Abstract:
The focusing non-linear Schrödinger (NLS) equation is the simplest universal model describing the modulation instability (MI) of quasimonochromatic waves in weakly non-linear media, and MI is considered to be the main physical mechanism for the appearance of anomalous (rogue) waves (AWs) in nature. In this paper the finite-gap method is used to study the NLS Cauchy problem for generic periodic initial perturbations of the unstable background solution of the NLS equation (here called the Cauchy problem of AWs) in the case of a finite number N of unstable modes. It is shown how the finite-gap method adapts to this specific Cauchy problem through three basic simplifications enabling one to construct the solution, to leading and relevant order, in terms of elementary functions of the initial data. More precisely, it is shown that, to leading order, i) the initial data generate a partition of the time axis into a sequence of finite intervals, ii) in each interval I of the partition only a subset of N(I)⩽N unstable modes are ‘visible’, and iii) for t∈I the NLS solution is approximated by the N(I)-soliton solution of Akhmediev type describing for these ‘visible’ unstable modes a non-linear interaction with parameters also expressed in terms of the initial data through elementary functions. This result explains the relevance of the m-soliton solutions of Akhmediev type with m⩽N in the generic periodic Cauchy problem of AWs in the case of a finite number N of unstable modes.
Bibliography: 118 titles.
Keywords:
focusing non-linear Schrödinger equation, periodic Cauchy problem for anomalous waves, asymptotics in terms of elementary functions, finite-gap approximation, Riemann surfaces close to degenerate ones.
The work of the first author was supported by the Russian Science Foundation, grant 18-11-00316. The second author was partially supported by the University “La Sapienza”, grant 2017.
Citation:
P. G. Grinevich, P. M. Santini, “The finite-gap method and the periodic NLS Cauchy problem of anomalous waves for a finite number of unstable modes”, Russian Math. Surveys, 74:2 (2019), 211–263
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\paper The finite-gap method and the periodic NLS Cauchy problem of anomalous waves for a finite number of unstable modes
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\pages 211--263
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This publication is cited in the following 27 articles:
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Liming Ling, Xuan Sun, “Elliptic-rogue waves and modulational instability in nonlinear soliton equations”, Phys. Rev. E, 109:6 (2024)
Yu-Chen Lee, Markus Brühl, Dong-Jiing Doong, Sander Wahls, Boris Malomed, “Nonlinear Fourier classification of 663 rogue waves measured in the Philippine Sea”, PLoS ONE, 19:5 (2024), e0301709
P. G. Grinevich, “Riemann Surfaces Close to Degenerate Ones in the Theory of Rogue Waves”, Proc. Steklov Inst. Math., 325 (2024), 86–110
B. Prinari, “Inverse scattering transform for nonlinear Schrödinger systems on a nontrivial background: A survey of classical results, new developments and future directions”, J. Nonlinear Math. Phys., 30:2 (2023), 317
N. Sinthuja, S. Rajasekar, M. Senthilvelan, “Instability of single- and double-periodic waves in the fourth-order nonlinear Schrödinger equation”, Nonlinear Dyn., 111:17 (2023), 16497
P. G. Grinevich, P. M. Santini, “The finite-gap method and the periodic Cauchy problem for $(2+1)$-dimensional anomalous waves for the focusing Davey–Stewartson $2$ equation”, Russian Math. Surveys, 77:6 (2022), 1029–1059
F. Coppini, P. G. Grinevich, P. M. Santini, “Periodic rogue waves and perturbation theory”, Perturbation Theory, Encyclopedia of Complexity and Systems Science Series, Springer, Berlin–Heidelberg, 2022, 565–584
F. Coppini, P. G. Grinevich, P. M. Santini, “Periodic rogue waves and perturbation theory”, Encyclopedia of Complexity and Systems Science, Springer, Berlin–Heidelberg, 2022, 1–22
P. G. Grinevich, P. M. Santini, “The linear and nonlinear instability of the Akhmediev breather”, Nonlinearity, 34:12 (2021), 8331–8358
D. E. Pelinovsky, “Instability of double-periodic waves in the nonlinear Schrödinger equation”, Front. Physics, 9 (2021), 599146
F. Coppini, P. G. Grinevich, P. M. Santini, Encyclopedia of Complexity and Systems Science, 2021, 1
Sebastian Klein, Martin Kilian, “On Closed Finite Gap Curves in Spaceforms I”, SIGMA, 16 (2020), 011, 29 pp.
A. Tikan, “Effect of local peregrine soliton emergence on statistics of random waves in the one-dimensional focusing nonlinear Schrodinger equation”, Phys. Rev. E, 101:1 (2020), 012209
F. Coppini, P. G. Grinevich, P. M. Santini, “Effect of a small loss or gain in the periodic nonlinear Schrodinger anomalous wave dynamics”, Phys. Rev. E, 101:3 (2020), 032204
A. N. Kulikov, D. A. Kulikov, “Odnofazovye i dvukhfazovye resheniya fokusiruyuschego nelineinogo uravneniya Shredingera”, Vestnik TvGU. Seriya: Prikladnaya matematika, 2020, no. 2, 18–34
J. Chen, R. Zhang, “The complex Hamiltonian systems and quasi-periodic solutions in the derivative nonlinear Schrödinger equations”, Stud. Appl. Math., 145:2 (2020), 153–178
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