Abstract
In this paper, we introduce a first-order low-regularity integrator for the Davey–Stewartson system in the elliptic-elliptic case. It only requires the boundedness of one additional derivative of the solution to be first-order convergent. By rigorous error analysis, we show that the scheme provides first-order accuracy in for rough initial data in with .
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 11901120
Funding statement: C. Ning is partially supported by NSFC 11901120.
References
[1] C. Besse, Schéma de relaxation pour l’équation de Schrödinger non linéaire et les systèmes de Davey et Stewartson, C. R. Acad. Sci. Paris Sér. I Math. 326 (1998) no. 12, 1427–1432. 10.1016/S0764-4442(98)80405-9Search in Google Scholar
[2] C. Besse, N. J. Mauser and H. P. Stimming, Numerical study of the Davey–Stewartson system, M2AN Math. Model. Numer. Anal. 38 (2004), no. 6, 1035–1054. 10.1051/m2an:2004049Search in Google Scholar
[3] T. Cazenave, Semilinear Schrödinger Equations, Courant Lect. Notes Math. 10, American Mathematical Society, Providence, 2003. 10.1090/cln/010Search in Google Scholar
[4] R. Cipolatti, On the existence of standing waves for a Davey–Stewartson system, Comm. Partial Differential Equations 17 (1992), no. 5–6, 967–988. 10.1080/03605309208820872Search in Google Scholar
[5] R. Cipolatti, On the instability of ground states for a Davey–Stewartson system, Ann. Inst. H. Poincaré Phys. Théor. 58 (1993), no. 1, 85–104. Search in Google Scholar
[6] A. Davey and K. Stewartson, On three-dimensional packets of surface waves, Proc. Roy. Soc. Lond. Ser. A 338 (1974), 101–110. 10.1098/rspa.1974.0076Search in Google Scholar
[7] J.-M. Ghidaglia and J.-C. Saut, On the initial value problem for the Davey–Stewartson systems, Nonlinearity 3 (1990), no. 2, 475–506. 10.1088/0951-7715/3/2/010Search in Google Scholar
[8] N. Hayashi, Local existence in time of solutions to the elliptic-hyperbolic Davey–Stewartson system without smallness condition on the data, J. Anal. Math. 73 (1997), 133–164. 10.1007/BF02788141Search in Google Scholar
[9] N. Hayashi and H. Hirata, Global existence and asymptotic behaviour in time of small solutions to the elliptic-hyperbolic Davey–Stewartson system, Nonlinearity 9 (1996), no. 6, 1387–1409. 10.1088/0951-7715/9/6/001Search in Google Scholar
[10] N. Hayashi and J.-C. Saut, Global existence of small solutions to the Davey–Stewartson and the Ishimori systems, Differential Integral Equations 8 (1995), no. 7, 1657–1675. 10.57262/die/1368397751Search in Google Scholar
[11] M. Hofmanová and K. Schratz, An exponential-type integrator for the KdV equation, Numer. Math. 136 (2017), no. 4, 1117–1137. 10.1007/s00211-016-0859-1Search in Google Scholar
[12] C. Klein, K. McLaughlin and N. Stoilov, Spectral approach to the scattering map for the semi-classical defocusing Davey–Stewartson II equation, Phys. D 400 (2019), Article ID 132126. 10.1016/j.physd.2019.05.006Search in Google Scholar
[13] C. Klein and N. Stoilov, Numerical study of blow-up mechanisms for Davey–Stewartson II systems, Stud. Appl. Math. 141 (2018), no. 1, 89–112. 10.1111/sapm.12214Search in Google Scholar
[14] C. Klein and N. Stoilov, Numerical scattering for the defocusing Davey–Stewartson II equation for initial data with compact support, Nonlinearity 32 (2019), no. 11, 4258–4280. 10.1088/1361-6544/ab28c6Search in Google Scholar
[15] M. Knöller, A. Ostermann and K. Schratz, A Fourier integrator for the cubic nonlinear Schrödinger equation with rough initial data, SIAM J. Numer. Anal. 57 (2019), no. 4, 1967–1986. 10.1137/18M1198375Search in Google Scholar
[16] H. Leblond, Electromagnetic waves in ferromagnets: A Davey–Stewartson-type model, J. Phys. A 32 (1999), no. 45, 7907–7932. 10.1088/0305-4470/32/45/308Search in Google Scholar
[17] F. Linares and G. Ponce, On the Davey–Stewartson systems, Ann. Inst. H. Poincaré C Anal. Non Linéaire 10 (1993), no. 5, 523–548. 10.1016/s0294-1449(16)30203-7Search in Google Scholar
[18] J. Lu and Y. Wu, Sharp threshold for scattering of a generalized Davey–Stewartson system in three dimension, Commun. Pure Appl. Anal. 14 (2015), no. 5, 1641–1670. 10.3934/cpaa.2015.14.1641Search in Google Scholar
[19] G. M. Muslu, Numerical study of blow-up to the purely elliptic generalized Davey–Stewartson system, J. Comput. Appl. Math. 317 (2017), 331–342. 10.1016/j.cam.2016.12.003Search in Google Scholar
[20] A. Nachman, I. Regev and D. Tataru, A nonlinear Plancherel theorem with applications to global well-posedness for the defocusing Davey–Stewartson equation and to the inverse boundary value problem of Calderón, Invent. Math. 220 (2020), no. 2, 395–451. 10.1007/s00222-019-00930-0Search in Google Scholar
[21] A. C. Newell and J. V. Moloney, Nonlinear Optics, Adv. Topics Interdiscip. Math. Sci., Addison-Wesley, Redwood, 1992. Search in Google Scholar
[22] K. Nishinari and J. Satsuma, Multi-dimensional localized behavior of electrostatic ion wave in a magnetized plasma, Phys. Plasmas 1 (1994), Article ID 2559. 10.1063/1.870583Search in Google Scholar
[23] M. Ohta, Instability of standing waves for the generalized Davey–Stewartson system, Ann. Inst. H. Poincaré Phys. Théor. 62 (1995), no. 1, 69–80. 10.57262/die/1368397756Search in Google Scholar
[24] A. Ostermann and K. Schratz, Low regularity exponential-type integrators for semilinear Schrödinger equations, Found. Comput. Math. 18 (2018), no. 3, 731–755. 10.1007/s10208-017-9352-1Search in Google Scholar
[25] C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation, Self-Focusing and Wave Collapse, Appl. Math. Sci. 139, Springer, New York, 1999. Search in Google Scholar
[26] M. Tsutsumi, Decay of weak solutions to the Davey–Stewartson systems, J. Math. Anal. Appl. 182 (1994), no. 3, 680–704. 10.1006/jmaa.1994.1113Search in Google Scholar
[27] P. W. White, The Davey–Stewartson equations: A numerical study, Ph.D. Thesis, Oregon State University, 1994. Search in Google Scholar
[28] P. W. White and J. A. C. Weideman, Numerical simulation of solitons and dromions in the Davey–Stewartson system, Math. Comput. Simulation 37 (1994), 469–479. 10.1016/0378-4754(94)00032-8Search in Google Scholar
[29] Y. Wu and F. Yao, Embedded exponential-type low-regularity integrators for KdV equation under rough data, preprint (2020), https://arxiv.org/abs/2008.07053v2. 10.1007/s10543-021-00895-8Search in Google Scholar
[30] Y. Wu and X. Zhao, Optimal convergence of a second order low-regularity integrator for the KdV equation, IMA J. Numer. Anal. (2021), 10.1093/imanum/drab054. 10.1093/imanum/drab054Search in Google Scholar
[31] Y. Wu and X. Zhao, Embedded exponential-type low-regularity integrators for KdV equation under rough data, BIT Numer. Math. (2021), 10.1007/s10543-021-00895-8. 10.1007/s10543-021-00895-8Search in Google Scholar
[32] V. E. Zakharov, S. L. Musher and A. M. Rubenchik, Hamiltonian approach to the description of nonlinear plasma phenomena, Phys. Rep. 129 (1985), no. 5, 285–366. 10.1016/0370-1573(85)90040-7Search in Google Scholar
[33] V. E. Zakharov and E. I. Schulman, Integrability of nonlinear systems and perturbation theory, What is Integrability?, Springer Ser. Nonlinear Dynam., Springer, Berlin (1991), 185–250. 10.1007/978-3-642-88703-1_5Search in Google Scholar
© 2022 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- On the Spectrum of an Operator Associated with Least-Squares Finite Elements for Linear Elasticity
- Anisotropic Adaptive Finite Elements for an Elliptic Problem with Strongly Varying Diffusion Coefficient
- DPG Methods for a Fourth-Order div Problem
- Stable Implementation of Adaptive IGABEM in 2D in MATLAB
- Optimal Convergence of the Newton Iterative Crank–Nicolson Finite Element Method for the Nonlinear Schrödinger Equation
- Partially Discontinuous Nodal Finite Elements for 𝐻(curl) and 𝐻(div)
- Improvement of the Constructive A Priori Error Estimates for a Fully Discretized Periodic Solution of Heat Equation
- An 𝐿𝑝-DPG Method with Application to 2D Convection-Diffusion Problems
- An Improvement on a Class of Fixed Point Iterative Methods for Solving Absolute Value Equations
- Low-Regularity Integrator for the Davey–Stewartson System: Elliptic-Elliptic Case
- The Numerical Approximation to a Stochastic Age-Structured HIV/AIDS Model with Nonlinear Incidence Rates
- Qualitative Properties of Space-Dependent SIR Models with Constant Delay and Their Numerical Solutions
- A Staggered Discontinuous Galerkin Method for Quasi-Linear Second Order Elliptic Problems of Nonmonotone Type
Articles in the same Issue
- Frontmatter
- On the Spectrum of an Operator Associated with Least-Squares Finite Elements for Linear Elasticity
- Anisotropic Adaptive Finite Elements for an Elliptic Problem with Strongly Varying Diffusion Coefficient
- DPG Methods for a Fourth-Order div Problem
- Stable Implementation of Adaptive IGABEM in 2D in MATLAB
- Optimal Convergence of the Newton Iterative Crank–Nicolson Finite Element Method for the Nonlinear Schrödinger Equation
- Partially Discontinuous Nodal Finite Elements for 𝐻(curl) and 𝐻(div)
- Improvement of the Constructive A Priori Error Estimates for a Fully Discretized Periodic Solution of Heat Equation
- An 𝐿𝑝-DPG Method with Application to 2D Convection-Diffusion Problems
- An Improvement on a Class of Fixed Point Iterative Methods for Solving Absolute Value Equations
- Low-Regularity Integrator for the Davey–Stewartson System: Elliptic-Elliptic Case
- The Numerical Approximation to a Stochastic Age-Structured HIV/AIDS Model with Nonlinear Incidence Rates
- Qualitative Properties of Space-Dependent SIR Models with Constant Delay and Their Numerical Solutions
- A Staggered Discontinuous Galerkin Method for Quasi-Linear Second Order Elliptic Problems of Nonmonotone Type