating the growth of higher-order sidebands is elucidated explicitly for these non-local, conjugate PT-symmetric equations. These mechanisms provide a theoretical confirmation of the initial amplification phase of the growth-and-decay cycles of breathers. Such repeated patterns will serve as a manifestation of the classical Fermi-Pasta-Ulam-Tsingou recurrence.

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Dynamics of Complex Singularities of Nonlinear PDEs

Weideman

2022

SEMA SIMAI Springer Series

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Solutions to nonlinear evolution equations exhibit a wide range of interesting phenomena such as shocks, solitons, recurrence, and blow-up. As an aid to understanding some of these features, the solutions can be viewed as analytic functions of a complex space variable. The dynamics of poles and branch point singularities in the complex plane can often be associated with the aforementioned features of the solution. Some of the computational and analytical results in this area are surveyed here. This includes a first attempt at computing the poles in the famous Zabusky–Kruskal experiment that lead to the discovery of the soliton.

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A connection between the maximum displacements of rogue waves and the dynamics of poles in the complex plane

Abstract: . (2017) A connection between the maximum displacements of rogue waves and the dynamics of poles in the complex plane.Chaos: An Interdisciplinary Journal of Nonlinear Science, 27 (9). 091103. ISSN 1054-1500. DOIhttps://doi.org/10.1063/1.5001007 Link to record in KAR

Cited by 9 publications

References 24 publications

The Dynamics of Pole Trajectories in the Complex Plane and Peregrine Solitons for Higher-Order Nonlinear Schrödinger Equations: Coherent Coupling and Quintic Nonlinearity

The Dynamics of Pole Trajectories in the Complex Plane and Peregrine Solitons for Higher-Order Nonlinear Schrödinger Equations: Coherent Coupling and Quintic Nonlinearity

Rational solitons for non-local Hirota equations: Robustness and cascading instability

Dynamics of Complex Singularities of Nonlinear PDEs

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