# American Institute of Mathematical Sciences

December  2014, 34(12): 5181-5209. doi: 10.3934/dcds.2014.34.5181

## Multi-hump solutions of some singularly-perturbed equations of KdV type

 1 Department of Mathematics, Korea University, Seoul, South Korea 2 Hyein Engineering and Construction, Changwon Kyungnam, South Korea 3 Coastal Development and Ocean Energy Research Department, Korea Institute of Ocean Science and Technology, Ansan, South Korea 4 Department of Mathematics, Virginia Tech, Blacksburg, VA 24061, United States 5 Department of Mathematics, Ajou University, Suwon, South Korea

Received  August 2013 Revised  May 2014 Published  June 2014

This paper studies the existence of multi-hump solutions with oscillations at infinity for a class of singularly perturbed 4th-order nonlinear ordinary differential equations with $\epsilon > 0$ as a small parameter. When $\epsilon =0$, the equation becomes an equation of KdV type and has solitary-wave solutions. For $\epsilon > 0$ small, it is proved that such equations have single-hump (also called solitary wave or homoclinic) solutions with small oscillations at infinity, which approach to the solitary-wave solutions for $\epsilon = 0$ as $\epsilon$ goes to zero. Furthermore, it is shown that for small $\epsilon > 0$ the equations have two-hump solutions with oscillations at infinity. These two-hump solutions can be obtained by patching two appropriate single-hump solutions together. The amplitude of the oscillations at infinity is algebraically small with respect to $\epsilon$ as $\epsilon \rightarrow 0$. The idea of the proof may be generalized to prove the existence of symmetric solutions of $2^n$-humps with $n=2,3,\dots,$ for the equations. However, this method cannot be applied to show the existence of general nonsymmetric multi-hump solutions.
Citation: J. W. Choi, D. S. Lee, S. H. Oh, S. M. Sun, S. I. Whang. Multi-hump solutions of some singularly-perturbed equations of KdV type. Discrete & Continuous Dynamical Systems, 2014, 34 (12) : 5181-5209. doi: 10.3934/dcds.2014.34.5181
##### References:
 [1] C. J. Amick and K. Kirchgässner, Solitary water-waves in the presence of surface tension, Arch. Rational Mech. Anal., 105 (1989), 1-49. doi: 10.1007/BF00251596.  Google Scholar [2] C. J. Amick and J. F. Toland, Solitary waves with surface tension $I$: Trajectories homoclinic to periodic orbits in four dimensions, Arch. Rational Mech. Anal., 118 (1992), 37-69. doi: 10.1007/BF00375691.  Google Scholar [3] J. T. Beale, Exact solitary water waves with capillary ripples at infinity, Comm. Pure Appl. Math., 44 (1991), 211-257. doi: 10.1002/cpa.3160440204.  Google Scholar [4] P. Bolle and B. Buffoni, Multibump homoclinic solutions to a centre equilibrium in a class of autonomous Hamiltonian systems, Nonlinearity, 12 (1999), 1699-1716. doi: 10.1088/0951-7715/12/6/317.  Google Scholar [5] B. Buffoni, Infinitely many large amplitude homoclinic orbits for a class of autonomous Hamiltonian systems, J. Differential Equations, 121 (1995), 109-120. doi: 10.1006/jdeq.1995.1123.  Google Scholar [6] B. Buffoni, A. R. Champneys and J. F. Toland, Bifurcation and coalescence of a plethora of homoclinic orbits for a Hamiltonian system, J. Dynam. Differential Equations, 8 (1996), 221-279. doi: 10.1007/BF02218892.  Google Scholar [7] B. Buffoni and M. D. Groves, A multiplicity result for solitary gravity-capillary waves in deep water via critical-point theory, Arch. Ration. Mech. Anal., 146 (1999), 183-220. doi: 10.1007/s002050050141.  Google Scholar [8] B. Buffoni, M. D. Groves and J. F. Toland, A plethora of solitary gravity-capillary water waves with nearly critical Bond and Froude numbers, Philos. Trans. Roy. Soc. London Ser. A, 354 (1996), 575-607. doi: 10.1098/rsta.1996.0020.  Google Scholar [9] A. R. Champneys and M. Groves, A global investigation of solitary-wave solutions to a two-parameter model for water waves, J. Fluid Mech., 342 (1997), 199-229. doi: 10.1017/S0022112097005193.  Google Scholar [10] A. R. Champneys and J. F. Toland, Bifurcation of a plethora of multi-modal homoclinic orbits for autonomous Hamiltonian systems, Nonlinearity, 6 (1993), 665-721. doi: 10.1088/0951-7715/6/5/002.  Google Scholar [11] F. Dias and G. Iooss, Water-waves as a spatial dynamical system, in Handbook of mathematical fluid dynamics, North-Holland, Amsterdam, II (2003), 443-499. doi: 10.1016/S1874-5792(03)80012-5.  Google Scholar [12] M. D. Groves and B. Sandstede, A plethora of three-dimensional periodic travelling gravity-capillary water waves with multipulse transverse profiles, J. Nonlinear Sci., 14 (2004), 297-340. doi: 10.1007/BF02666024.  Google Scholar [13] J. Hunter and J. Scheurle, Existence of perturbed solitary wave solutions to a model equation for water waves, Phys. D, 32 (1988), 253-268. doi: 10.1016/0167-2789(88)90054-1.  Google Scholar [14] G. Iooss and M. C. Pérouème, Perturbed homoclinic solutions in reversible 1:1 resonance vector fields, J. Diff. Equ., 102 (1993), 62-88. doi: 10.1006/jdeq.1993.1022.  Google Scholar [15] R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 455 (2002), 63-82. doi: 10.1017/S0022112001007224.  Google Scholar [16] E. Lombardi, Homoclinic orbits to exponentially small periodic orbits for a class of reversible systems. Application to water waves, Arch. Rat. Mech. Anal., 137 (1997), 227-304. doi: 10.1007/s002050050029.  Google Scholar [17] E. Lombardi, Oscillatory Integrals And Phenomena Beyond All Algebraic Orders, With Applications to Homoclinic Orbits in Reversible Systems, Lecture Notes in Mathematics, 1741, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0104102.  Google Scholar [18] Y. Pomeau, A. Ramani and B. Grammaticos, Structural stability of the Korteweg-de Vries solitons under a singular perturbation, Phys. D, 31 (1988), 127-134. doi: 10.1016/0167-2789(88)90018-8.  Google Scholar [19] R. L. Sachs, Bifurcation for semi-linear elliptic problems on an infinite strip via the Nash-Moser technique, in Analysis, et. cetere (Eds P. H. Rabinowitz and E. Zehnder), Academic Press, (1990), 563-572.  Google Scholar [20] S. M. Sun, Existence of a generalized solitary wave solution for water with positive Bond number smaller than 1/3, J. Math. Anal. Appl., 156 (1991), 471-504. doi: 10.1016/0022-247X(91)90410-2.  Google Scholar [21] S. M. Sun, On the oscillatory tails with arbitrary phase shift for solutions of the perturbed Korteweg-de Vries equation, SIAM J. Appl. Math., 58 (1998), 1163-1177. doi: 10.1137/S0036139996299212.  Google Scholar [22] S. M. Sun and M. C. Shen, Exponentially small estimate for the amplitude of capillary ripples of a generalized solitary wave, J. Math. Anal. Appl., 172 (1993), 533-566. doi: 10.1006/jmaa.1993.1042.  Google Scholar [23] S. M. Sun and M. C. Shen, Solitary waves in a two-layer fluid with surface tension, SIAM J. Math. Anal., 24 (1993), 866-891. doi: 10.1137/0524054.  Google Scholar [24] S. M. Sun and M. C. Shen, Exponentially small estimate for a generalized solitary wave solution to the perturbed K-dV equation, Nonlinear Anal., 23 (1994), 545-564. doi: 10.1016/0362-546X(94)90093-0.  Google Scholar

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##### References:
 [1] C. J. Amick and K. Kirchgässner, Solitary water-waves in the presence of surface tension, Arch. Rational Mech. Anal., 105 (1989), 1-49. doi: 10.1007/BF00251596.  Google Scholar [2] C. J. Amick and J. F. Toland, Solitary waves with surface tension $I$: Trajectories homoclinic to periodic orbits in four dimensions, Arch. Rational Mech. Anal., 118 (1992), 37-69. doi: 10.1007/BF00375691.  Google Scholar [3] J. T. Beale, Exact solitary water waves with capillary ripples at infinity, Comm. Pure Appl. Math., 44 (1991), 211-257. doi: 10.1002/cpa.3160440204.  Google Scholar [4] P. Bolle and B. Buffoni, Multibump homoclinic solutions to a centre equilibrium in a class of autonomous Hamiltonian systems, Nonlinearity, 12 (1999), 1699-1716. doi: 10.1088/0951-7715/12/6/317.  Google Scholar [5] B. Buffoni, Infinitely many large amplitude homoclinic orbits for a class of autonomous Hamiltonian systems, J. Differential Equations, 121 (1995), 109-120. doi: 10.1006/jdeq.1995.1123.  Google Scholar [6] B. Buffoni, A. R. Champneys and J. F. Toland, Bifurcation and coalescence of a plethora of homoclinic orbits for a Hamiltonian system, J. Dynam. Differential Equations, 8 (1996), 221-279. doi: 10.1007/BF02218892.  Google Scholar [7] B. Buffoni and M. D. Groves, A multiplicity result for solitary gravity-capillary waves in deep water via critical-point theory, Arch. Ration. Mech. Anal., 146 (1999), 183-220. doi: 10.1007/s002050050141.  Google Scholar [8] B. Buffoni, M. D. Groves and J. F. Toland, A plethora of solitary gravity-capillary water waves with nearly critical Bond and Froude numbers, Philos. Trans. Roy. Soc. London Ser. A, 354 (1996), 575-607. doi: 10.1098/rsta.1996.0020.  Google Scholar [9] A. R. Champneys and M. Groves, A global investigation of solitary-wave solutions to a two-parameter model for water waves, J. Fluid Mech., 342 (1997), 199-229. doi: 10.1017/S0022112097005193.  Google Scholar [10] A. R. Champneys and J. F. Toland, Bifurcation of a plethora of multi-modal homoclinic orbits for autonomous Hamiltonian systems, Nonlinearity, 6 (1993), 665-721. doi: 10.1088/0951-7715/6/5/002.  Google Scholar [11] F. Dias and G. Iooss, Water-waves as a spatial dynamical system, in Handbook of mathematical fluid dynamics, North-Holland, Amsterdam, II (2003), 443-499. doi: 10.1016/S1874-5792(03)80012-5.  Google Scholar [12] M. D. Groves and B. Sandstede, A plethora of three-dimensional periodic travelling gravity-capillary water waves with multipulse transverse profiles, J. Nonlinear Sci., 14 (2004), 297-340. doi: 10.1007/BF02666024.  Google Scholar [13] J. Hunter and J. Scheurle, Existence of perturbed solitary wave solutions to a model equation for water waves, Phys. D, 32 (1988), 253-268. doi: 10.1016/0167-2789(88)90054-1.  Google Scholar [14] G. Iooss and M. C. Pérouème, Perturbed homoclinic solutions in reversible 1:1 resonance vector fields, J. Diff. Equ., 102 (1993), 62-88. doi: 10.1006/jdeq.1993.1022.  Google Scholar [15] R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 455 (2002), 63-82. doi: 10.1017/S0022112001007224.  Google Scholar [16] E. Lombardi, Homoclinic orbits to exponentially small periodic orbits for a class of reversible systems. Application to water waves, Arch. Rat. Mech. Anal., 137 (1997), 227-304. doi: 10.1007/s002050050029.  Google Scholar [17] E. Lombardi, Oscillatory Integrals And Phenomena Beyond All Algebraic Orders, With Applications to Homoclinic Orbits in Reversible Systems, Lecture Notes in Mathematics, 1741, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0104102.  Google Scholar [18] Y. Pomeau, A. Ramani and B. Grammaticos, Structural stability of the Korteweg-de Vries solitons under a singular perturbation, Phys. D, 31 (1988), 127-134. doi: 10.1016/0167-2789(88)90018-8.  Google Scholar [19] R. L. Sachs, Bifurcation for semi-linear elliptic problems on an infinite strip via the Nash-Moser technique, in Analysis, et. cetere (Eds P. H. Rabinowitz and E. Zehnder), Academic Press, (1990), 563-572.  Google Scholar [20] S. M. Sun, Existence of a generalized solitary wave solution for water with positive Bond number smaller than 1/3, J. Math. Anal. Appl., 156 (1991), 471-504. doi: 10.1016/0022-247X(91)90410-2.  Google Scholar [21] S. M. Sun, On the oscillatory tails with arbitrary phase shift for solutions of the perturbed Korteweg-de Vries equation, SIAM J. Appl. Math., 58 (1998), 1163-1177. doi: 10.1137/S0036139996299212.  Google Scholar [22] S. M. Sun and M. C. Shen, Exponentially small estimate for the amplitude of capillary ripples of a generalized solitary wave, J. Math. Anal. Appl., 172 (1993), 533-566. doi: 10.1006/jmaa.1993.1042.  Google Scholar [23] S. M. Sun and M. C. Shen, Solitary waves in a two-layer fluid with surface tension, SIAM J. Math. Anal., 24 (1993), 866-891. doi: 10.1137/0524054.  Google Scholar [24] S. M. Sun and M. C. Shen, Exponentially small estimate for a generalized solitary wave solution to the perturbed K-dV equation, Nonlinear Anal., 23 (1994), 545-564. doi: 10.1016/0362-546X(94)90093-0.  Google Scholar
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