On exact solutions of the Schäfer–Wayne short pulse equation: WKI eigenvalue problem

Abstract

We consider a new equation recently found by Schäfer and Wayne, hereafter named Schäfer–Wayne short pulse equation (SWSPE), describing the propagation of an ultrashort pulse in nonlinear media. Using some vanishing boundary conditions, we construct and discuss the N-soliton solutions to the previous equation by means of the Wadati–Konno–Ichikawa (WKI) method, which is arguably more direct than the map through the sine-Gordon equation investigated much earlier by Sakovich and Sakovich (2005 J. Phys. Soc. Japan 74 239, 2006 J. Phys. A: Math. Gen. 39 L361). We particularly focus our attention on the two-soliton solution as an application. As a result, the collision process for two-soliton solutions with ‘similar’ amplitudes exhibits very different behaviour from the case when the amplitudes are ‘dissimilar’.

Towards a Unified Method for Exact Solutions of Evolution Equations. An Application to Reaction Diffusion Equations with Finite Memory Transport

“…Thus methods which are not equivalent to the GMM are the "bridge method" [49][50][51][52][53], which gives rise to the existence of parametric solutions of evolution equations, the homogeneous balance method [10][11][12][13][14] for finding exact solutions of (1.1) when it depends explicitly on x and t . In this respect the GMM method will be extended to treat equations with variable coefficients in a future work.…”

“…Very recently a "bridge transformation" had been first introduced in [49]. It had been simplified and used in [50] to connect the short pulse equation, which is completely integrable, to the sine-Gordon equation (see also [51][52][53]). By this "bridge method" the independent variables in (1.1) are transformed to new ones, namely…”

Symmetry reductions and conservation laws of the short pulse equation

“…Under this algebra (1), in terms of invariance variables , reduces to the ODE which after rearrangement can be written as (11) where D = (d/dy). Now, we consider α = 0, for which (11) reduces to (12) which has the particular solution . Thus in terms of its variables, the corresponding solution of the SPE (1) turns out to be (13) The time dependent solution (13) is independent of β.…”

“…The short pulse equation (SPE) serves as a basic governing equation for many physical/mathematical models describing different processes in many scientific areas [4][5][6][7]. Several authors have analyzed the SPE in different contexts by utilizing various methods and obtained diverse classes of solutions [8][9][10][11][12][13][14][15][16][17][18][19]. References [11,15] have obtained traveling wave solutions.…”

On the Variational Approach for Analyzing the Stability of Solutions of Evolution Equations

“…Different approaches had been proposed which are based on various aspects; Approaches of topological (or geometrical structure) invariance of solutions (due to small disturbances when linearizing an EE) [12,13,14,20].…”