Solitary Wave Solutions of the Short Pulse Equation

An exact nonsingular solitary wave solution of the Schäfer–Wayne short pulse equation is derived from the breather solution of the sine-Gordon equation by means of a transformation between these two integrable equations. The short pulse equation (SPE), which has the form uxt = u+ 1 6 ( u ) xx (1) up to scale transformations of its variables, was derived by Schäfer and Wayne [1] as a model equation describing the propagation of ultra-short light pulses in silica optical fibers. In contrast to the nonlinear Schrödinger equation (NLSE) which models slowly varying wave trains, the SPE approximates Maxwell’s equations in the case when the pulse spectrum is not narrowly localized around the carrier frequency, that is when the pulse is as short as a few cycles of the central frequency. Numerical simulations made in [2] showed that the accuracy of the SPE approximation to Maxwell’s equations steadily increases as the pulse shortens, whereas the NLSE approximation fails to be accurate for ultra-short pulses. Like the NLSE, the SPE is an integrable nonlinear partial differential equation. Actually, the equation (1) appeared first a long time ago, as one of Rabelo’s equations possessing a zero-curvature representation with a parameter [3] (we are grateful to Prof. E.G. Reyes for this essential reference; see also [4]). Recently, the Lax pair of the SPE, of the Wadati–Konno–Ichikawa (WKI) type, was rediscovered in [5], where also the second-order recursion operator of the SPE was presented and the chain of transformations v(x, t) = ( u2x + 1 ) −1/2 ; x = w(y, t), v(x, t) = wy(y, t); z(y, t) = arccoswy (2) was constructed which relates the SPE (1) with the sine-Gordon equation (SGE) zyt = sin z. (3) Very recently, the bi-Hamiltonian structure of the SPE was discovered and the corresponding hierarchy and conservation laws were studied in [6, 7]. The soliton solutions of the NLSE have played an important role in the recent development of fiber-optic communications. For this reason, one would definitely like to have exact mathematical expressions for the ultra-short light pulses governed by the SPE. In the present paper, we derive exact solitary wave solutions of the SPE (1) from the well-known ones of the SGE (3) by means of the transformation (2) between these two integrable equations. 1 2 ANTON SAKOVICH AND SERGEI SAKOVICH Schäfer and Wayne [1] proved the nonexistence theorem that the SPE does not possess any solution representing a smooth localized pulse moving with constant shape and speed. Actually, it is easy to find the “wave of translation” solution of the SPE directly, in order to see that this solution is singular. Substituting u = u(ξ), ξ = x+ ct, c = constant (4) into (1), we get the second-order ordinary differential equation