Wave breaking in the short-pulse equation

Abstract

Sufficient conditions for wave breaking are found for the short-pulse equation describing wave packets of few cycles on the ultra-short pulse scale. The analysis relies on the method of characteristics and conserved quantities of the short-pulse equation and holds both on an infinite line and in a periodic domain. Numerical illustrations of the finite-time wave breaking are given in a periodic domain.

Wave-Breaking and Peakons for a Modified Camassa–Holm Equation

“…Well-posedness and wave breaking for the short pulse equation have been studied in [54] and [43], respectively. It was shown in [5] that the short pulse equation and the WKI equation [58] u t = u xx…”

Wellposedness of bounded solutions of the non-homogeneous initial boundary for the short pulse equation

“…Well-posedness and wave breaking for the short pulse equation have been studied in [21] and [16], respectively.…”

Well-posedness and small data scattering for the generalized Ostrovsky equation

“…On the other hand, the short-pulse equation (SPE) proved to be extremely interesting from a mathematical point of view due to the existence of an infinite hierarchy of conserved quantities [10], an ingenious transformation that related it to the integrable sine-Gordon equation and illustrated its complete integrability [11] and which, in turn, allowed the calculation of explicit analytical solutions of loop-and of breather-form for this model [12]. More recently, on the analysis side, the global well-posedness question [13] and wave-breaking phenomena in this equation were studied [16], while interesting generalizations such as the regularized version of the SPE [17] and applications including the emergence of SPE in descriptions of nonlinear metamaterials [18] have also emerged. Notice that while we are not aware of applications presently of this equation for p 4, we will consider the equation in its generalized form presented above, keeping our results as general as possible in what follows.…”