The short pulse equation: Bäcklund transformations and applications

Abstract

A Bäcklund transformation (BT), which involves both independent and dependent variables, is established and studied for the short pulse (SP) equation. Based it, the nonlinear superposition formulae for 2-, 3-, and 4-BT are presented. The general result for the composition of-BTs is achieved and given in terms of determinants. As applications, various solutions including loop solitons, breather solutions, and their interaction solutions are worked out.

On Construction of Solutions of Linear Differential Systems with Argument Deviations of Mixed Type

“…The function u satisfies the integral Equation ( 15) and, in particular, is absolutely continuous. By (11) and (13), Equation ( 15) is rewritten as…”

“…The techniques are rather flexible and can be used in relation to other problems. Although the approach is not explicitly designed for partial differential equations, it still might be used in the cases where systems of equations with one independent variable arise (e.g., equations obtained when a symmetry reduction is possible, equations related to the inverse scattering transform or discretization [10,11]). Unlike methods used for integrable systems (e.g., the Hirota bilinear method [12]), the technique is aimed at getting approximate solutons without previous knowledge of the structure of the solution set.…”

“…Furthermore, better approximations are obtained when we proceed to higher order iterations according to formula (14). For problem (68), (69), by virtue of (11), this formula means that the functions u m = col(u m,1 , u m,2 , u m,3 ), m ≥ 1, are constructed according to the recurrence relations…”

Backlund transformation of the Geng-Xue system

“…Recently, Rasin and Schiff discussed BT for the CH equation with the help of reciprocal transformation and concluded that it involves not only the dependent variables but also the independent spatial variables [2]. Later on Mao, Liu et al construct BTs for the DP, the Novikov and the short pulse equations [31,32,33,34]. As far as we know, there is no results on the BT of the GX system.…”

Bäcklund transformation and nonlinear superposition formula for the two-component short pulse equation

“…which was proposed as a nonlinear differential equation model to describe the propagation of ultra-short pulses in isotropic optical fibers [1,2]. Similar to other completely integrable partial differential equations, the properties of the SP equation such as Lax pair, bi-Hamiltonian structure, Hirota bilinear method, Darboux transformation, Bäcklund transformation, Riemann-Hilbert approach and so on have been studied from various perspectives [3][4][5][6][7][8][9][10][11][12][13]. In addition, since the stability of the solution is vital to physical consideration, there are several related research works for SP equation (see [14,15] and the references therein).…”