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On a Keller-Segel system with logarithmic sensitivity and non-diffusive chemical
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 |
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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