A generalised short pulse equation: Darboux transformation and exact solutions
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A generalised short pulse equation: Darboux transformation and exact solutions CHENDI ZHU and LIHUA WU∗ Fujian Province University Key Laboratory of Computational Science, School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, Fujian, People’s Republic of China ∗ Corresponding author. E-mail: [email protected] MS received 20 March 2020; revised 12 May 2020; accepted 1 July 2020 Abstract. We construct infinitely many conservation laws of the generalised short pulse equation with the help of its Lax pairs. By a reciprocal transformation, the generalised short pulse equation was transformed to the first negative flow of Sawada–Kotera hierarchy. On the basis of Darboux and reciprocal transformations, we obtain some exact solutions of the generalised short pulse equation. Keywords. Generalised short pulse equation; conservation laws; Darboux transformation; exact solutions. PACS Nos 02.30.Ik; 04.20.Jb; 05.45.Yv
1. Introduction The integrable short pulse equation u xt = u + 16 (u 3 )x x
(1)
was first discovered in differential geometry to describe pseudospherical surfaces [1,2]. Later, it was found again in physics to model the propagation of ultra-short infrared light pulses in silica optical fibres [3,4]. After that, various aspects of the short pulse equation have been studied, including Lax pair [5], bi-Hamiltonian structure [6], solitons [7,8], generalisations [9,10] and many others. Recently, by considering the classification of integrable generalised short pulse (GSP) equations with quadratic and cubic nonlinear terms, Hone et al [11] found two new integrable GSP equations. One of them is u xt = u + (u 2 − 4u 2 u x )x
(2)
which was shown to have reciprocal link with the first negative flow of Sawada–Kotera hierarchy ((S K )−1 ). Using the direct method, Matsuno [12] obtained parametric solutions of this GSP equation. As is well-known, Darboux transformation [13,14] is one of the most direct and powerful methods for finding the solutions of nonlinear integrable systems. The purpose of the present 0123456789().: V,-vol
paper is to apply the reciprocal and Darboux transformations to study the exact solutions of the GSP equation (2) under the zero boundary condition as |x| → ∞. The paper is organised as follows. In §2, we establish infinite conservation laws of the GSP equation. In §3, resorting to the reciprocal transformation, we relate the GSP equation to (S K )−1 equation. In §4, by virtue of the Darboux transformation of (S K )−1 equation, reciprocal transformation and the asymptotic behaviours of wave functions, we arrive at exact solutions of the GSP equation. Section 5 gives conclusions.
2. Infinite conservation laws In this section, we shall construct infinite conservation laws of the GSP equation. To this end, we present the 3 × 3 matrix Lax pairs of the GSP equation (2) [11] ⎞ 0 0 1−2u x 0 0 1 + 4u x ⎠ ψ, ψx =⎝ λ(1−2u x ) 0 0 ⎛ ⎞ ψ1 ψ= ⎝ψ2 ⎠ , ψ3 ⎛
(3)
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⎞ 1/3λ 0 −4u 2 (1−2u x ) ψt =⎝ 1/3 2u −4u 2 (1 + 4u x )⎠ψ. 2 −4λu
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