Laguerre polynomials and transitional asymptotics of the modified Korteweg–de Vries equation for step-like initial data
Tóm tắt
We consider the compressive wave for the modified Korteweg–de Vries equation with background constants $$c>0$$ for $$x\rightarrow -\infty $$ and 0 for $$x\rightarrow +\infty $$. We study the asymptotics of solutions in the transition zone $$4c^2t-\varepsilon t0,$$$$\sigma \in (0,1),$$$$\beta >0.$$ In this region we have a bulk of nonvanishing oscillations, the number of which grows as $$\frac{\varepsilon t}{\ln t}.$$ Also we show how to obtain Khruslov–Kotlyarov’s asymptotics in the domain $$4c^2t-\rho \ln t
Tài liệu tham khảo
Bertola, M., Buckingham, R., Lee, S.Y., Pierce, V.: Spectra of random Hermitian matrices with a small-rank external source: the supercritical and subcritical regimes. J. Stat. Phys. 153(4), 654–697 (2013). https://doi.org/10.1007/s10955-013-0845-2
Bertola, M., Lee, S.Y., Mo, M.Y.: Mesoscopic colonization in a spectral band. J. Phys. A 42, 41, 415204,17 (2009)
Bertola, M., Lee, S.Y.: First colonization of a spectral outpost in random matrix theory. Constr. Approx. 30(2), 225–263 (2009)
Bertola, M., Tovbis, A.: Universality for the focusing Schroedinger equation at the gradient catastrophe point: rational breathers and poles of the tritronquee solution to Painleve I. Commun. Pure Appl. Math. 66(5), 678–752 (2013)
Bothner, T., Deift, P., Its, A., Krasovsky, I.: On the asymptotic behavior of a log gas in the bulk scaling limit in the presence of a varying external potential I. Commun. Math. Phys. 337(3), 1397–1463 (2015). arxiv1407.2910
Bikbaev, R.F., Novokshenov, V.Yu.: The Korteveg–de Vries equation with finite gap boundary conditions and self-similar solutions of whitham equations Proc. In: III International Workshop “Nonlinear and Turbulent Processes in Physics” Kiev 1, 32–35 (1988)
Bikbaev, R.F., Novokshenov, Yu.V.: Existence and uniqueness of the solution of the Whitham equation (Russian). Asymptotic methods for solving problems in mathematical physics Akad. Nauk SSSR Ural. Otdel. Bashkir. Nauchn. Tsentr Ufa 81–95 (1989)
Bikbaev, R.F.: Structure of a shock wave in the theory of the Korteweg–de Vries equation. Phys. Lett. A 141(5–6), 289–293 (1989)
Bikbaev, R.F., Sharipov, R.A.: The asymptotic behaviour, as \(t\rightarrow \infty \), of the solution of the Cauchy problem for the Korteweg-de Vries equation in a class of potentials with finite-gap behaviour as \(x\rightarrow \pm \infty \). Teoret. Mat. Fiz. 78/3, 345–356 (1989). (translation in Theoret. and Math. Phys.78/3 244–252)
Bikbaev, R.F.: The Korteweg–de Vries equation with finite-gap boundary conditions and Whitham deformations of Riemann surfaces (Russian). Funktsional. Anal. i Prilozhen. 23/4, 1–10 (1990). (translation in Funct. Anal. Appl.23/4 257–266)
Bikbaev, R.F.: The influence of viscosity on the structure of shock waves in the MKdV model (Russian). Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) Voprosy Kvant. Teor. Polya Statist. Fiz. 11, 37–42 (1995). (184 translation in J. Math. Sci.772 3042–3045)
Bikbaev, R.F.: Complex Whitham deformations in problems with “integrable instability” (Russian). Teoret. Mat. Fiz. 104/3, 393–419 (1996). (translation in Theoret. and Math. Phys.104/3 1078–1097)
Bikbaev, R.F.: Modulational instability stabilization via complex Whitham deformations: nonlinear Schrodinger equation Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 215 Differentsialnaya Geom. Gruppy Li i Mekh. 14 65–76 310 translation in J. Math. Sci. (New York)85 (1997) 1 1596–1604
Boutet de Monvel, A., Kotlyarov, V.P.: Focusing nonlinear Schrodinger equation on the quarter plane with time-periodic boundary condition: a Riemann–Hilbert approach. J. Inst. Math. Jussieu 6(4), 579–611 (2007)
Boutet de Monvel, A., Its, A.R., Kotlyarov, V.P.: Long-time asymptotics for the focusing NLS equation with time-periodic boundary condition. C. R. Math. Acad. Sci. Paris 345(11), 615–620 (2007)
Boutet de Monvel, A., Its, A.R., Kotlyarov, V.P.: Long -time asymptotics for the focusing NLS equation with time—periodic boundary condition on the half line. Commun. Math. Phys. 290(2), 479–522 (2009)
Boutet de Monvel, A., Kotlyarov, V.P., Shepelsky, D.G.: Focusing NLS equation: long-time dynamics of step-like initial data. Int. Math. Res. Not. 7, 1613–1653 (2011)
Buckingham, R., Venakides, S.: Long-time asymptotics of the non-linear Schrodinger equation shock problem. Commun. Pure Appl. Math. 60(9), 1349–1414 (2007)
Claeys, T.: Birth of a cut in unitary random matrix ensembles. Int. Math. Res. Not. (IMRN) 2008(6), 40 (2008). https://doi.org/10.1093/imrn/rnm166
Claeys, T., Grava, T.: Solitonic asymptotics for the Korteweg–de Vries equation in the small dispersion limit. SIAM J. Math. Anal. 42(5), 2132–2154 (2010)
Cohen, A., Kappeler, T.: Scattering and inverse scattering for steplike potentials in the Schrdinger equation. Indiana Univ. Math. J. 34(1), 121–180 (1985)
Corless, R.M., Gonnet, G.H., Hare, D.E.G., Jeffrey, D.J., Knuth, D.E.: On the Lambert W function. Adv. Comput. Math. 5, 329–359 (1996)
Deift, P., Kriecherbauer, T., McLaughlin, K.T.-R., Venakides, S., Zhou, X.: Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. Commun. Pure Appl. Math. 52(11), 1335–1425 (1999)
Deift, P., Trubowitz, E.: Inverse scattering on the line. Commun. Pure Appl. Math. 32(2), 121–251 (1979)
Deift, P., Zhou, X.: A steepest descent method for oscillatory Riemann–Hilbert problems. Asymptot. MKdV Equ. Ann. Math. 137(2), 295–368 (1993)
Deift, P., Venakides, S., Zhou, X.: The collisionless shock region for the long-time behavior of solutions of the KdV equation. Commun. Pure Appl. Math. 47(2), 199–206 (1994)
Egorova, I., Gladka, Z., Kotlyarov, V., Teschl, G.: Long-time asymptotics for the Korteweg–de Vries equation with steplike initial data. Nonlinearity 26(7), 1839–1864 (2012)
Grunert, K., Teschl, G.: Long-time asymptotics for the Korteweg–de Vries equation via nonlinear steepest descent. Math. Phys. Anal. Geom. 12(3), 287324 (2009)
Gurevich, A.V., Pitaevskii, L.P.: Decay of initial discontinuity in the Korteweg–de Vries equation. JETP Lett. 17/5, 193 (1975)
Jakovleva, A.I.: Master thesis “Application of inverse scattering transform method to a Cauchy problem for the modified Korteweg-de Vries equation”, Kharkov (1980) (Russian)
Kamvissis, S.: Long time behavior for the focusing nonlinear Schroedinger equation with real spectral singularities. Commun. Math. Phys. 180(2), 325–341 (1996)
Khruslov, E.Ya.: Splitting of an initial step-like perturbation for the KdV equation. Lett. JETP 21(4), 469–472 (1975)
Khruslov, EYa.: Asymptotics of the solution of the Cauchy problem for the Korteweg de Vries equation with initial data of step type. Matem. Sbornik (New Series) 99(141):2, 261–281 (1976)
Khruslov, E.Ya., Kotlyarov, V.P.: Asymptotic solitons of the modified Korteweg–de Vries equation. Inverse Probl. 5(6), 1075–1088 (1989)
Khruslov, E. Ya., Kotlyarov, V. P.: Soliton asymptotics of nondecreasing solutions of nonlinear completely integrable evolution equations. Spectral operator theory and related topics. Adv. Soviet Math. 19 American Mathematical Society, Providence, RI 129–180 (1994)
Khruslov, E.Ya., Kotlyarov, V.P.: Generation of asymptotic solitons in an integrable model of stimulated Raman scattering by periodic boundary data. Mat. Fiz. Anal. Geom. 10(3), 366–384 (2003)
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, 093506 (2010)
Kotlyarov, V., Minakov, A.: Step-initial function to the MKdV equation: hyper-elliptic long-time asymptotics of the solution. J. Math. Phys. Anal. Geom. 8(1), 37–61 (2011)
Kotlyarov, V., Minakov, A.: Modulated elliptic wave and asymptotic solitons in a shock problem to the modified Kortweg–de Vries equation. J. Phys. A 48(30), 305201, 35 (2015)
Lavrent’ev, M.A., Sabat, B.V.: Metody teorii funkcii kompleksnogo peremennogo. (Russian) Methods of the theory of functions of a complex variable Gosudarstv. Izdat. Tehn.-Teor. Lit. Moscow-Leningrad (1951)
Leach, J.A.: An initial-value problem for the modified Korteweg–de Vries equation. IMA J. Appl. Math. 78(6), 1196–1213 (2013)
Marchant, T.R.: Undular bores and the initial-boundary value problem for the modified Korteweg–de Vries equation. Wave Motion 45(4), 540–555 (2008)
Marchenko, V.A.: Operatory Shturma-Liuvillya i ikh prilozheniya. (Russian) [Sturm-Liouville operators and their applications] Izdat. “Naukova Dumka”, Kiev (1977) 331 pp
Minakov, A.: Long-time behaviour of the solution to the mKdV equation with step-like initial data. J. Phys. A: Math. Theor. 44, 085206 (2011)
Minakov, A.: Asymptotics of rarefaction wave solution to the mKdV equation. J. Math. Phys. Anal. Geom. 7(1), 59–86 (2011)
Moskovchenko, E.A., Kotlyarov, V.P.: A new Riemann–Hilbert problem in a model of stimulated Raman scattering. J. Phys. A: Math. Gen. 39, 014591 (2006)
Moskovchenko, E.A.: Simple periodic boundary data and Riemann–Hilbert problem for integrable model of the stimulated Raman scattering. J. Math. Phys. Anal. Geom. 5/1, 82–103 (2009)
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(5), 31 (2010). https://doi.org/10.1088/1751-8113/43/5/055205
Minakov, A.: Ph.D. thesis. Riemann–Hilbert problems and the modified Korteweg de Vries equation: asymptotic analysis of solutions with step-like initial data (2013)
Mo, M.Y.: The Riemann–Hilbert approach to double scaling limit of random matrix eigenvalues near the “birth of a cut” transition. Int. Math. Res. Not (2008). https://doi.org/10.1093/imrn/rnn042
Novokshenov, VYu.: Time asymptotics for soliton equations in problems with step initial conditions (Russian) Sovrem. Mat. Prilozh., Asimptot. Metody Funkts. Anal. 5, 138–168 (2005). (translation in J. Math. Sci. (N. Y.) 1255 717–749)
Novoksenov, VYu.: Asymptotic behavior as \(t\rightarrow \infty \) of the solution of the Cauchy problem for a nonlinear Schrodinger equation (Russian). Dokl. Akad. Nauk SSSR 251(4), 799–802 (1980)
Novokshenov, VYu.: Asymptotic formulae for the solutions of the system of nonlinear Schrodinger equations. Uspekhi Matem Nauk 37(2), 215–216 (1982)
Novokshenov, VYu.: Asymptotics as \(t\rightarrow \infty \) of the Solution to a Two-Dimentional Generalisation of the Toda Lattice. Doklady AN SSSR 265/6, 1320–1324 (1982). (translation in Soviet Math. Dokl26/1 264–268)
Shabat, A.B.: An inverse scattering problem. (Russian). Differentsialnye Uravneniya 15(10), 1824–1834 (1918)
Trogdon, Thomas, Olver, Sheehan, Deconinck, Bernard: Numerical inverse scattering for the Kortewegde Vries and modified Kortewegde Vries equations. Physica D 241(11), 10031025 (2012)
Vanlessen, M.: Strong asymptotics of Laguerre-type orthogonal polynomials and applications in random matrix theory. Constr. Approx. 25, 125–175 (2007)
Wadati, M.: The modified Korteweg–de Vries equation. J. Phys. Soc. Jpn. 34, 1289–1296 (1973)
Zaharov, V.E., Shabat, A.B.: A plan for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. I. (Russian). Funkcional. Anal. i Prilozen. 8(3), 43–53 (1974)
Zhou, X.: The Riemann–Hilbert problem and inverse scattering. SIAM J. Math. Anal. 20, 966–986 (1989)