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Some new soliton-like solutions and periodic wave solutions with loop or without loop to a generalized KdV equation. (English) Zbl 1203.35242

Summary: By using the integral bifurcation method, we study a generalized KdV equation which was first derived by Fokas from physical considerations via a methodology of Fuchssteiner. All kinds of soliton-like or kink-like wave solutions and periodic wave solutions with loop or without loop are obtained. Smooth compacton-like periodic wave solution and non-smooth periodic cusp wave solution are also obtained. Their dynamic properties are investigated and their profiles are given by mathematical software.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K50 Bifurcation problems for infinite-dimensional Hamiltonian and Lagrangian systems
35C08 Soliton solutions
35B10 Periodic solutions to PDEs
35-04 Software, source code, etc. for problems pertaining to partial differential equations
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[1] Fokas, A.S., On a class of physically important integrable equations, Physica D, 87, 145-450, (1995) · Zbl 1194.35363
[2] Fuchssteiner, B.; Fokas, A.S.; structures, Symplectic, Their backlund transformations and hereditary symmetries, Physica D: nonlinear phenomen., 4, 47-66, (1981) · Zbl 1194.37114
[3] Li, Z.; Sibgatullin, N.R., An improved theory of long waves on the water surface, J. appl. math. mech., 61, 177-482, (1997)
[4] Fuchssteiner, B., Application of hereditary symmetries to nonlinear evolution equations, Nonlinear anal., 3, 849-862, (1981) · Zbl 0419.35049
[5] Rosenau, P.; Hyman, J.M., Compactons: solitons with finite wavelength, Phys. rev. lett., 70, 564-567, (1993) · Zbl 0952.35502
[6] Li, J.; Liu, Z., Smooth and non-smooth travelling waves in a nonlinearly dispersive equation, Appl. math. model., 25, 41-56, (2000) · Zbl 0985.37072
[7] Long, Y.; He, B.; Rui, W.; Chen, C., Compacton-like and kink-like waves for a higher-order wave equation of korteweg – de Vries type, Int. J. comput. math., 83, 12, 959-971, (2006) · Zbl 1134.35096
[8] He, B.; Rui, W.; Li, S.; Chen, C., Bounded travelling wave solutions for a modified form of generalized degasperis – procesi equation, Appl. math. comput., 206, 1, 113-123, (2008) · Zbl 1163.35473
[9] Li, J.; Rui, W.; Long, Y.; He, B., Travelling wave solutions for a higher order wave equation of KdV type (III), Math. bios. eng., 3, 1, 125-135, (2006) · Zbl 1136.35449
[10] Long, Y.; Li, J.; Rui, W.; He, B., Travelling wave solutions for a higher order wave equation of KdV type, Appl. math. mech., 28, 11, 1455-1465, (2007) · Zbl 1231.35035
[11] Long, Y.; Rui, W.; He, B., Travelling wave solutions for a higher order wave equations of KdV type (I), Chaos soliton fract., 23, 469-475, (2005) · Zbl 1069.35075
[12] He, B.; Meng, Q.; Rui, W.; Long, Y., Bifurcations of travelling wave solutions for the mk(n,n) equation, Commun. nonlinear sci. numer. simulat., 13, 10, 2114-2123, (2008) · Zbl 1221.35336
[13] He, B.; Li, J.; Long, Y.; Rui, W., Bifurcations of travelling wave solutions for a variant of camassa – holm equation, Nonlinear anal.: real world appl., 9, 222-232, (2008) · Zbl 1185.35217
[14] Meng, Q.; He, B.; Long, Y.; Rui, W., Bifurcations of travelling wave solutions for a general sine – gordon equation, Chaos soliton fract., 29, 483-489, (2006) · Zbl 1099.35116
[15] Liu, Z.; Long, Y., Compacton-like wave and kink-like wave of GCH equation, Nonlinear anal.: real world appl., 8, 136-155, (2007) · Zbl 1106.35065
[16] Bi, Q., Wave patterns associated with a singular line for a bi-Hamiltonian system, Phys. lett. A, 369, 5-6, 407-417, (2007) · Zbl 1209.37073
[17] Rui, W.; He, B.; Long, Y.; Chen, C., The integral bifurcation method and its application for solving a family of third-order dispersive pdes, Nonlinear anal., 69, 1256-1267, (2008) · Zbl 1144.35461
[18] Rui, W.; Xie, S.; He, B.; Long, Y., Integral bifurcation method and its application for solving the modified equal width wave equation and its variants, Rostock. math. kolloq., 62, 87-106, (2007) · Zbl 1148.35079
[19] Yomba, E., The extended F-expansion method and its application for solving the nonlinear wave, CKGZ, GDS, DS GZ equations, Phys. lett. A, 340, 149-160, (2005) · Zbl 1145.35455
[20] Ma, Y.; Li, B.; Wang, C., A series of abundant exact travelling wave solutions for a modified generalized Vakhnenko equation using auxiliary equation method, Appl. math. comput., 211, 102-107, (2009) · Zbl 1400.35203
[21] Rui, W.; He, B.; Long, Y., The binary F-expansion method and its application for solving the (n+1)-dimensional sine – gordon equation, Commun. nonlinear sci. numer. simulat., 14, 1245-1258, (2009) · Zbl 1221.35360
[22] Zhang, S., A generalized auxiliary equation method and its application to the (2+1)-dimensional KdV equations, Appl. math. comput., 188, 1-6, (2007) · Zbl 1114.65355
[23] Rui, W.; Long, Y.; He, B.; Li, Z., Integral bifurcation method combined with computer for solving a higher order wave equation of KdV type, Int. J. comput. math., (2008)
[24] Rui, W.; Long, Y.; He, B., Some new travelling wave solutions with singular or nonsingular character for the higher order wave equation of KdV type (III), Nonlinear anal., 70, 11, 3816-3828, (2009) · Zbl 1167.34006
[25] Rui, W.; Long, Y., New periodic loop solitons of the generalized KdV equation, Int. J. nonlinear sci. numer. simulat., 9, 4, 441-444, (2008)
[26] He, J.H., Preliminary report on the energy balance for nonlinear oscillations, Mech. res. commun., 29, 107-111, (2002) · Zbl 1048.70011
[27] Long, Y.; Li, Z.; Rui, W., New travelling wave solutions for a nonlinearly dispersive wave equation of camassa – holm equation type, Appl. math. comput., 217, 1315-1320, (2010) · Zbl 1203.35235
[28] Long, Y.; Rui, W., Variable wave-form periodic solutions for the dispersive wave equation of a generalized camassa – holm equation, Int. J. nonlinear sci. numer. simulat., 10, 4, 469-474, (2009)
[29] He, B.; Rui, W.; Chen, C.; Li, S., Exact travelling wave solutions of a generalized camassa – holm equation using the integral bifurcation method, Appl. math. comput., 206, 1, 141-149, (2008) · Zbl 1163.35472
[30] Bressan, A.; Constantin, A., Global conservative solutions of the camassa – holm equation, Arch. ration. mech. anal., 183, 215-239, (2007) · Zbl 1105.76013
[31] Constantin, A.; Strauss, W.A., Stability of peakons, Commun. pure appl. math., 53, 303-310, (2000) · Zbl 1049.35149
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