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Dynamical understanding of loop soliton solution for several nonlinear wave equations. (English) Zbl 1139.35076
The paper addresses the so-called loop solutions which were recently found, in an exact analytical form, in the Vakhnenko equation,
\[ u_{xt} + (uu_x)_x + u = 0, \] short-pulse equation (which may have some applications to nonlinear optics),
\[ u_{xt} = u + (1/6) (u^3)_{xx}, \] and in an extended Korteweg-de Vries equation. Actually, these are solutions to the respective ordinary differential equations (ODEs) obtained by substitution \(u = u(x - ct)\). The present paper analyzes the phase space of the ODE corresponding to each of the above-mentioned PDEs. The main conclusion is that, in each case, the loop solution is not a true single solution, but rather a combination of three different solutions, each of them beginning or ending at a fixed point of the saddle type.

MSC:
35L70 Second-order nonlinear hyperbolic equations
35Q53 KdV equations (Korteweg-de Vries equations)
35Q51 Soliton equations
35C05 Solutions to PDEs in closed form
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[1] Vakhnenko V O. High-frequency soliton-like waves in a relaxing medium. J Math Phys, 40: 2011–2020 (1999) · Zbl 0946.35094 · doi:10.1063/1.532847
[2] Vakhnenko V O, Parkes E J. The two loop soliton solution of the Vakhnenko equation. Nonlinearity, 11: 1457–1464 (1998) · Zbl 0914.35115 · doi:10.1088/0951-7715/11/6/001
[3] Morrison T P, Parkes E J, Vakhnenko V O. The N-loop soliton solution of the Vakhnenko equation. Nonlinearity, 12: 1427–1437 (1999) · Zbl 0935.35129 · doi:10.1088/0951-7715/12/5/314
[4] Morrison T P, Parkes E J. The N-loop soliton solution of the modified Vakhnenko equation (a new nonlinear evolution equation). Chaos, Solitons and Fractals, 16: 13–26 (2003) · Zbl 1048.35104 · doi:10.1016/S0960-0779(02)00314-4
[5] Sakovich A, Sakovich S. Solitary wave solutions of the short pulse equation. J Phys A Math Gen, 39: L361–367 (2006) · Zbl 1092.81531 · doi:10.1088/0305-4470/39/22/L03
[6] Schafer T, Wayne C E. Propagation of ultra-short opical pulses in cubic nonlinear media. Physica D, 196: 90–105 (2004) · Zbl 1054.81554 · doi:10.1016/j.physd.2004.04.007
[7] Tzirtzilakis E, Marinakis V, Apokis C, Bountis T. Soliton-like solutions of higher order wave equations of the Korteweg-de-Vries Type. J Math Phys, 43: 6151–6161 (2002) · Zbl 1060.35127 · doi:10.1063/1.1514387
[8] Tzirtzilakis E, Xenos M, Marinakis V, Bountis T. Interactions and stability of solitary waves in shallow water. Chaos, Solitons and Fractals, 14: 87–95 (2002) · Zbl 1068.76011 · doi:10.1016/S0960-0779(01)00211-9
[9] Fokas A S. On class of physically important integrable equations. Physica D, 87: 145–150 (1995) · Zbl 1194.35363 · doi:10.1016/0167-2789(95)00133-O
[10] Li J B, Wu J H, Zhu H P. Travelling waves for an Integrable Higher Order KdV Type Wave Equations. International Journal of Bifurcation and Chaos, 16(8): 2235–2260 (2006) · Zbl 1192.37100 · doi:10.1142/S0218127406016033
[11] Li J B, Pai H H. On the Study of Sigular Nonlinear Travelling Wave Equations: Dynamical Syotem Appwach. Beijing: Science Press, 2007
[12] Byrd P F, Fridman M D. Handbook of Elliptic Integrals for Engineers and Sciensists. Berlin: Springer, 1971
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