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. *

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.

Whitham-Broer-Kaup (WBK) equations describing the propagation of shallow-water waves, with a variable transformation, are transformed into a generalized Ablowitz-Kaup-Newell-Segur system, the bilinear forms of which are obtained via the rational transformations. Employing the matrix extension and symbolic computation, we derive types of solutions of the WBK equations through the selection of different canonical matrices, including solitons, rational solutions, and complexitons. Furthermore, dynamic properties of the solutions are discussed graphically and a novel phenomenon is observed, i.e., the coexistence of the elastic-inelastic interactions without disturbing each other.

By symbolic computation we study a variable-coefficient derivative nonlinear Schrödinger (vc-DNLS) equation describing nonlinear Alfvén waves in inhomogeneous plasmas. Based on the Lax pair of the vc-DNLS equation, the N-fold Darboux transformation is constructed via a gauge transformation and the reduction technique. Multi-solitonic solutions in terms of the double Wronskian for the vc-DNLS equation are obtained. Two-and three-solitonic interactions are analyzed graphically, i.e., overtaking, head-on and parallel interactions. Plasma streaming and inhomogeneous magnetic field control the amplitudes and velocities of the solitonic waves, respectively. The nonuniform density affects the amplitudes of the solitonic waves. The effects of the spectral parameters on the dynamics of the two-solitonic waves are discussed.Our results might facilitate the analytic investigation on certain inhomogeneous systems in the Earth's magnetosphere, solar winds, planetary bow shocks, dusty cometary tails and interplanetary shocks.

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