Initial-Boundary Value Problem for Stimulated Raman Scattering Model: Solvability of Whitham Type System of Equations Arising in Long-Time Asymptotic Analysis

Rustem R. Aydagulov   ; Alexander A. Minakov

Symmetry, integrability and geometry: methods and applications, Tome 14 (2018)

Résumé

An initial-boundary value problem for a model of stimulated Raman scattering was considered in [Moskovchenko E.A., Kotlyarov V.P., J. Phys. A: Math. Theor. 43 (2010), 055205, 31 pages]. The authors showed that in the long-time range $t\to+\infty$ the $x>0$, $t>0$ quarter plane is divided into 3 regions with qualitatively different asymptotic behavior of the solution: a region of a finite amplitude plane wave, a modulated elliptic wave region and a vanishing dispersive wave region. The asymptotics in the modulated elliptic region was studied under an implicit assumption of the solvability of the corresponding Whitham type equations. Here we establish the existence of these parameters, and thus justify the results by Moskovchenko and Kotlyarov.

Détail

Keywords: stimulated Raman scattering; Riemann–Hilbert problem; Whitham modulation theory; integrable systems.

Bibliographie

[1] Ablowitz M. J., Biondini G., Wang Q., “Whitham modulation theory for the Kadomtsev–Petviashvili equation”, Proc. A., 473 (2017), 20160695, 23 pp., arXiv: 1610.03478 | DOI | MR

[2] Bikbaev R. F., “The Korteweg–de Vries equation with finite-gap boundary conditions, and Whitham deformations of Riemann surfaces”, Funct. Anal. Appl., 23 (1989), 257–266 | DOI | MR | Zbl

[3] Bikbaev R. F., “Structure of a shock wave in the theory of the Korteweg–de Vries equation”, Phys. Lett. A, 141 (1989), 289–293 | DOI | MR

[4] Bikbaev R. F., “The influence of viscosity on the structure of shock waves in the MKdV model”, J. Math. Sci., 199 (1992), 3042–3045 | DOI | MR | Zbl

[5] Bikbaev R. F., “Complex Whitham deformations in problems with “integrable instability””, Theoret. and Math. Phys., 104 (1995), 1078–1097 | DOI | MR | Zbl

[6] Bikbaev R. F., “Saturation of modulational instability via complex whitham deformations: nonlinear Schrödinger equation”, J. Math. Sci., 85 (1997), 1596–1604 | DOI | MR

[7] Bikbaev R. F., Novokshenov V. Yu., “The Korteweg–de Vries equation with finite-gap boundary conditions, and one-parameter solutions of the Whitham equation”, Asymptotic Methods for Solving Problems in Mathematical Physics, Akad. Nauk SSSR Ural. Otdel., Bashkir. Nauchn. Tsentr, Ufa, 1989, 9–23 | MR

[8] Bikbaev R. F., Sharipov R. A., “The asymptotic behavior, as $t\to\infty$, of the solution of the Cauchy problem for the Korteweg–de Vries equation in a class of potentials with finite-gap behavior as $x\to\pm\infty$”, Theoret. and Math. Phys., 78 (1989), 244–252 | DOI | MR | Zbl

[9] Biondini G., Fagerstrom E., “The integrable nature of modulational instability”, SIAM J. Appl. Math., 75 (2015), 136–163 | DOI | MR | Zbl

[10] Biondini G., Mantzavinos D., “Long-time asymptotics for the focusing nonlinear Schrödinger equation with nonzero boundary conditions at infinity and asymptotic stage of modulational instability”, Comm. Pure Appl. Math., 70 (2017), 2300–2365 | DOI | MR | Zbl

[11] Boutet de Monvel A., Its A., Kotlyarov V., “Long-time asymptotics for the focusing NLS equation with time-periodic boundary condition on the half-line”, Comm. Math. Phys., 290 (2009), 479–522 | DOI | MR | Zbl

[12] Bridges T. J., Ratliff D. J., “On the elliptic-hyperbolic transition in Whitham modulation theory”, SIAM J. Appl. Math., 77 (2017), 1989–2011 | DOI | MR | Zbl

[13] Buckingham R., Venakides S., “Long-time asymptotics of the nonlinear Schrödinger equation shock problem”, Comm. Pure Appl. Math., 60 (2007), 1349–1414 | DOI | MR | Zbl

[14] Claeys T., “Asymptotics for a special solution to the second member of the Painlevé I hierarchy”, J. Phys. A: Math. Theor., 43 (2010), 434012, 18 pp., arXiv: 1001.2213 | DOI | MR | Zbl

[15] Deift P., Zhou X., “A steepest descent method for oscillatory Riemann–Hilbert problems. Asymptotics for the MKdV equation”, Ann. of Math., 137 (1993), 295–368 | DOI | MR | Zbl

[16] Egorova I., Gladka Z., Kotlyarov V., Teschl G., “Long-time asymptotics for the Korteweg–de Vries equation with step-like initial data”, Nonlinearity, 26 (2013), 1839–1864, arXiv: 1210.7434 | DOI | MR | Zbl

[17] Fokas A. S., Menyuk C. R., “Integrability and self-similarity in transient stimulated Raman scattering”, J. Nonlinear Sci., 9 (1999), 1–31 | DOI | MR | Zbl

[18] Grava T., “Whitham modulation equations and application to small dispersion asymptotics and long time asymptotics of nonlinear dispersive equations”, Rogue and Shock Waves in Nonlinear Dispersive Media, Lecture Notes in Phys., 926, Springer, Cham, 2016, 309–335, arXiv: 1701.00069 | DOI

[19] Grava T., Klein C., “Numerical solution of the small dispersion limit of Korteweg–de Vries and Whitham equations”, Comm. Pure Appl. Math., 60 (2007), 1623–1664, arXiv: math-ph/0511011 | DOI | MR | Zbl

[20] Grava T., Pierce V. U., Tian F.-R., “Initial value problem of the Whitham equations for the Camassa–Holm equation”, Phys. D, 238 (2009), 55–66, arXiv: 0805.2558 | DOI | MR | Zbl

[21] Kotlyarov V., Minakov A., “Riemann–Hilbert problem to the modified Korteveg–de Vries equation: long-time dynamics of the steplike initial data”, J. Math. Phys., 51 (2010), 093506, 31 pp. | DOI | MR | Zbl

[22] Kotlyarov V., Minakov A., “Riemann–Hilbert problems and the mKdV equation with step initial data: short-time behavior of solutions and the nonlinear Gibbs-type phenomenon”, J. Phys. A: Math. Theor., 45 (2012), 325201, 17 pp. | DOI | MR | Zbl

[23] Kotlyarov V., Minakov A., “Step-initial function to the MKdV equation: hyper-elliptic long-time asymptotics of the solution”, J. Math. Phys. Anal. Geometry, 8 (2012), 38–62 | MR | Zbl

[24] Kuznetsov E. A., “Solitons in a parametrically unstable plasma”, Sov. Phys. Dokl., 22 (1977), 507–508

[25] Kuznetsov E. A., “Fermi–Pasta–Ulam recurrence and modulation instability”, JETP Lett., 105 (2017), 125–129, arXiv: 1605.05080 | DOI

[26] Kuznetsov E. A., Spektor M. D., “Modulation instability of soliton trains in fiber communication systems”, Theoret. and Math. Phys., 120 (1999), 997–1008 | DOI | MR | Zbl

[27] Minakov A., “Asymptotics of rarefaction wave solution to the mKdV equation”, J. Math. Phys. Anal. Geometry, 7 (2011), 59–86 | MR | Zbl

[28] Minakov A., “Long-time behavior of the solution to the mKdV equation with step-like initial data”, J. Phys. A: Math. Theor., 44 (2011), 085206, 31 pp. | DOI | MR | Zbl

[29] Minakov A., “Riemann–Hilbert problem for Camassa–Holm equation with step-like initial data”, J. Math. Anal. Appl., 429 (2015), 81–104, arXiv: 1401.6777 | DOI | MR | Zbl

[30] Minakov A., “Asymptotics of step-like solutions for the Camassa–Holm equation”, J. Differential Equations, 261 (2016), 6055–6098, arXiv: 1512.04762 | DOI | MR | Zbl

[31] Moskovchenko E. A., “Simple periodic boundary data and Riemann–Hilbert problem for integrable model of the stimulated Raman scattering”, J. Math. Phys. Anal. Geometry, 5 (2009), 82–103 | MR

[32] Moskovchenko E. A., Kotlyarov V. P., “A new Riemann–Hilbert problem in a model of stimulated Raman scattering”, J. Phys. A: Math. Gen., 39 (2006), 14591–14610 | DOI | MR | Zbl

[33] Moskovchenko E. A., Kotlyarov V. P., “Long-time asymptotic behavior of an integrable model of the stimulated Raman scattering with periodic boundary data”, J. Math. Phys. Anal. Geometry, 5 (2009), 386–395 | MR

[34] Moskovchenko E. A., Kotlyarov V. P., “Periodic boundary data for an integrable model of stimulated Raman scattering: long-time asymptotic behavior”, J. Phys. A: Math. Theor., 43 (2010), 055205, 31 pp. | DOI | MR | Zbl

[35] Novokshenov V. Yu., “Temporal asymptotics for soliton equations in problems with step initial conditions”, J. Math. Sci., 125 (2005), 717–749 | DOI | MR

[36] Potëmin G. V., “Algebro-geometric construction of self-similar solutions of the Whitham equations”, Russian Math. Surveys, 43 (1988), 252–253 | DOI | MR