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 在這篇論文，我們探討線性BBM和KdV方程對於初始邊界問題解的週期性。 根據[10]的方法，我們對線性BBM和KdV方程作奇函數擴充及偶函數擴充並藉由傅利葉轉換求得BBM和KdV方程的解，最後並探討BBM和KdV方程的週期性。
 In this paper we study the eventual periodicity of solutions to the initial-boundary value problem for the linear generalized BBM-Burgers and KdV-Burgers equation in the quarter plane with periodic boundary data and force term. We derive a representation formula for solutions to the equations by the Fourier transform thought the odd and even extension. Finial, we rigorously establish the eventual periodicity of these solutions.
 中文摘要 ........................................................................................i英文摘要 .......................................................................................iiContents ........................................................................................iiiIntroduction ....................................................................................12. Linear generalized BBM equation ...............................................3 2.1 Representation of solutions to BBM equation .........................3 2.2 Eventual periodicity of BBM equation.....................................93. Linear generalized KdV equation...............................................11 3.1 Representation of solutions to KdV equation..........................11 3.2 Eventual periodicity of KdV equation.....................................15References.......................................................................................16Appendix.........................................................................................17
 [1] C.J. Amick, J.L. Bona and M.E. Schonbek, Decay of solutions of some nonlinear wave equations, J. Di . Eq. 81 (1989), 1-49.[2] T. B. Benjamin, J. L. Bona, J. J. Mahony, Model Equations for Long Waves in Nonlinear Dispersive Systems, Philos. Royal Soc. London Series A, 272, (1972), 47-78.[3] B. Boczar-Karakiewicz, J.L. Bona and D. Cohen, Interaction of shallow-water waves and bottom topography, In Dynamical problems in continuum physics, IMA Series in Mathematics and Its Aplications, 4, Springer-Verlag (1987), 131-176.[4] B. Boczar-Karakiewicz, J.L. Bona and B. Pelchat, Interaction of internal waves with the sea bed on continental shelves, Continental Shelf Res. 11 (1991), 1181-1197.[5] J.L. Bona, H. Chen, S. Sun and B. Zhang, Comparison of quarter-plane and two-point boundary value problems: the BBM-equation, Discrete Contin. Dyn. Sys. 13 (2005), 921-940.[6] J.L. Bona and L. Luo, Initial-boundary value problems for model equations for the propagation of long waves, in: G. Gerrayra, G. Goldstein and F. Neubrander (Ed.), Lecture Notes in Pure and Appl. Math., 168, Dekker, New York, (1995), 65-94.[7] J.L. Bona, W.G. Pritchard and L.R. Scott, An evaluation of a model equation for water waves, Philos. Royal Soc. London Series A 302 (1981), 457-510.[8] J.L. Bona, S. Sun and B.-Y. Zhang, Forced oscillations of a damped Korteweg-de Vries equation in a quarter plane, Commun. Contemp. Math. 5 (2003), 369-400.[9] J.L. Bona and J.Wu, Temporal growth and eventual periodicity for dispersive wave equations in a quarter plane, to appear in Discrete Contin. Dyn. Sys.[10] J. M. Hong, J. Wu and J.-M. Yuan, Explicit solution representation for the BBM equation in a quarter plane and the eventual periodicity.[11] E.H. Lieb and M. Loss, Analysis (second edition), American Mathematical Society, Providence, RI (2001).[12] J. Shen, J. Wu and J.-M. Yuan, Eventual periodicity for the KdV equation on a half-line, Physica D 227 (2007), 105-119.
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