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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]

In this paper, we consider the discrete soliton hierarchy with self-consistent sources by using Lie algebra sl(3). Then a Blaszak-Marciniak equations hierarchy with self-consistent sources is obtained through the discrete zero curvature equation. [ABSTRACT FROM AUTHOR]

Abstract: A generalized Hamiltonian structure of the fractional soliton equation hierarchy is presented by using differential forms and exterior derivatives of fractional orders. We construct the generalized fractional trace identity through the Riemann–Liouville fractional derivative. An example of the fractional KN soliton equation hierarchy and Hamiltonian structure is presented, which is a new integrable hierarchy and possesses Hamiltonian structure. [Copyright &y& Elsevier]

Abstract: We present a kind of integrable couplings of soliton equations hierarchy with self-consistent sources by using the loop algebra . As an application, a new integrable couplings of the generalized Wadati–Konono–Ichikawa (G-WKI) equations hierarchy with self-consistent sources is derived through the enlarged loop algebra . [Copyright &y& Elsevier]

A generalized fractional trace identity and Hamiltonian structure of the fractional soliton equation hierarchy are presented by using the modified Riemann-Liouville derivative and exterior derivatives of fractional orders. Example of the fractional Kaup Newell (KN) soliton equation hierarchy and Hamiltonian system are considered, which is a new integrable hierarchy and possesses Hamiltonian structure. [ABSTRACT FROM AUTHOR]

Abstract: A kind of new non-isospectral integrable couplings of discrete soliton equations hierarchy with self-consistent sources associated with is presented. As an application example, the integrable coupling hierarchy of non-isospectral Ablowitz–Ladik with self-consistent sources is derived by using of the loop algebra . [Copyright &y& Elsevier]

Abstract: The generalized zero curvature equation is presented, which is an important tool to construct the (2+1)-dimensional integrable soliton equation hierarchy. A few of new (2+1)-dimensional integrable soliton equation hierarchies are obtained through the generalized zero curvature equation. [Copyright &y& Elsevier]

Abstract: In [W.X. Ma, J. Phys. A: Math. Theor. 40 (2007) 15055], Prof. Ma gave a beautiful result (a discrete variational identity). In this Letter, based on a discrete block matrix spectral problem, a new hierarchy of Lax integrable lattice equations with four potentials is derived. By using of the discrete variational identity, we obtain Hamiltonian structure of the discrete soliton equation hierarchy. Finally, an integrable coupling system of the soliton equation hierarchy and its Hamiltonian structure are obtained through the discrete variational identity. [Copyright &y& Elsevier]

Abstract: In this article by considering a multi-dimensional Lie algebra system, a new multi-function integrable equation hierarchy is presented with the help of MAPLE. Then the integrable coupling system of the new multi-function integrable equation hierarchy is obtained through enlarging matrix spectral problem. [Copyright &y& Elsevier]

Abstract: In this Letter by considering a -dimensional discrete non-isospectral linear problem, a new -dimensional integrable lattice hierarchy is constructed. It shows that generalization of the Blaszak–Marciniak lattice hierarchy can be obtained as a reduction. Then an extended algebraic system of X is presented, from which the integrable coupling system of the -dimensional discrete non-isospectral Blaszak–Marciniak lattice equations are obtained. [Copyright &y& Elsevier]

A new loop algebra is constructed, which is devoted to establishing an isospectral problem. By making use of Tu scheme, the multi-component coupled Burgers hierarchy is obtained. Furthermore, an expanding loop algebra of the loop algebra is presented. It follows that the multi-component integrable coupling system of the multi-component coupled Burgers hierarchy is worked out. In general, the method can be applied to other evolution equations hierarchies. [Copyright &y& Elsevier]

The Hirota bilinear method is studied in a lot of local equations, but there are few work to solve nonlocal equations with external potential by Hirota bilinear method. In this letter, we succeed to bilinearize the generalized nonlocal Gross–Pitaevskii (NGP) equation with an arbitrary time-dependent linear potential through a nonstandard procedure and present more general bright soliton solutions, which describes the dynamics of soliton solutions in quasi-one-dimensional Bose–Einstein condensations. Under some reasonable assumptions, one-bright-soliton and two-bright-soliton solutions are constructed analytically by the improved Hirota method. From the gauge equivalence, we can see the difference between the solutions of the nonlocal GP equation and the solutions of the local GP equation. [ABSTRACT FROM AUTHOR]

A Gross-Pitaevskii (GP) equation with a parabola external potential is considered, and is transformed into a standard nonlinear Schrödinger (NLS) equation. By using the homogeneous balance principle and F-expansion method, we study non-autonomous wave solutions of the GP equation with a parabola external potential. In particular, based on the similarity transformation, several families of non-autonomous wave solutions of the GP equation are presented with snaking behaviors and different amplitude surfaces. These obtained bright-dark soliton solutions can give some potential applications in Bose-Einstein condensates. • A Gross-Pitaevskii (GP) equation with a parabola external potential is considered. • We study the non-autonomous wave solutions of GP equation with a parabola external potential. • These obtained bright-dark soliton solutions can give some potential applications in Bose-Einstein condensates. [ABSTRACT FROM AUTHOR]

Abstract The Hirota bilinear method has been studied in a lot of local equations, but there are few of works to solve nonlocal equations by Hirota bilinear method. In this letter, we show that the nonlocal integrable complex modified Korteweg-de Vries (MKdV) equation admits multiple complex soliton solutions. A variety of exact solutions including the single bright soliton solutions and two bright soliton solutions are derived via constructing an improved Hirota bilinear method for nonlocal complex MKdV equation. From the gauge equivalence, we can see the difference between the solution of nonlocal integrable complex MKdV equation and the solution of local complex MKdV equation. Highlights • An improved Hirota bilinear method for nonlocal complex MKdV equation is constructed. • We show that the nonlocal integrable complex MKdV equation admit multiple complex soliton solutions. • A variety of exact solutions including the single bright soliton solutions and the bright two soliton solutions are derived. • The difference between the solutions of nonlocal and ocal integrable complex MKdV equations is considered. [ABSTRACT FROM AUTHOR]