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In this paper, we propose and investigate the reverse-space–time nonlocal nonlinear Fokas–Lenells equation by the idea of Ablowitz and Musslimani. The reverse-space–time Fokas–Lenells equation, associated with a 2 × 2 matrix Lax pair, is the important integrable system, which can be reduced to the nonlocal Fokas–Lenells equation. Based on its Lax pair, we construct nonlocal version of N-fold Darboux transformation (DT) for the Fokas–Lenells equation, and obtain two kinds of soliton solutions from vanishing and plane wave backgrounds. Further some novel one-soliton and two-soliton are derived with the zero and nonzero seed solutions through complex computations, including the bright soliton, kink soliton and breather wave soliton. Moreover, various graphical analyses on the presented solutions are made to reveal the dynamic behaviors, which display the elastic interactions between two solitons and their amplitudes keeping unchanged after the interactions except for the phase shifts. It is clearly shown that these solutions have new properties which differ from ones of the classical Fokas–Lenells equation. [ABSTRACT FROM AUTHOR]

The generalized nonlocal nonlinear Hirota (GNNH) equation has been widely concerned, it can be regarded as the generalization of the nonlocal Schrödinger equation, and can be reduced to a nonlocal Hirota equation. In this paper, we mainly study a GNNH equation and its determinant representation of the N-fold Darboux transformation. Then we derive some novel exact solutions including the breather wave solitons, bright solitons, some characteristics of solitary wave and interactions are considered. In particularly, the dynamic features of one-soliton, two-soliton solutions and the elastic interactions between the two solitons are displayed. We find that unlike the local case, the q(x,t) and q ∗ (− x , t) $q^{*}(-x,t)$ of the GNNH equation have some novel characteristics of solitary wave, which are different form the classical Hirota equation. [ABSTRACT FROM AUTHOR]

Starting from a 3×3 spectral problem, a Darboux transformation (DT) method for coupled Schrödinger (CNLS) equation is constructed, which is more complex than 2×2 spectral problems. A scheme of soliton solutions of an integrable CNLS system is realised by using DT. Then, we obtain the breather solutions for the integrable CNLS system. The method is also appropriate for more non-linear soliton equations in physics and mathematics. [ABSTRACT FROM AUTHOR]

How to construct a variable coefficient integrable coupling equation hierarchy is an important problem. In this paper, we present new Lax pairs with some arbitrary functions and generate a variable coefficient integrable coupling of Ablowitz-Kaup-Newell-Segur hierarchy from a zero-curvature equation. Then the Hamiltonian structure of the variable coefficient coupling equation hierarchy is derived from the variational trace identity. It is also indicated that this method is an efficient and straightforward way to construct the variable coefficient integrable coupling equation hierarchy. [ABSTRACT FROM AUTHOR]

It is shown that the Kronecker product can be applied to construct new nonlinear integrable coupling system of soliton equation hierarchy in this paper. A direct application to the KN spectral problem leads to a novel KN soliton equation hierarchy of nonlinear integrable coupling system, which is different from any one obtained before. Furthermore, we present the Hamiltonian structures of nonlinear integrable couplings of KN hierarchy by using a pairs from special non-semisimple matrix Lie algebras and variational identities over the associated loop algebras. It is also indicated that the study of nonlinear integrable couplings using the Kronecker product is an efficient and straightforward method. [ABSTRACT FROM AUTHOR]