Binary Darboux transformation and new soliton solutions of the focusing nonlocal nonlinear Schrödinger equation

For the focusing nonlocal nonlinear Schrödinger (NNLS) equation, we establish the N-fold binary Darboux transformation (BDT) and give a complete proof on the form invariance of Lax pair. Then, by choosing the tanh-function solution as a seed, we implement the one-fold BDT and obtain two new types of soliton solutions including the exponential and exponential-and-rational types. Both two types of solutions are nonsingular with a wide range of parameter regimes, and they can describe the elastic soliton interactions over the nonzero background with the asymptotic phase difference π as x → ± ∞ . Also, we derive the expressions of all asymptotic solitons and discuss the parametric conditions for different soliton profiles. It turns out that each asymptotic soliton can display both the dark and antidark soliton profiles or exhibit no spatial localization, which confirms that the focusing NNLS equation admits a rich variety of soliton interactions like the defocusing NNLS case.