Numerical Study of Non - linear Dispersive Partial Differential Equations

15 Acknowledgements 17 Introduction 19 1 Dispersive Partial Differential Equations and Integrable Systems 29 1.1 Linear Dispersive Partial Differential Equations . . . . . . . . . . . . 29 1.2 Semilinear Dispersive PDEs . . . . . . . . . . . . . . . . . . . . . . . 32 1.2.1 Equations of NLS Type. . . . . . . . . . . . . . . . . . . . . . 33 1.2.2 Equations of KdV Type. . . . . . . . . . . . . . . . . . . . . . 34 1.3 Complete Integrability . . . . . . . . . . . . . . . . . . . . . . . . . . 35 1.3.1 From the Finite to the Infinite Dimensional Notion of Complete Integrability . . . . . . . . . . . . . . . . . . . . . . . . . 36 1.3.2 Solitonic Solutions, an Equilibrium between Dispersion and Nonlinearity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 1.4 Related Physical Phenomena . . . . . . . . . . . . . . . . . . . . . . . 42 1.4.1 Dispersive Shock Waves . . . . . . . . . . . . . . . . . . . . . 42 1.4.2 Blow-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 1.5 The KP and the DS II Equations, (2+1)-Dimensional PDEs . . . . . 49 1.5.1 Analytic Properties of the KP Equations . . . . . . . . . . . . 49 1.5.2 Analytic Properties of the DS Equations . . . . . . . . . . . . 52 1.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 2 Numerical Methods 55 2.1 Space Discretization: Spectral Method . . . . . . . . . . . . . . . . . 55 2.1.1 A Fourier Spectral Method . . . . . . . . . . . . . . . . . . . . 55 2.1.2 Advantages of Spectral Methods with respect to alternative Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 2.2 Stiff Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 2.2.1 Concept of Stiffness . . . . . . . . . . . . . . . . . . . . . . . . 61 2.2.2 Absolute Stability and Stiff Problems . . . . . . . . . . . . . . 63 3 te l-0 06 92 54 9, v er si on 1 30 A pr 2 01 2 2.3 Choice of Time-Integration Schemes . . . . . . . . . . . . . . . . . . . 65 2.3.1 Efficiency of Runge-Kutta and Multistep Methods in solving Stiff Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 2.3.2 Exponential Integrators . . . . . . . . . . . . . . . . . . . . . 68 2.3.3 Other possible Approaches . . . . . . . . . . . . . . . . . . . . 73 2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3 Comparison of Time Stepping Schemes 77 3.