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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

show all references

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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