Via the Nth Darboux transformation, a chain of nonsingular localized-wave solutions is derived for a nonlocal nonlinear Schrödinger equation with the self-induced parity-time (PT) -symmetric potential. It is found that the Nth iterated solution in general exhibits a variety of elastic interactions among 2N solitons on a continuous-wave background and each interacting soliton could be the dark or antidark type. The interactions with an arbitrary odd number of solitons can also be obtained under different degenerate conditions. With N=1 and 2, the two-soliton and four-soliton interactions and their various degenerate cases are discussed in the asymptotic analysis. Numerical simulations are performed to support the analytical results, and the stability analysis indicates that the PT-symmetry breaking can also destroy the stability of the soliton interactions.

In this paper, via the generalized Darboux transformation, rational soliton solutions are derived for the parity-time-symmetric nonlocal nonlinear Schrödinger (NLS) model with the defocusing-type nonlinearity. We find that the first-order solution can exhibit the elastic interactions of rational antidarkantidark, dark-antidark, and antidark-dark soliton pairs on a continuous wave background, but there is no phase shift for the interacting solitons. Also, we discuss the degenerate case in which only one rational dark or antidark soliton survives. Moreover, we reveal that the second-order rational solution displays the interactions between two solitons with combined-peak-valley structures in the near-field regions, but each interacting soliton vanishes or evolves into a rational dark or antidark soliton as |z| → ∞. In addition, we numerically examine the stability of the first-and second-order rational soliton solutions. *

Applicable in arterial mechanics, Bose gases of impenetrable bosons and Bose–Einstein condensates, a variable-coefficient Korteweg–de Vries (vcKdV) equation is investigated in this paper with symbolic computation. Based on the Ablowitz–Kaup–Newell–Segur system, the Lax pair and auto-Bäcklund transformation are constructed. Furthermore, the nonlinear superposition formula and an infinite number of conservation laws for the vcKdV equation are also derived. Special attention is paid to the analytic one- and two-solitonic solutions with their physical properties and possible applications discussed.

In this paper, the Wronskian technique is applied to the vector nonlinear Schrödinger equation with arbitrary m components, which arises from some applications in the multimode fibers, photorefractive materials, and Bose–Einstein condensates. Via the iterative algorithm based on the Darboux transformation, the (m+1)-component Wronskian solution is generated from the zero solution. The verification of the solution is finished by using the (m+1)-component Wronskian notation and new determinantal identities. With a set of N linearly independent solutions of the zero-potential Lax pair, the (m+1)-component Wronskian solution is found to be the representation of the bright N-soliton solution which contains (m+1)N parameters. For characterizing the asymptotic behavior of the generic bright N-soliton solution, an algebraic procedure is derived to obtain the explicit expressions of asymptotic solitons as t→∓∞.

With the stationary solution assumption, we establish the connection between the nonlocal nonlinear Schrödinger (NNLS) equation and an elliptic equation. Then, we obtain the general stationary solutions and discuss the relevance of their smoothness and boundedness to some integral constants. Those solutions, which cover the known results in the literature, include the unbounded elliptic-function and hyperbolic-function solutions, the bounded sn-, cn-and dn-function solutions, as well as the bright and dark soliton solutions. By the imaginary translation invariance of the NNLS equation, we also derive the complex-amplitude stationary solutions, in which all the bounded cases obey either the PT -or anti-PT -symmetric relation. In particular, the complex tanh-function solution can exhibit no spatial localization in addition to the dark and anti-dark soliton profiles, where is sharp contrast with the common dark soliton. Considering the physical relevance to PT -symmetric systems, we show that the complex-amplitude stationary solutions can yield a wide class of complex and time-independent PT -symmetric potentials, and the symmetry breaking does not occur in the PT -symmetric linear systems with the associated potentials.

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The singular manifold method from the Painlevé analysis can be used to investigate many important integrable properties for the nonlinear partial differential equations. In this paper, the two-singular-manifold method is applied to the (2+1)-dimensional Gardner equation with two Painlevé expansion branches to determine the Hirota bilinear form, Bäcklund transformation, Lax pairs and Darboux transformation. Based on the obtained Lax pairs, the binary Darboux transformation is constructed and the N × N Grammian solution is also derived by performing the iterative algorithm N times with symbolic computation.