Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 070, 26 pages      arXiv:2312.16406      https://doi.org/10.3842/SIGMA.2024.070
Contribution to the Special Issue on Evolution Equations, Exactly Solvable Models and Random Matrices in honor of Alexander Its' 70th birthday

Soliton Condensates for the Focusing Nonlinear Schrödinger Equation: a Non-Bound State Case

Alexander Tovbis a and Fudong Wang bc
a) University of Central Florida, Orlando FL, USA
b) School of Sciences, Great Bay University, Dongguan, P.R. China
c) Great Bay Institute for Advanced Study, Dongguan, P.R. China

Received December 30, 2023, in final form July 16, 2024; Published online July 31, 2024

Abstract
In this paper, we study the spectral theory of soliton condensates - a special limit of soliton gases - for the focusing NLS (fNLS). In particular, we analyze the kinetic equation for the fNLS circular condensate, which represents the first example of an explicitly solvable fNLS condensate with nontrivial large scale space-time dynamics. Solution of the kinetic equation was obtained by reducing it to Whitham type equations for the endpoints of spectral arcs. We also study the rarefaction and dispersive shock waves for circular condensates, as well as calculate the corresponding average conserved quantities and the kurtosis. We want to note that one of the main objects of the spectral theory - the nonlinear dispersion relations - is introduced in the paper as some special large genus (thermodynamic) limit the Riemann bilinear identities that involve the quasimomentum and the quasienergy meromorphic differentials.

Key words: soliton condensate; focusing nonlinear Schrödinger equation; kurtosis; nonlinear dispersion relations; dispersive shock wave.

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References

  1. Bertola M., Tovbis A., Meromorphic differentials with imaginary periods on degenerating hyperelliptic curves, Anal. Math. Phys. 5 (2015), 1-22.
  2. Byrd P.F., Friedman M.D., Handbook of elliptic integrals for engineers and scientists, 2nd ed., Grundlehren Math. Wiss., Vol. 67, Springer, New York, 1971.
  3. Congy T., El G.A., Roberti G., Tovbis A., Dispersive hydrodynamics of soliton condensates for the Korteweg-de Vries equation, J. Nonlinear Sci. 33 (2023), 104, 45 pages, arXiv:2208.04472.
  4. Congy T., El G.A., Roberti G., Tovbis A., Randoux S., Suret P., Statistics of extreme events in integrable turbulence, Phys. Rev. Lett. 132 (2024), 207201, 7 pages, arXiv:2307.08884.
  5. Dafermos C.M., Hyperbolic conservation laws in continuum physics, 3rd ed., Grundlehren Math. Wiss., Vol. 325, Springer, Berlin, 2010.
  6. Doyon B., Dubail J., , Konik R., Yoshimura T., Large-scale description of interacting one-dimensional bose gases: generalized hydrodynamics supersedes conventional hydrodynamics, Phys. Rev. Lett. 119 (2017), 195301, 7 pages, arXiv:1704.04151.
  7. El G.A., The thermodynamic limit of the Whitham equations, Phys. Lett. A 311 (2003), 374-383, arXiv:nlin.SI/0310014.
  8. El G.A., Soliton gas in integrable dispersive hydrodynamics, J. Stat. Mech. Theory Exp. 2021 (2021), 114001, 69 pages, arXiv:2104.05812.
  9. El G.A., Tovbis A., Spectral theory of soliton and breather gases for the focusing nonlinear Schrödinger equation, Phys. Rev. E 101 (2020), 052207, 21 pages, arXiv:1910.05732.
  10. Flaschka H., Forest M.G., McLaughlin D.W., Multiphase averaging and the inverse spectral solution of the Korteweg-de Vries equation, Comm. Pure Appl. Math. 33 (1980), 739-784.
  11. Forest M.G., Lee J.-E., Geometry and modulation theory for the periodic nonlinear Schrödinger equation, in Oscillation Theory, Computation, and Methods of Compensated Compactness (Minneapolis, Minn., 1985), IMA Vol. Math. Appl., Vol. 2, Springer, New York, 1986, 35-69.
  12. Gurevich A.V., Pitaevskii L.P., Nonstationary structure of a collisionless shock wave, Sov. Phys. JETP 38 (1974), 291-297.
  13. Kuijlaars A., Tovbis A., On minimal energy solutions to certain classes of integral equations related to soliton gases for integrable systems, Nonlinearity 34 (2021), 7227-7254, arXiv:2101.03964.
  14. Lax P.D., Levermore C.D., The small dispersion limit of the Korteweg-de Vries equation. II, Comm. Pure Appl. Math. 36 (1983), 571-593.
  15. Saff E.B., Totik V., Logarithmic potentials with external fields, Grundlehren Math. Wiss., Vol. 316, Springer, Berlin, 1997.
  16. Tovbis A., Wang F., Recent developments in spectral theory of the focusing NLS soliton and breather gases: the thermodynamic limit of average densities, fluxes and certain meromorphic differentials; periodic gases, J. Phys. A 55 (2022), 424006, 50 pages, arXiv:2203.03566.
  17. Wadati M., Sanuki H., Konno K., Relationships among inverse method, Bäcklund transformation and an infinite number of conservation laws, Progr. Theoret. Phys. 53 (1975), 419-436.
  18. Zakharov V.E., Kinetic equation for solitons, Sov. Phys. JETP 33 (1971), 538-541.

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