Abstract
In this paper, we firstly deduce a reverse space-time Fokas–Lenells equation which can be derived from a rather simple but extremely important symmetry reduction of corresponding local equation. Next, the determinant representations of one-fold Darboux transformation and N-fold Darboux transformation are expressed in detail by special eigenfunctions of spectral problem. Depending on zero seed solution and nonzero seed solution, exact solutions, including bright soliton solutions, kink solutions, periodic solutions, breather solutions, rogue wave solutions and several types of mixed soliton solutions, can be presented. Furthermore, the dynamical behaviors are discussed through some figures. It should be mentioned that the solutions of nonlocal Fokas–Lenells equation possess new characteristics different from the ones of local case. Besides, we also demonstrate the integrability by providing infinitely many conservation laws. The above results provide an alternative possibility to understand physical phenomena in the field of nonlinear optics and related fields.
Highlights
In this paper, we firstly deduce a reverse space-time Fokas-Lenells equation which can be derived from a rather simple but extremely important symmetry reduction of corresponding local equation
The FL equation has been derived as a model to describe femtosecond pulse propagation through single mode optical silica fiber when suitable higher-order linear and nonlinear optical effects were taken into account [2]
Another remarkable feature of the FL system is that it corresponds to the first negative flow of the integrable hierarchy associated with the derivative nonlinear Schrodinger (NLS) equation in the same way that the Camassa-Holm equation is related to the Korteweg-de Vries equation [3]
Summary
Under the new symmetry reduction r(x, t) = q(−x, −t), the compatible system (2.1) leads to the reverse spacetime FL equation (1.3), which is different from previous classic FL equation (1.1). The following Lax pair of a completely integrable Eq (1.3) can be given by the FL spectral problem. Where λ is called spectral parameter, and Ψ is the vector eigenfunction associated with λ of the nonlocal FL system. It is shown that the compatibility condition Ut − Vx + [U, V] = 0 can precisely yield Eq (1.3). In what follows, based on the DT of FL system and nonlocal system, we will study DT method for the reverse space-time FL equation (1.3)
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