TY - JOUR

T1 - Effective dispersion in the focusing nonlinear Schrödinger equation

AU - Leisman, Katelyn Plaisier

AU - Zhou, Douglas

AU - Banks, J. W.

AU - Kovačič, Gregor

AU - Cai, David

N1 - Funding Information:
We are grateful to G. Biondini, K. Newhall, M. Schwartz, and V. Zharnitsky for helpful discussions. K.L. was partly supported by the U.S. Department of Education Graduate Assistance in Areas of National Need (GAANN), the NSF Research Training Group Grant No. DMS-0636358, and the NSF Research Training Group Grant No. DMS-1344962. G.K. was partly supported by the NSF Grant No. DMS-1615859. J.W.B. was partly supported by a U.S. Presidential Early Career Award for Scientists and Engineers. J.W.B. and G.K. were partly supported by a Simons foundation grant for collaboration on weak turbulence. D.Z. was partly supported by National Science Foundation of China with Grants No. 11671259, No. 11722107, and No. 91630208. D.C. was partly supported by National Science Foundation of China with Grant No. 31571071. D.Z. and D.C. were partly supported by NYU Abu Dhabi Institute Grant No. G1301 and the SJTU-UM Collaborative Research Program.
Publisher Copyright:
© 2019 American Physical Society.

PY - 2019/8/19

Y1 - 2019/8/19

N2 - For waves described by the focusing nonlinear Schrödinger equation (FNLS), we present an effective dispersion relation (EDR) that arises dynamically from the interplay between the linear dispersion and the nonlinearity. The form of this EDR is parabolic for a robust family of "generic" FNLS waves and equals the linear dispersion relation less twice the total wave action of the wave in question multiplied by the square of the nonlinearity parameter. We derive an approximate form of this EDR explicitly in the limit of small nonlinearity and confirm it using the wave-number-frequency spectral (WFS) analysis, a Fourier-transform based method used for determining dispersion relations of observed waves. We also show that it extends to the FNLS the universal EDR formula for the defocusing Majda-McLaughlin-Tabak (MMT) model of weak turbulence. In addition, unexpectedly, even for some spatially periodic versions of multisolitonlike waves, the EDR is still a downward shifted linear-dispersion parabola, but the shift does not have a clear relation to the total wave action. Using WFS analysis and heuristic derivations, we present examples of parabolic and nonparabolic EDRs for FNLS waves and also waves for which no EDR exists.

AB - For waves described by the focusing nonlinear Schrödinger equation (FNLS), we present an effective dispersion relation (EDR) that arises dynamically from the interplay between the linear dispersion and the nonlinearity. The form of this EDR is parabolic for a robust family of "generic" FNLS waves and equals the linear dispersion relation less twice the total wave action of the wave in question multiplied by the square of the nonlinearity parameter. We derive an approximate form of this EDR explicitly in the limit of small nonlinearity and confirm it using the wave-number-frequency spectral (WFS) analysis, a Fourier-transform based method used for determining dispersion relations of observed waves. We also show that it extends to the FNLS the universal EDR formula for the defocusing Majda-McLaughlin-Tabak (MMT) model of weak turbulence. In addition, unexpectedly, even for some spatially periodic versions of multisolitonlike waves, the EDR is still a downward shifted linear-dispersion parabola, but the shift does not have a clear relation to the total wave action. Using WFS analysis and heuristic derivations, we present examples of parabolic and nonparabolic EDRs for FNLS waves and also waves for which no EDR exists.

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U2 - 10.1103/PhysRevE.100.022215

DO - 10.1103/PhysRevE.100.022215

M3 - Article

C2 - 31574653

AN - SCOPUS:85072104009

SN - 1063-651X

VL - 100

JO - Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

JF - Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

IS - 2

M1 - 022215

ER -