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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.

35L70 Second-order nonlinear hyperbolic equations
35Q53 KdV equations (Korteweg-de Vries equations)
35Q51 Soliton equations
35C05 Solutions to PDEs in closed form
Full Text: DOI
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