Abstract
We consider the periodic standing waves in the derivative nonlinear Schrödinger (DNLS) equation arising in plasma physics. By using a newly developed algebraic method with two eigenvalues, we classify all periodic standing waves in terms of eight eigenvalues of the Kaup–Newell spectral problem located at the end points of the spectral bands outside the real line. The analytical work is complemented with the numerical approximation of the spectral bands, this enables us to fully characterize the modulational instability of the periodic standing waves in the DNLS equation.
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References
Biagioni, H.A., Linares, F.: Ill-posedness for the derivative Schrödinger and generalized Benjamin–Ono equations. Trans. Am. Math. Soc. 353, 3649–3659 (2001)
Bronski, J.C., Hur, V.M., Johnson, M.A.: Modulational instability in equations of KdV type. New Approaches to Nonlinear Waves. Lecture Notes in Phys., vol. 908, pp. 83–133. Springer, Cham (2016)
Cao, C.W., Geng, X.G.: Classical integrable systems generated through nonlinearization of eigenvalue problems. Nonlinear Physics (Shanghai, 1989). Research Reports in Physics, pp. 68–78. Springer, Berlin (1990)
Cao, C.W., Yang, X.: A (2+1)-dimensional derivative Toda equation in the context of the Kaup-Newell spectral problem. J. Phys. A Math. Theor. 41, 025203 (2008). 19 pages
Chen, J., Pelinovsky, D.E.: Rogue periodic waves in the modified Korteweg-de Vries equation. Nonlinearity 31, 1955–1980 (2018)
Chen, J., Pelinovsky, D.E.: Rogue periodic waves in the focusing nonlinear Schrödinger equation. Proc. R. Soc. Lond. A 474, 20170814 (2018). 18 pages
Chen, J., Pelinovsky, D.E.: Periodic travelling waves of the modified KdV equation and rogue waves on the periodic background. J. Nonlinear Sci. 29, 2797–2843 (2019)
Chen, J., Pelinovsky, D.E., White, R.E.: Periodic standing waves in the focusing nonlinear Schrödinger equation: Rogue waves and modulation instability. Phys. D 405, 132378 (2020). 13 pages
Chen, J., Zhang, R.: The complex Hamiltonian systems and quasi-periodic solutions in the derivative nonlinear Schrödinger equations. Stud. Appl. Math. 145, 153–178 (2020)
Chen, X.J., Yang, J.: Direct perturbation theory for solitons of the derivative nonlinear Schrödinger equation and the modified nonlinear Schrödinger equation. Phys. Rev. E 65, 066608 (2002). 12 pages
Chow, K.W., Ng, T.W.: Periodic solutions of a derivative nonlinear Schrödinger equation: elliptic integrals of the third kind. J. Comput. Appl. Math. 235, 3825–3830 (2011)
Colin, M., Ohta, M.: Stability of solitary waves for derivative nonlinear Schrödinger equation. Ann. I.H. Poincaré-AN 23, 753–764 (2006)
Curtis, C.W., Deconinck, B.: On the convergence of Hill’s method. Math. Comput. 79, 169–187 (2010)
Deconinck, B., Kutz, J.N.: Computing spectra of linear operators using the Floquet–Fourier–Hill method. J. Comput. Phys. 219, 296–321 (2006)
Deconinck, B., Segal, B.L.: The stability spectrum for elliptic solutions to the focusing NLS equation. Phys. D 346, 1–19 (2017)
Deconinck, B., Upsal, J.: The orbital stability of elliptic solutions of the focusing nonlinear Schrödinger equation. SIAM J. Math. Anal. 52, 1–41 (2020)
Deconinck, B., Upsal, J.: Real Lax spectrum implies spectral stability. Stud. Appl. Math. 145, 765–790 (2020)
Fukaya, N., Hayashi, M., Inui, T.: A sufficient condition for global existence of solutions to a generalized derivative nonlinear Schrödinger equation. Anal. PDEs 10, 1149–1167 (2017)
Geng, X.G., Li, Z., Xue, B., Guan, L.: Explicit quasi-periodic solutions of the Kaup–Newell hierarchy. J. Math. Anal. Appl. 425, 1097–1112 (2015)
Guo, B.L., Wu, Y.: Orbital stability of solitary waves for the nonlinear derivative Schrödinger equation. J. Differ. Equ. 123, 35–55 (1995)
Hakkaev, S., Stanislavova, M., Stefanov, A.: All non-vanishing bell-shaped solutions for the cubic derivative NLS are stable. arXiv:2006.13658 (2020)
Hayashi, M.: Long-period limit of exact periodic traveling wave solutions for the derivative nonlinear Schrödinger equation. Ann. lÍnst. Henri Poincaré C Anal. Non linéaire 36, 1331–1360 (2019)
Hayashi, N., Ozawa, T.: On the derivative nonlinear Schrödinger equation. Phys. D 55, 14–36 (1992)
Hayashi, N., Ozawa, T.: Finite energy solution of nonlinear Schrödinger equations of derivative type. SIAM J. Math. Anal. 25, 1488–1503 (1994)
Jenkins, R., Liu, J., Perry, P.A., Sulem, C.: Global well-posedness for the derivative nonlinear Schrödinger equation. Commun. Partial Differ. Equ. 43, 1151–1195 (2018)
Johnson, M.A., Zumbrun, K.: Convergence of Hill’s method for nonselfadjoint operators. SIAM J. Numer. Anal. 50, 64–78 (2012)
Kamchatnov, A.M.: On improving the effectiveness of periodic solutions of the NLS and DNLS equations. J. Phys. A Math. Gen. 23, 2945–2960 (1990)
Kamchatnov, A.M.: New approach to periodic solutions of integrable equations and nonlinear theory of modulational instability. Phys. Rep. 286, 199–270 (1997)
Kamchatnov, A.M.: Evolution of initial discontinuities in the DNLS equation theory. J. Phys. Commun. 2, 025027 (2018). 22 pages
Kaup, D.J., Newell, A.C.: An exact solution for a derivative nonlinear Schrödinger equation. J. Math. Phys. 19, 798–801 (1978)
Kuchment, P.A.: Floquet Theory for Partial Differential Equations. Birkhäuser, Basel (1993)
Kwon, S., Wu, Y.: Orbital stability of solitary waves for derivative nonlinear Schrödinger equation. J. d’ Anal. Math. 135, 473–486 (2018)
Lax, P.D.: Integrals of nonlinear equation of evolution and solitary waves. Commun. Pure Appl. Math. 21, 467–490 (1968)
Liu, J., Perry, P.A., Sulem, C.: Global existence for the derivative nonlinear Schrödinger equation by the method of inverse scattering. Commun. Partial Differ. Equ. 41, 1692–1760 (2016)
Miao, C., Wu, Y., Xu, G.: Global well-posedness for Schrödinger equation with derivative in \(H^{1/2}({{\mathbb{R}}})\). J. Differ. Equ. 251, 2164–2195 (2011)
Miao, C., Tang, X., Xu, G.: Stability of the traveling waves for the derivative Schrödinger equation in the energy space. Calc. Var. PDEs 56, 45 (2017). 20 pages
Mio, W., Ogino, T., Minami, K., Takeda, S.: Modified nonlinear Schrödinger equation for Alfvén waves propagating along the magnetic field in cold plasmas. J. Phys. Soc. Jpn. 41, 265–271 (1976)
Mjolhus, E.: On the modulational instability of hydromagnetic waves parallel to the magnetic field. J. Plasma Phys. 16, 321–334 (1976)
Ma, W.X., Zhou, R.: On the relationship between classical Gaudin models and BC-type Gaudin models. J. Phys. A Math. Gen. 34, 3867–880 (2001)
Pelinovsky, D.E., Saalmann, A., Shimabukuro, Y.: The derivative NLS equation: global existence with solitons. Dyn. PDEs 14, 271–294 (2017)
Pelinovsky, D.E., Shimabukuro, Y.: Existence of global solutions to the derivative NLS equation with the inverse scattering transform method. Int. Math. Res. Notices 2018, 5663–5728 (2018)
Pelinovsky, D.E., White, R.E.: Localized structures on librational and rotational travelling waves in the sine–Gordon equation. Proc. R. Soc. Lond. A 476, 20200490 (2020). 18 pages
Qiao, Z.: A new completely integrable Liouville’s system produced by the Kaup–Newell eigenvalue problem. J. Math. Phys. 34, 3110–3120 (1993)
Takaoka, H.: Well-posedness for the one dimensional Schrödinger equation with the derivative nonlinearity. Adv. Differ. Equ. 4, 561–680 (1999)
Weinstein, M.I.: Liapunov stability of ground states of nonlinear dispersive evolution equations. Commun. Pure Appl. Math. 39, 51–68 (1986)
Wright, O.C.: Maximal amplitudes of hyperelliptic solutions of the derivative nonlinear Schrödinger equation. Stud. Appl. Math. 144, 1–30 (2020)
Wu, Y.: Global well-posedness for the nonlinear Schrödinger equation with derivative in energy space. Anal. PDE 6, 1989–2002 (2013)
Wu, Y.: Global well-posedness on the derivative nonlinear Schrödinger equation. Anal. PDE 8, 1101–1112 (2015)
Zhao, P., Fan, E.G.: Finite gap integration of the derivative nonlinear Schrödinger equation: a Riemann–Hilbert method. Phys. D 402, 132213 (2020). 31 pages
Zhou, R.G.: An integrable decomposition of the derivative nonlinear Schrödinger equation. Chin. Phys. Lett. 24, 589–591 (2007)
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This work was supported in part by the National Natural Science Foundation of China (No. 11971103).
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Chen, J., Pelinovsky, D.E. & Upsal, J. Modulational Instability of Periodic Standing Waves in the Derivative NLS Equation. J Nonlinear Sci 31, 58 (2021). https://doi.org/10.1007/s00332-021-09713-5
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DOI: https://doi.org/10.1007/s00332-021-09713-5
Keywords
- Derivative nonlinear Schrödinger equation
- Periodic standing waves
- Kaup–Newell spectral problem
- Spectral stability
- Modulational stability