Persistence of solitary wave solutions of singularly perturbed Gardner equation. (English) Zbl 1143.35359
Summary: The paper studies the singularly perturbed Gardner equation. Based on the relation between solitary wave solution and homoclinic orbits of the associated ordinary differential equations, the persistence of the solitary wave solution for the singularly perturbed Gardner equation is investigated using a geometric singular perturbation method. We show that the solitary wave solution exists when the perturbation parameter is sufficiently small.
MSC:
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
| 35Q51 | Soliton equations |
| 37K55 | Perturbations, KAM theory for infinite-dimensional Hamiltonian and Lagrangian systems |
Keywords:
singularly perturbed Gardner equation; solitary wave solutions; geometric singular perturbation methods; persistenceReferences:
| [1] | Shen, J. W.; Xu, W., Bifurcations of sooth and non-smooth traveling wave solutions of Degasperis-Procesi equation, Int J Nonlinear Sci Numer Simul, 5, 4, 397-402 (2004) · Zbl 1401.35276 |
| [2] | Shen, J. W.; Xu, W., Smooth and non-smooth traveling wave solutions in a nonlinearly dispersive Boussinesq equation, Chaos, Solition & Fractals, 23, 1, 117-130 (2005) · Zbl 1062.35088 |
| [3] | Shen, J. W., Traveling wave solutions in the generalized Hirota-Satsuma coupled KdV system, Appl Math Comput, 161, 2, 365-383 (2005) · Zbl 1092.35094 |
| [4] | Shen, J. W.; Xu, W., Bifurcations of sooth and non-smooth traveling wave solutions in the generalized Camassa-Holm equation, Chaos, Solition & Fractals, 26, 4, 1149-1162 (2005) · Zbl 1072.35579 |
| [5] | He, J. H., Variational principles for some nonlinear partial differential equations with variable coefficients, Chaos, Solition & Fractals, 19, 4, 847-851 (2004) · Zbl 1135.35303 |
| [6] | He, J. H., Application of homotopy perturbation method to nonlinear wave equations, Chaos, Solition & Fractals, 26, 3, 695-700 (2005) · Zbl 1072.35502 |
| [7] | He, J. H., Homotopy perturbation method for bifurcation of nonlinear problems, Int J Nonlinear Sci Numer Simul, 6, 2, 207-208 (2005) · Zbl 1401.65085 |
| [8] | Fan, X. H.; Tian, L. X., The existence of solitary waves of singularly perturbed mKdV-KS equation, Chaos, Solitons & Fractals, 26, 1111-1118 (2005) · Zbl 1072.35575 |
| [9] | Gianne, Derks; Arjen, Doelman; van Gils, Stephan A.; Timco, Visser, Travelling waves in a singularly perturbed sine-Gordon equation, Physica D, 180, 40-70 (2003) · Zbl 1040.35093 |
| [10] | Fenichel, N., Geometric singular perturbation theory for ordinary differential equations, J Differ Equations, 31, 53-98 (1979) · Zbl 0476.34034 |
| [11] | Jones CKRT. Geometric singular perturbation theory. Lecture Notes Math 1609, Springer-Verlag; 1994, 45-118.; Jones CKRT. Geometric singular perturbation theory. Lecture Notes Math 1609, Springer-Verlag; 1994, 45-118. · Zbl 0840.58040 |
| [12] | Akveld, M. E.; Hulshof, J., Travelling wave solutions of a fourth-order semilinear diffusion equation, Appl Math Lett, 11, 115-120 (1998) · Zbl 0932.35017 |
| [13] | Kyrychko, Y. N.; Bartuccell, M. V.; Blyuss, K. B., Persistence of travelling wave solutions of a fourth order diffusion system, J Comput Appl Math, 176, 433-443 (2005) · Zbl 1065.35035 |
| [14] | Guo, B.; Chen, H., Homoclinic orbit in a six-dimensional model of a perturbed order nonlinear SchrÖdinger equation, Commun Nonlinear Sci Numer Simul, 9, 4, 431-442 (2004) · Zbl 1055.35110 |
| [15] | Fu, Z. T.; Liu, S. D.; Liu, S. K., New kinds of solutions to Gardner equation, Chaos, Solitons & Fractals, 20, 301-309 (2004) · Zbl 1046.35097 |
| [16] | Camassa, R.; Kovacic, G.; Tin, S. K., A Melnikov method for homoclinic orbits with applications (1996), Springer: Springer New York |
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