**AUTHORS:**Chung-Hsien Tsai, Shy-Jen Guo

**Download as PDF**

**ABSTRACT:**
The objective of this paper is to construct bounded olutions of a model equation, which governs two
dimensional steady capillary gravity waves of an ideal fluid flow with Bond number near 1/3 and Froude
number close to one.

**KEYWORDS:**
Lyapunov’s Center Theorem, Schauder fixed point theorem , Bounded Solution.

**REFERENCES:**

[1] Amick, C.J. & Mcleod, J.B., ”A singular per urbation problem in waterwaves,” Stability Appl. Anal Continuous Media 1, 127 148 (1991)

[2] Amick, C.J. & Toland, J.F., ”Solitary waves with surface tension I: Trajectories homoclinic to periodic orbits in four dimensions,” Arch. Rat. Mech. Anal. 118, 37 69 (1992)

[3] Amick, C.J. & Toland, J.F., ”Homoclinic orbits in the dynamic phasespace analogy of an elastic strut,” Euro. Jnl of Applied Mathematics 3, 97 114 (1992)

[4] B. Buffoni, A.R. Champneys, A.R., Toland, J.F., ”Bifurcation and colescence of a plethora of homoclinic orbits for a Hamiltonian system,” Dyn. Differential Equations 8, 221 281 (1996)

[5] Champneys, A.R. ”Homoclinic orbits in reversible systems and their applications in mechanics, fluids and optics,” Physica D 112, 158 186 (1998)

[6] Coppel, W.A. ”Stability and Asymptotic Behavior of Differential equations,” D.C. HEATH AND COMPANY, (1965)

[7] Coppola, V.T. & Rand, R.H. ”Computer Algebra Implemention of Lie Transforms for Hamiltonian Systems: Application to the Nonlinear Stability of L4,” ZAMM.Z.angew. Math. Mech. 69, 275 284 (1989)

[8] Hunter J.K., and Vanden Broeck, J.M ., ”Solitary and periodic gravity capillary waves of finite amplitude,” . Fluid Mech. 134, 205 219 (1983)

[9] Iooss, G. & Kirchgässner, K. ”Water waves for small surface tension: an approach via norma form,” Proceedings of the Royal of Edinburgh 122A, 267 299 (1992)

[10] Korteweg, P.J. and de Vries, G. ”On the change of the form of long waves advancing in a rectangular canal, and on a new type of long stationary waves,” Phil. Mag. 39, 422 443 (1895)

[11] Markeev, A.P. ”Stability of a canonical system with two degrees of freedom in the presence of resonance,” Journals of Applied Mathematics and Mechanics 32, 766 722 (1968)

[12] Merkin, D.R. ”Intrduction to the Theory of Stability,” Springer Verlag, TAM 24, (1997)

[13] Meyer, K.R. & Hall, G.R. ”Intrduction to Hami ltonian Dynamical Systems and the N Body Problem,” Springer Verlag, 90, (1990)

[14] Peters, A.D.; Stoker, J.J. ”Solitary waves in liquids having non constant density.,” Comm. Pure Appl. Math. 13, 115 164 (1960)

[15] Sokol’skii, A.G. ”On the stability of an autono mous Hamiltonian system with two degrees of freedom in the case of equal frequencies,” J. Appl. Math.Mech., 38, 741 749 (1975)

[16] Tsai, Chung Hsien & Guo, Shy Jen ” The stability of zero solution of a model equation for steady capillary gravity waves with the bond number close to 1/3.,” Proceedings of the th WSEAS International Conference on FLUID MECHANICS (FLUIDS'09), pp.90 93, Ningbo, China anuary 10 12 2009

[17] Tsai, Chung Hsien & Guo, Shy Jen ” Unsymmetric Solitary Wave Solutions of a Fifth Order Model Equation for Steady Capillary Gravity Waves over a Bump with the Bond Number Near 1/3 ” Proceedings of the th WSEAS International Conference on FLUID MECHANICS (FLUIDS'09 pp.94 99, Ningbo, China anuary 10 12 2009

[18] Tsai, Chung Hsien & Guo, Shy Jen ”Stability of Fixed Points of a Fifth Order Equation,” 18th International Conference on APPLIED MATHEMATICS (AMATH '13), pp.94 99, Budapest, Hungary December 89 93 (2013).

[19] Zufiria, J.A. ”Weakly nonlinear non symmetric gravity waves on water of finite depth,” J. fluid Mech. 180, 371 385 (1987)