Darboux transformation, exact solutions and conservation laws for the reverse space-time Fokas–Lenells equation

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.

Field and reverse field solitons in wave-operator nonlinear Schrödinger equation with space-time reverse: Modulation instability

“…Here, we are confined to obtain the rational solutions of (8) and (9). By using the integrability condition mentioned in the above, we write, When substituting from (10) into (8) and (9) leads to the balance condition…”

“…Comparison of the method used here with the known methods in the literature, we find the following. In the UM, exact solutions are found by inserting (10) into (8) and (9) and by stetting the coefficients ( ) ( )…”

“…These are simple approaches to model potentially negative-value stationary space-time models, and for such models, it was shown that they possess a reverse (pointing upward) dimple [8]. An STR Fokas-Lenells (FL) equation was derived from a rather simple, but extremely important symmetry reduction of the corresponding local equation [9]. In [10], a general coupled integrable dispersionless system and nontrivial solutions in terms of the ratio of determinants were obtained using matrix DT.…”

Breathers, Transformation Mechanisms and Their Molecular State of a (3+1)-Dimensional Generalized Yu–Toda–Sasa–Fukuyama Equation

“…In the field of physics and mechanics, many scholars have become devoted to investigating the exact solutions of nonlinear integrable systems, which include solitons [1][2][3][4], breathers [5][6][7], lumps [8][9][10][11][12], quasi-periodic wave solutions [13][14][15][16], and so on. Scholars have undertaken a lot of work on exact solutions, and have summarized some effective methods, such as the Bäcklund transformation [17][18][19], inverse scattering method [20][21][22], Darboux transformation [23,24], algebraic geometry theory [25,26], Hirota's bilinear method [27,28] and Lie symmetry analysis [29][30][31][32].…”