非線性薛丁格方程在許多物理領域的研究中扮演相當重要的角色。尤其一維的非線性薛丁格方程屬於可積系統，故隱含著相當豐富理論特性及守恆性。除此之外，此方程也同時屬於漢彌爾頓系統，故具有辛結構之性質。因為這些豐富的理論特性及廣大的工程運用，非線性薛丁格方程一直是吾人感興趣的主題之一，且凸顯出其數值研究的重要性。
本論文提出具有較好的頻散關係及辛結構守恆的時域有限差分法來求解非線性薛丁格方程。在本文所提出的方法中，先將該方程拆解成線性及非線性之偏微分方程。在求解線性方程時，對於時間微分項，本文所提出的方法採取四階準確度且具辛結構守恆之特性的離散方法；而對於空間微分項，本文所提出的方法則對於其數值的頻散關係做了最佳化。另一方面，在求解此方程非線性部分時，由於其對於時間具有不變性，故可以求得其「實解」。為了驗證本文中所提出的方法之可行性，本論文測試了許多具實解及典型的測試問題。由結果可知，本論文所提出之方法，在所有的測試問題中均能保有相當好的準確度。
本文接著將所提出之數值方法應用於探討瘋狗浪及非線性薛丁格方程的解之漸進行為。憑藉著本文所提出的方法，瘋狗浪的形成機制及其結構、特性可以被更多的了解及探討。而對於薛丁格方程的解的漸進行為，此種複雜的解的特性及不同解之間的過渡行為也可以透過本文所提出的方法得以了解。

Nonlinear Schrodinger (NLS) equation appears in many studies of theoretical physics and possesses many fascinating properties. This equation in one space dimension is an example of integrable model, therefore, permitting an infinite number of conserved quantities such as the momentum and energy. This classical field equation can be rewritten as a system of equations involving Hamiltonian functions. This equation also possesses multisymplectic geometric structure and can be therefore constructed in a multi-symplectic form. Because of both these remarkable properties and wide physical and engineering applications, the nonlinear Schrodinger (NLS) equation has been the subjects of intensive study. Therefore, numerical study on the Schrodinger equation with cubic nonlinearity is essential.

In this dissertation, developed scheme with a better dispersion-relation-equation error reducing and symplecticity for the cubic nonlinear Schrodinger equation is proposed. Over one time step from tn to tn+1, the linear part of Schrodinger equation is solved firstly through four time integration steps. In this part of simulation, the explicit symplectic scheme of fourth order accuracy is adopted to approximate the temporal derivative
term. The second-order spatial derivative term in the linear Schrodinger equation is approximated by centered scheme. The resulting symplectic and space centered difference scheme renders an optimized numerical dispersion relation equation. In the second part of simulation, the solution of the nonlinear equation can be computed exactly thanks to the embedded invariant nature within each time increment. The proposed semidiscretized symplectic scheme underlying the modified equation analysis of second kind and the method of dispersion error minimization has been assessed in terms of the spatial modified wavenumber or the temporal angular frequency resolution.

Furthermore, several applications of the proposed new finite difference scheme for the calculation of Schrodinger equations are included such as the rogue waves in deep-water and the asymptotic problems accompanied with many remarkable quantities of Painleve equations. One of the objectives of this dissertation is to increase the knowledge about the rogue waves by following two steps. The first one is to explore the solution nature
in localized region near the point of gradient catastrophe. The second one is to enlighten the solution in the transitional region bounded by the breaking curves that separate two completely different smooth and oscillatory regions, which are manifested by the modulated plane waves and the two-phase nonlinear waves, respectively. Another objective of this study is to understand how the solution of cubic nonlinear Schrodinger (NSL) equation behaves at large time. The study on long-time asymptotics for cubic nonlinear Schrodinger (NSL) equation, in particularly in the transition zones separating the solution into the different regions, is carried out. Through the proposed scheme, the intricate phenomena in the transitional zone can be clearly visualized, which facilitates one to realize how the solution translate between different solution regions. The connection between the Painleve equations of types II and IV and the nonlinear Schrodinger (NSL) equation will be particularly addressed as well.

Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
1 Introduction 1
1.1 Motivation and objectives . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Dispersion-relation-equation (DRE) error reducing theory . . . . 5
1.2.2 Rogue wave in deep-water ocean . . . . . . . . . . . . . . . . . . 5
1.2.3 Long-time Asymptotic problems and the Painleve equation . . . . 6
1.3 Outlines of the study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
I Scheme development and its analysis 9
2 DRE error reducing scheme for cubic NLS equation 11
2.1 Cubic nonlinear Schrodinger (CNLS) equation and its mathematical properties
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Time-Splitting method for CNLS equation . . . . . . . . . . . . . . . . 16
2.2.1 Numerical method for the linear term in Schrodinger equation . . 16
2.2.2 Numerical method for the nonlinear term in Schrodinger equation 24
2.3 Extension to Two-dimensional scheme development for NLS equation . . 25
3 Fundamental analysis on the proposed scheme 33
3.1 Von Neumann (or Fourier) stability analysis of the proposed scheme . . . 33
3.1.1 One-Dimensional stability analysis . . . . . . . . . . . . . . . . 34
3.1.2 Two-Dimensional stability analysis . . . . . . . . . . . . . . . . 34
3.2 Verification and validation study . . . . . . . . . . . . . . . . . . . . . . 35
3.2.1 One-dimensional analytical problem . . . . . . . . . . . . . . . 35
3.2.2 Two-Dimensional Schrodinger equation with exact solution . . . 37
II Applications in the deep-water ocean rogue wave 55
4 Rogue wave in deep-water ocean 57
4.1 Analogue between rogue wave and NLS equation . . . . . . . . . . . . . 57
4.2 Mechanism of rogue wave . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.2.1 Formation of the rogue wave . . . . . . . . . . . . . . . . . . . . 61
4.2.2 Some key features of rogue wave evolution . . . . . . . . . . . . 61
4.3 The simulation of random ocean wave statistics . . . . . . . . . . . . . . 62
4.3.1 Introduction of JONSWAP spectrum . . . . . . . . . . . . . . . 62
4.3.2 Simulation results of random ocean wave . . . . . . . . . . . . . 64
III Connection between the NLS equation and the Painleve equations
77
5 Asymptotic problem: long-time solution behavior of NLS equation 79
5.1 Correlation between Painleve equations and NLS equation . . . . . . . . 80
5.2 Problem description and simulation results . . . . . . . . . . . . . . . . . 82
5.2.1 The long-time asymptotics for Dirichlet initial boundary value
(IBV) problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.2.2 The long-time asymptotics for two colliding initial value problem 85
5.2.3 The long-time asymptotics for a step-like initial value problem . 86
5.3 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6 Concluding remarks 99
iv
A Introduction of perfectly matched layer 101
B Introduction of filter 103
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

[1] J. B. Chen, M. Z. Qin, Y. F. Tang, Symplectic and multi-symplectic methods for the nonlinear Schrodinger equation, Computers and Mathematics with Applications, 43 (2002), 1095-1106.
[2] S. Reich, Multi-symplectic Runge-Kutta collocation methods for Hamiltonian wave equations, Journal Computational Physics, 157 (2000), 473-499.
[3] C. K. W. Tam, J. C. Webb, Dispersion-relation-preserving finite difference schemes for computational acoustics, Journal of Computational Physics, 107(1993), 262-281.
[4] F. Q. Hu, M. Y. Hussaini, J. L. Manthey, Low-dissipation and low-dispersion Runge-Kutta schemes for computational acoustics, Journal of Computational Physics, 124 (1996), 177-191.
[5] Q. Changa, E. Jiaa, W. Sunb, Difference Schemes for Solving the Generalized Nonlinear Schrodinger Equation, Journal of Computational Physics, 148 (2) (1999), 397-415.
[6] T. R. Taha and M. J. Ablowitz, Analytical and numerical aspects of certain nonlinear evolution equations. II. numerical, nonlinear Schrodinger equation, Journal of Computational Physics, 55 (2) (1984), 203-230.
[7] R. H. J. Grimshaw, A. Tovbis, Rogue waves: analytical predictions, Proceding of the Royal Society A, 469 (2157) (2013), 20130094.
[8] D. H. Peregrine, Water waves, nonlinear Schrodinger equations and their solutions, The Journal of the Australian Mathematical Society, 25 (1) (1983), 16-43.
[9] A. Kundu, A. Mukherjee, T. Naskar, Modelling rogue waves through exact dynamical lump soliton controlled by ocean currents, Proceding of the Royal Society A, 2014.
[10] C. Kharif, E. Pelinovsky, Physical mechanisms of the rogue wave phenomenon, European Journal of Mechanics - B/Fluids, 22 (6) (2003), 603-634.
[11] K. Dysthe, H. E. Krogstad, P. Muller, Oceanic Rogue Waves, Annual Review of Fluid Mechanics, 40 (2008), 287-310.
[12] C. Kharif, E. Pelinovsky, A. Slunyaev, Rogue Waves in the Ocean, Springer Berlin Heidelberg, 2009.
[13] C. C. Young, C. H. Wu, An efficient and accurate non-hydrostatic model with embedded Boussinesq-type like equations for surface wave modeling, International Journal for Numerical Methods in Fluids, 60 (1) (2009), 27-53.
[14] C. C. Young, C. H. Wu, Nonhydrostatic modeling of nonlinear deep-water wave groups, Journal of Engineering Mechanics, 136 (2) (2010); 155-167.
[15] A. Osborne, Nonlinear Ocean Waves and the Inverse Scattering Transform, Vol. 97. International Geophysics, Academic Press, 2010.
[16] P. A. Dieft, X. Zhou, A steepest descent method for oscillatory Riemann-Hilbert problems, American Mathematical Society, 26 (1) (1992), 119-124.
[17] P. A. Dieft, X. Zhou, Long-time asymptotics for integrable systems. High order Theory, Communications in Mathematical pphysics, 165 (1994), 175-191.
[18] A. B. de monvel, A. Its, D. Shepelsky, Painleve-type asymptotics for the Camassa-Holm eqution, SIAM Journal on Mathematical Analysis, 42 (4) (2010), 1854-1873.
[19] P. A. Clarkson, The Painleve equation - Nonlinear Special Functions, Journal of Computational and Applied Mathematics, 153 (2003), 127-140.
[20] M. Boiti , F. Pempinelli, Nonlinear Schrodinger equation, Backlund transformations and Painleve transcendents, Nuovo Cimento B, 59 (1980), 40-58.
[21] K. Iwasaki, H. Kimura, S. Shimomura, M. Yoshida, From Gauss to Painleve: A Modern Theory of Special Functions (in Aspect of Mathematics), Vieweg-Verlag, Braunschweig, Germany, E 16, 1991.
[22] C.M. Cosgrove, G. Scoufis, Painleve classification of a class of differential equations of the second order and second degree, Studies in Applied Mathematics, 88 (1993), 25-87.
[23] C. Heitzinger, C. Ringhofer, A note on the symplectic integration of the nonlinear Schrodinger equation, Journal of Computational Electronics, 3 (1) (2004), 33-44.
[24] L. Lee, G. Lyng, I. Vankova, The Gaussian semiclassical soliton ensenble and numerical methods for the focusing nonlinear Schrodinger equation, Physica D, 241 (2012),1767-1781.
[25] G. Fibich, Self-focusing in the damped nonlinear Schrodinger equation, SIAM Journal on Applied Mathematics, 61 (15) (2001), 1680-1705.
[26] T. J. Bridges, Multi-symplectic structures and wave propagation, Mathematical Proceedings of the Cambridge Philsophical Society, 121 (1997), 147-190.
[27] X. S. Liu, Y. Y. Qi, J. F. He, P. Z. Ding, Recent progress in symplectic algorithm for use in quantum systems, Communications in Computational Physics, 2 (1) (2007), 1-53.
[28] X. S. Liu, L.W. Su, P. Z. Ding, Symplectic algorithm for use in computing the timeindependent Schrodinger equation, International Journal of Quantum Chemistry, 87 (1) (2002), 1-11.
[29] T. J. Bridges, S. Reich, Numerical methods for Hamiltonian PDEs, Journal of Physics A: Mathematical and General, 39 (2006), 5287-5320.
[30] X. Liu, Y. C. Sun, Y. Tang, Conservativity of symplectic methods for the Ablowitz-Ladik discrete nonlinear Schrodinger equation, Master Thesis, Chinese Academy of Science, 2004.
[31] V. E. Zakharov, A. B. Shabat, Interaction between solitons in a stable medium, Soviet Physics - JETP, 64 (1973), 1627-1639.
[32] A. Aydin, B. Barasozen, Symplectic and multi-symplectic methods for coupled nonlinear Schrodinger equations with periodic solutions, Computer Physics Communications, 177 (2007), 566-583.
[33] T. J. Bridges, S. Reich, Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserves symplecticity, Physics Letter A, 284 (2001), 184-193.
[34] E. Hairer, C. Lubich, G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations (2nd edition), Springer Series in Computational Mathematics, 31, 2006.
[35] J. M. Sanz-Serna, A. Portillo, A classical numerical integrators for wave-packet dynamics, Journal of Chemical Physics, 104 (6) (1996), 2349-2355.
[36] N. N. Yanenko, The Methods of Fractional Steps: The Solution of Problems of Mathematical Physics in Several Variables (M:Holt; edit), Springer-Verlag, New York, 1971.
[37] Y. M. Chen, H. J. Zhu, S. H. Song, Multi-symplectic splitting method for two-dimensional nonlinear Schrodinger equation, Communications in Theoretical Physics, 56 (2011), 617-622.
[38] Q. Chang, E. Jia, W. Sun, Difference schemes for solving the generalized nonlinear Schrodinger equation, Journal of Computational Physics, 148 (1999), 397-415.
[39] W. Bao, S. Jin, P. A. Markowich, On time-splitting spectral approximations for the Schrodinger equation in the semiclassical regime, Journal of Computational Physics, 175 (2002), 487-524.
[40] P. A. Markowich, P. Pietra, C. Pohl, Numerical approximation of quadratic observables of Schrodinger equations in the semi-classical limit, Numerische Mathematik, 81 (1999), 595-630.
[41] S. Jin, C. D. Levermore, D.W. McLaughlin, The behavior of solutions of the NLS equation in the semiclassical limit, in Singular Limits of Dispersive waves (Plenum, New York, Londan), 1994.
[42] J. C. Bronski, D. W. McLaughlin, Semiclassical behavior in the NLS equation: OPtical shocks-focusing instabilities, in Singular Limits of Dispersive waves (Plenum, New York, Londan), 1994.
[43] S. Jin, C. D. Levermore, D.W. McLaughlin, The semiclassical limit of defocusing NLS hierarchy, Communications on Pure and Applied Mathematics 52 (5) (1999), 613-654.
[44] A. Ankiewicz, P. A. Clarkson, N. Akhmediev, Rogue waves, rational solutions, the patterns of their zeros and integral relations, Journal of Physics A: Mathematical and Theoretical, 43 (12) (2010), 122002(9pp).
[45] G. B. Whitham, Non-linear dispersion of water waves, Journal of Fluid Mechanics, 27 (2) (1967), 399-412.
[46] V. H. Chu, C. C. Mei, The non-linear evolution of Stokes waves in deep water, Journal of Fluid Mechanics, 47 (2) (1970), 337-351.
[47] H. Hasimoto, H. Ono, Nonlinear modulation of gravity waves, Journal of the Physical Society of Japan, 33 (3) (1972), 805-811.
[48] F. Dias, C. Kharif, Nonlinear gravity and capillary-gravity waves, Annual Review of Fluid Mechanics, 31 (1999), 301-346.
[49] N. K. Vitanov, Deep-water waves: On the nonlinear Schrodinger equation and its solutions, Journal of Theoretical and Applied Mechanics, 43 (2) (2013), 43-54.
[50] N. Akhmediev, A. Ankiewicz, M. Taki, Waves that appear from nowhere and disappear without a trace, Physics Letters A, 373 (2006), 675-678.
[51] G. J. Komen, L. Cavaleri, M. Donelan, K. Hasselmann, S. Hasselmann, P. A. E. M. Janssen, Dynamics and modelling of ocean waves, in Cambridge University Press, New York, 1994.
[52] K. Hasselmann, T. P. Barnett,E. Bouws, H. Carlson, D. E. Cartwright, K. Enke, J. A. Ewing, H. Gienapp, D. E. Hasselmann, P. Kruseman, A. Meerburg, P. Muller, D. J. Olbers, K. Richter, W. Sell, H. Walden, Measurements of wind-wave growth and swell decay during the Joint North Sea Wave Project (JONSWAP), Deutches Hydrographisches Institut, A. (8) (12) (1973), 95 pp.
[53] V. E. Zakharov, A. B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional of waves in nonlinear media, Journal of Experimental and Theoretical Physics, 34 (1972), 62-69.
[54] F. Bureau, Equations differentielles du second ordre en Y et du second degre en &;#376; dont l’integrale generale est a points critiques fixes, Annali di Matematica Pura ed Applicata, 91 (1972), 163-281.
[55] J. Chazy, Sur les equations differentielles du troisieme ordre et d’ordre superieur dont l’integrale generale a ses points critiques fixes, Acta Math, 34 (1911), 317-385.
[56] L. Gagnon, B. Grammaticos, A. Ramani, P. Winternitz, Lie symmetries of a generalised nonlinear Schrodinger equation: III. Reductions to third-order ordinary differential equations, Journal of Physics A: Mathematical and General, 22 (1989), 499-509.
[57] E.L. Ince, Ordinary Differential Equations, Dover, New York, (1956).
[58] A. B. de Monvel, V. P. Kotlyarov, D. Shepelsky, C. Zheng, Initial boundary value problems for integrable systems: towards the long time asymptotics, Nonlinearity, 23 (10) (2010), 2483-2499.
[59] R. Buckingham, S. Venakides, Long-time asymptotics of the nonlinear Schrodinger equation shock problem, Communications on Pure and Applied Mathematics, 60 (9) (2007), 1349-1414.
[60] A. B. de Monvel, V. P. Kotlyarov, D. Shepelsky, Focusing NLS equation: long-time dynamics of step-like initial data, International Mathematics Research Notices, 2011 (7) (2010), 1613-1653.
[61] C. Zheng, A perfectly matched layer approach to the nonlinear Schrodinger wave equations, Journal of Computational Physics, 227 (1) (2007), 537-556.
[62] C. Bogey, C. Bailly, A family of low dispersive and low dissipative explicit schemes for flow and noise computations, Journal of Computational Physics, 194 (2004), 194-214.
[63] A. Jameson, W. Schmidt, E. Turkel, Numerical solution of the Euler equations by finite volume methods using Runge-Kutta time stepping schemes, American Institute of Aeronautics and Astronautics Jounary, 1981-1259, 1981.
[64] M.R. Visbal, D.V. Gaitonde, High-order-accurate methods for complex unsteady subsonic flows, American Institute of Aeronautics and Astronautics Jounary, 37 (10) (1999), 1231-1239.
[65] C.A. Kennedy, M.H. Carpenter, Several new numerical methods for compressible shear-layer simulations, Applied Numerical Mathematics, 14 (2) (1994), 397-433.