The d-bar formalism for the modified Veselov-Novikov equation on the half-plane

• Autorzy: Hwang Guenbo , Moon Byungsoo

Identyfikatory

DOI

10.7494/OpMath.2022.42.2.179

• Języki publikacji: EN

Abstrakty

EN

We study the modified Veselov–Novikov equation (mVN) posed on the half-plane via the Fokas method, considered as an extension of the inverse scattering transform for boundary value problems. The mVN equation is one of the most natural (2+1)-dimensional generalization of the (1+1)-dimensional modified Korteweg-de Vries equation in the sense as to how the Novikov–Veselov equation is related to the Korteweg–de Vries equation. In this paper, by means of the Fokas method, we present the so-called global relation for the mVN equation, which is an algebraic equation coupled with the spectral functions, and the d-bar formalism, also known as Pompieu’s formula. In addition, we characterize the d-bar derivatives and the relevant jumps across certain domains of the complex plane in terms of the spectral functions.

Słowa kluczowe

EN

initial-boundary value; integrable nonlinear PDE; spectral analysis; d-bar

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2022
• Tom: Vol. 42, no. 2
• Strony: 179--217
• Opis fizyczny: Bibliogr. 30 poz.

Twórcy

autor

Hwang Guenbo

• Department of Mathematics and Institute of Natural Sciences, Daegu University, Gyeongsan Gyeongbuk 38453, Korea

autor

Moon Byungsoo

• Department of Mathematics, Incheon National University, Incheon 22012, Korea

Bibliografia

• [1] M.J. Ablowitz, D. BarYaacov, A.S. Fokas, On the inverse scattering transform for the Kadomtsev–Petviasvhili equation, Stud. Appl. Math. 69 (1983), 135–143.
• [2] R. Beals, R.R. Coifman, Linear spectral problems, nonlinear equations and the ¯∂-method, Inverse Problems 5 (1989), 87–130.
• [3] L.V. Bogdanov, The Veselov–Novikov equation as a natural two-dimensional generalization of the Korteweg-de Vries equation, Theoret. and Math. Phys. 70 (1987), 219–223.
• [4] A. Boutet de Monvel, D. Shepelsky, L. Zielinski, A Riemann–Hilbert approach for the Novikov equation, Symmetry Integr. Geom.: Methods Appl. 12 (2016), 095.
• [5] R.M. Chen, W. Lian, D. Wang, R. Xu, A rigidity property for the Novikov equation and the asymptotic stability of peakons, Arch. Rational Mech. Anal. 241 (2021), 497–533.
• [6] R. Croke, J.L. Muller, A. Stahel, Transverse instability of plane wave soliton solutions of the Novikov–Veselov equation, Contemporary Mathematics 635 (2015).
• [7] P. Deift, X. Zhou, A steepest descent method for oscillatory Riemann–Hilbert problems, Bull. Amer. Math. Soc. 26 (1992), 119–123.
• [8] P. Deift, S. Venakides, X. Zhou, New results in small dispersion KdV by an extension of the steepest descent method for Riemann–Hilbert problems, Int. Math. Res. Notices 6 (1997), 286–299.
• [9] A.S. Fokas, A unified transform method for solving linear and certain nonlinear PDEs, Proc. Roy. Soc. London A 453 (1997), 1411–1443.
• [10] A.S. Fokas, Integrable nonlinear evolution equations on the half-line, Comm. Math. Phys. 230 (2002), 1–39.
• [11] A.S. Fokas, The generalized Dirichlet-to-Neumann map for certain nonlinear evolution PDEs, Comm. Pure Appl. Math. LVIII (2005), 639–670.
• [12] A.S. Fokas, A Unified Approach to Boundary Value Problems, CBMS-NSF Regional Conference Series in Applied Mathematics SIAM, 2008.
• [13] A.S. Fokas, The Davey–Stewartson equation on the half-plane, Commun. Math. Phys. 289 (2009), 957–993.
• [14] A.S. Fokas, M.J. Ablowitz, On the scattering transform of multi-dimensional nonlinear equations related to first order systems in the plane, J. Math. Phys. 25 (1984), 2494–2505.
• [15] A.S. Fokas, L.Y. Sung, On the solvability of the N-wave the Davey–Stewartson and the Kadomtsev–Petviashvili equation, Inverse Problems 8 (1992), 673–708.
• [16] G. Hwang, The modified Korteweg–de Vries equation on the quarter plane with t-periodic data, J. Nonlinear Math. Phys. 24 (2017), 620–634.
• [17] G. Hwang, The elliptic sinh-Gordon equation in a semi-strip, Adv. Nonlinear Anal. 8 (2019), 533–544.
• [18] G. Hwang, A.S. Fokas, The modified Korteweg-de Vries equation on the half-line with a sine-wave as Dirichlet datum, J. Nonlinear Math. Phys. 20 (2013), 135–157.
• [19] M. Lassas, J.L. Muller, S. Siltanen, A. Stahel, The Novikov–Veselov equation and the inverse scattering method, Part I: Analysis, Physica D 241 (2012), 1322–1335.
• [20] M. Lassas, J.L. Muller, S. Siltanen, A. Stahel, The Novikov–Veselov equation and the inverse scattering method: II. Computation, Nonlinearity 25 (2012), 1799–1818.
• [21] J. Lenells, A.S. Fokas, The nonlinear Schrödinger equation with t-periodic data: I. Exact results, Proc. R. Soc. A 471 (2015), 20140925.
• [22] J. Lenells, A.S. Fokas, The nonlinear Schrödinger equation with t-periodic data: II. Perturbative results, Proc. R. Soc. A 471 (2015), 20140926.
• [23] D. Mantzavinos, A.S. Fokas, The Kadomtsev–Petviashvili II equation on the half-plane, Physica D 240 (2011), 477–511.
• [24] A.V. Mikhailov, V.S. Novikov, Perturbative symmetry approach, J. Phys. A: Math. Gen. 35 (2002), 4775–4790.
• [25] B. Moon, Single peaked traveling wave solutions to a generalized μ–Novikov equation, Adv. Nonlinear Anal. 10 (2021), 66–75.
• [26] I.A. Taimanov, Modified Veselov–Novikov equation and differential geometry of surface, Trans. Amer. Math. Soc. 197 (1997), 133–151.
• [27] D.-S. Wang, X. Wang, Long-time asymptotics and the bright N-soliton solutions of the Kundu–Eckhaus equation via the Riemann–Hilbert approach, Nonlinear Anal.: Real World Appl. 41 (2018), 334–361.
• [28] D.-S. Wang, B. Guo, X. Wang, Long-time asymptotics of the focusing Kundu–Eckhaus equation with nonzero boundary conditions, J. Differential Equations 266 (2019), 5209–5253.
• [29] D. Yu, Q. P. Liu, S. Wang, Darboux transformation for the modified Veselov–Novikov equation, J. Phys. A: Math. Gen. 35 (2002), 3779–3785.
• [30] X. Zhou, Inverse scattering transform for the time dependent Schrödinger equation with application to the KPI equation, Commun. Math. Phys. 128 (1990), 551–564.

Uwagi

Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023).