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

Abstract

Abstract. The short pulse equation provides a model for the propagation of ultra-short light pulses in silica optical fibers. It is a nonlinear evolution equation. In this paper the wellposedness of bounded solutions for the inhomogeneous initial boundary value problem associated to this equation is studied.

A note on the non-homogeneous initial boundary problem for an Ostrovsky-Hunter type equation

“…This equation generalizes the Korteweg-deVries equation that corresponds to γ = 0. In [8,15], the authors proved the wellposedness of the entropy solution of the Cauchy problem for (13), while, following [7,38], in [6], the convergence of the solutions of (14) to the entropy solutions of (13) is proven.…”

“…Instead, if γ < 0, (13), known as the Vakhnenko equation [27,42], describes the short-wave perturbations in a relaxing medium when the equations of motion are closed by the dynamic equation of state (see [4,43]).…”

“…In [9,16,20], the authors study the well-posedness for the initial boundary value problem for (13). In particular, in [9], they prove the well-posedness of the entropy solution of non-homogeneous initial boundary value problem for (13), assuming on the initial datum (4) and (6), while, on the boundary datum they assume (10).…”

“…Observe that, if we have (13) with γ < 0, [9, Theorem 1.2] is not true, because, from a mathematical point of view, [9, Lemma 2.6] does not hold (see also Lemmas 2.4,3.7. ) In this paper, we improve the result of [9], when γ > 0, assuming on the boundary datum (7).…”

Well-Posedness Results for the Continuum Spectrum Pulse Equation

“…The Cauchy problem for (9) was studied in [42][43][44] in the context of energy spaces, [4,5,45,46] in the context of entropy solutions. The homogeneous initial boundary value problem was studied in [47][48][49][50]. Nonlocal formulations of (9) were analyzed in [15,51] and the convergence of a finite difference scheme proved in [52].…”