\def\cydot{\leavevmode\raise.4ex\hbox{.}} \def\cprime{$'$}
\begin{thebibliography}{10}

\bibitem{Ar2000}
Enrico Arbarello.
\newblock Sketches of {K}d{V}.
\newblock In {\em Symposium in Honor of C. H. Clemens (Salt Lake City, UT,
  2000)}, volume 312 of {\em Contemp. Math.}, pages 9--69. Amer. Math. Soc.,
  Providence, RI, 2002.

\bibitem{Be1972}
T.~B. Benjamin.
\newblock The stability of solitary waves.
\newblock {\em Proc. Roy. Soc. (London) Ser. A}, 328:153--183, 1972.

\bibitem{BeLi1983}
H.~Berestycki and P.-L. Lions.
\newblock Nonlinear scalar field equations. {I}. {E}xistence of a ground state.
\newblock {\em Arch. Rational Mech. Anal.}, 82(4):313--345, 1983.

\bibitem{Bo1975}
J.~Bona.
\newblock On the stability theory of solitary waves.
\newblock {\em Proc. Roy. Soc. London Ser. A}, 344(1638):363--374, 1975.

\bibitem{BoDo95}
J.~L. Bona, V.~A. Dougalis, O.~A. Karakashian, and W.~R. McKinney.
\newblock Conservative, high-order numerical schemes for the generalized
  {K}orteweg-de {V}ries equation.
\newblock {\em Philos. Trans. Roy. Soc. London Ser. A}, 351(1695):107--164,
  1995.

\bibitem{BoDo96}
J.~L. Bona, V.~A. Dougalis, O.~A. Karakashian, and W.~R. McKinney.
\newblock The effect of dissipation on solutions of the generalized
  {K}orteweg-de {V}ries equation.
\newblock {\em J. Comput. Appl. Math.}, 74(1-2):127--154, 1996.
\newblock TICAM Symposium (Austin, TX, 1995).

\bibitem{BoSm1975}
J.~L. Bona and R.~Smith.
\newblock The initial-value problem for the {K}orteweg-de {V}ries equation.
\newblock {\em Philos. Trans. Roy. Soc. London Ser. A}, 278(1287):555--601,
  1975.

\bibitem{BoSo94}
J.~L. Bona and A.~Soyeur.
\newblock On the stability of solitary-waves solutions of model equations for
  long waves.
\newblock {\em J. Nonlinear Sci.}, 4(5):449--470, 1994.

\bibitem{BoDo86}
Jerry~L. Bona, Vassilios~A. Dougalis, and Ohannes~A. Karakashian.
\newblock Fully discrete {G}alerkin methods for the {K}orteweg-de {V}ries
  equation.
\newblock {\em Comput. Math. Appl. Ser. A}, 12(7):859--884, 1986.

\bibitem{BoDo91}
Jerry~L. Bona, Vassilios~A. Dougalis, Ohannes~A. Karakashian, and William~R.
  McKinney.
\newblock Fully-discrete methods with grid refinement for the generalized
  {K}orteweg-de {V}ries equation.
\newblock In {\em Viscous profiles and numerical methods for shock waves
  (Raleigh, NC, 1990)}, pages 1--11. SIAM, Philadelphia, PA, 1991.

\bibitem{BrJe2000}
J.~C. Bronski and R.~L. Jerrard.
\newblock Soliton dynamics in a potential.
\newblock {\em Math. Res. Lett.}, 7(2-3):329--342, 2000.

\bibitem{BuPe1992}
V.~S. Buslaev and G.~S. Perel{\cprime}man.
\newblock Scattering for the nonlinear {S}chr\"odinger equation: states that
  are close to a soliton.
\newblock {\em Algebra i Analiz}, 4(6):63--102, 1992.

\bibitem{BuSu2003}
Vladimir~S. Buslaev and Catherine Sulem.
\newblock On asymptotic stability of solitary waves for nonlinear
  {S}chr\"odinger equations.
\newblock {\em Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 20(3):419--475,
  2003.

\bibitem{CoKe2001}
J.~Colliander, M.~Keel, G.~Staffilani, H.~Takaoka, and T.~Tao.
\newblock Global well-posedness for {K}d{V} in {S}obolev spaces of negative
  index.
\newblock {\em Electron. J. Differential Equations}, pages No. 26, 7 pp.
  (electronic), 2001.

\bibitem{CoSt1999}
J.~Colliander, G.~Staffilani, and H.~Takaoka.
\newblock Global wellposedness for {K}d{V} below {$L\sp 2$}.
\newblock {\em Math. Res. Lett.}, 6(5-6):755--778, 1999.

\bibitem{CrGu2004}
W.~Craig, P.~Guyenne, D.P. Nicholls, and C.~Sulem.
\newblock Hamiltonian long wave expansions for water waves over a rough bottom.
\newblock {\em Proceedings of the Royal Society of London A, (to appear)}.

\bibitem{CrGr1994}
Walter Craig and Mark~D. Groves.
\newblock Hamiltonian long-wave approximations to the water-wave problem.
\newblock {\em Wave Motion}, 19(4):367--389, 1994.

\bibitem{CrSu2000}
Walter Craig and Catherine Sulem.
\newblock The water-wave problem and its long-wave and modulational limits.
\newblock In {\em Mathematical and numerical aspects of wave propagation
  (Santiago de Compostela, 2000)}, pages 14--23. SIAM, Philadelphia, PA, 2000.

\bibitem{DeZh93}
P.~Deift and X.~Zhou.
\newblock A steepest descent method for oscillatory {R}iemann-{H}ilbert
  problems. {A}symptotics for the {MK}d{V} equation.
\newblock {\em Ann. of Math. (2)}, 137(2):295--368, 1993.

\bibitem{DeJo2004}
S.~I. Dejak and B.~L.~G. Jonsson.
\newblock Long time dynamics of {mKdV} solitons (in preparation 2004).

\bibitem{FrGu2003}
J.~Fr\"{o}hlich, S.~Gustafson, B.L.G Jonsson, and I.M. Sigal.
\newblock Solitary wave dynamics in an external potential (in-print).
\newblock {\em Comm. Math. Phys.}, 2003.

\bibitem{FrTs2003}
J\"{u}rg Fro\"{o}hlich, Tai-Peng Tsai, and Horng-Tzer Yau.
\newblock On the point particle ({N}ewtonian) limit of the non-linear {H}artree
  equation.
\newblock {\em Comm. Math. Phys.}, 225(2):223--274, 2002.

\bibitem{GaSi2004}
Zhou Gang and I.M. Sigal.
\newblock Asymptotic stability of nonlinear {Sch\"{o}dinger} equation with
  potential ({Preprint, Toronto} 2004).

\bibitem{GrShSt87}
Manoussos Grillakis, Jalal Shatah, and Walter Strauss.
\newblock Stability theory of solitary waves in the presence of symmetry. {I}.
\newblock {\em J. Funct. Anal.}, 74(1):160--197, 1987.

\bibitem{GuSi2003}
S.~Gustafson and Sigal I.M.
\newblock {\em Mathematical Concepts of Quantum Mechanics}.
\newblock Springer-Verlag, New York, 2003.

\bibitem{JeKa1970}
Alan Jeffrey and Tsunehiko Kakutani.
\newblock Stability of the {B}urgers shock wave and the {K}orteweg-de {V}ries
  soliton.
\newblock {\em Indiana Univ. Math. J.}, 20:463--468, 1970/1971.

\bibitem{Jo1973}
R.~S. Johnson.
\newblock On the development of a solitary wave moving over an uneven bottom.
\newblock {\em Proc. Cambridge Philos. Soc.}, 73:183--203, 1973.

\bibitem{Ka1983}
Tosio Kato.
\newblock On the {C}auchy problem for the (generalized) {K}orteweg-de {V}ries
  equation.
\newblock In {\em Studies in applied mathematics}, volume~8 of {\em Adv. Math.
  Suppl. Stud.}, pages 93--128. Academic Press, New York, 1983.

\bibitem{KePoVe1993}
Carlos~E. Kenig, Gustavo Ponce, and Luis Vega.
\newblock Well-posedness and scattering results for the generalized
  {K}orteweg-de {V}ries equation via the contraction principle.
\newblock {\em Comm. Pure Appl. Math.}, 46(4):527--620, 1993.

\bibitem{KePoVe1996}
Carlos~E. Kenig, Gustavo Ponce, and Luis Vega.
\newblock A bilinear estimate with applications to the {K}d{V} equation.
\newblock {\em J. Amer. Math. Soc.}, 9(2):573--603, 1996.

\bibitem{Ke02}
Sahbi Keraani.
\newblock Semiclassical limit of a class of {S}chr\"odinger equations with
  potential.
\newblock {\em Comm. Partial Differential Equations}, 27(3-4):693--704, 2002.

\bibitem{KoDe1895}
D.J. Korteweg and F.~{de Vries}.
\newblock On the change of form of long waves advancing in a rectangular canal,
  and on a new type of long stationary waves.
\newblock {\em Philos. Mag.}, 39:422--443, 1895.

\bibitem{Mi1979}
John~W. Miles.
\newblock On the {K}orteweg-de\thinspace {V}ries equation for a gradually
  varying channel.
\newblock {\em J. Fluid Mech.}, 91(1):181--190, 1979.

\bibitem{ReSiI}
Simon Reed and Barry Simon.
\newblock {\em Methods of Modern Mathematical Physics, I. Functional Analysis}.
\newblock Academic Press, San Diego, second edition, 1980.

\bibitem{ReSiIV}
Simon Reed and Barry Simon.
\newblock {\em Methods of Modern Mathematical Physics, IV. Analysis of
  Operators}.
\newblock Academic Press, San Diego, 1980.

\bibitem{RoScSoPreprintII}
I.~Rodnianski, W.~Schlag, and Soffer A.
\newblock Asymptotic stability of n-soliton state of nls, arxiv:math.ap.

\bibitem{RoScSoPreprint}
I.~Rodnianski, W.~Schlag, and Soffer A.
\newblock Dispersive analysis of charge transfer models, arxiv:math.ap.

\bibitem{ScWa2000}
Guido Schneider and C.~Eugene Wayne.
\newblock The long-wave limit for the water wave problem. {I}. {T}he case of
  zero surface tension.
\newblock {\em Comm. Pure Appl. Math.}, 53(12):1475--1535, 2000.

\bibitem{SoWe90}
A.~Soffer and M.~I. Weinstein.
\newblock Multichannel nonlinear scattering for nonintegrable equations.
\newblock {\em Comm. Math. Phys.}, 133(1):119--146, 1990.

\bibitem{St1977}
Walter~A. Strauss.
\newblock Existence of solitary waves in higher dimensions.
\newblock {\em Comm. Math. Phys.}, 55(2):149--162, 1977.

\bibitem{TsYa2002III}
Tai-Peng Tsai and Horng-Tzer Yau.
\newblock Asymptotic dynamics of nonlinear {S}chr\"odinger equations:
  resonance-dominated and dispersion-dominated solutions.
\newblock {\em Comm. Pure Appl. Math.}, 55(2):153--216, 2002.

\bibitem{TsYa2002II}
Tai-Peng Tsai and Horng-Tzer Yau.
\newblock Relaxation of excited states in nonlinear {S}chr\"odinger equations.
\newblock {\em Int. Math. Res. Not.}, (31):1629--1673, 2002.

\bibitem{TsYa2002I}
Tai-Peng Tsai and Horng-Tzer Yau.
\newblock Stable directions for excited states of nonlinear {S}chr\"odinger
  equations.
\newblock {\em Comm. Partial Differential Equations}, 27(11-12):2363--2402,
  2002.

\bibitem{GrPu1993}
E.~van Groesen and S.~R. Pudjaprasetya.
\newblock Uni-directional waves over slowly varying bottom. {I}. {D}erivation
  of a {K}d{V}-type of equation.
\newblock {\em Wave Motion}, 18(4):345--370, 1993.

\bibitem{We1985}
M.I. Weinstein.
\newblock Modulational stability of ground states of nonlinear
  {{Schr\"{o}dinger}} equations.
\newblock {\em SIAM J. Math. Anal.}, 16(3):472--491, 1985.

\bibitem{Wi1974}
G.B. Whitham.
\newblock {\em Linear and Nonlinear Waves}.
\newblock Wiley, New York, 1974.

\bibitem{YoLi1994}
Sung~B. Yoon and Philip L.-F. Liu.
\newblock A note on {H}amiltonian for long water waves in varying depth.
\newblock {\em Wave Motion}, 20(4):359--370, 1994.

\end{thebibliography}