# zbMATH — the first resource for mathematics

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
##### Keywords:
Vakhnenko equation; short-pulse equation; reduction to ODEs
Full Text:
##### References:
 [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.