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An integrable variable coefficient nonlocal nonlinear Schrödinger equation (NNLS) is studied; by employing the Hirota’s bilinear method, the bilinear form is obtained, and the N -soliton solutions are constructed. In addition, some singular solutions and period solutions of the addressed equation with specific coefficients are shown. Finally, under certain conditions, the asymptotic behavior of the two-soliton solution is analyzed to prove that the collision of the two-soliton is elastic.

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In this paper, a [Formula: see text]-dimensional variable-coefficients Calogero–Bogoyavlenskii–Schiff (vcCBS) equation is studied. The infinitesimal generators and symmetry groups are obtained by using the Lie symmetry analysis on vcCBS. The optimal system of one-dimensional subalgebras of vcCBS is computed for determining the group-invariant solutions. On this basis, the vcCBS is reduced to two-dimensional partial differential equations (PDEs) by similarity reductions. Furthermore, the reduced PDEs are solved to obtain the two-soliton interaction solution, the soliton-kink interaction solution and some other exact solutions by the [Formula: see text]-expansion method. Moreover, it is shown that vcCBS is nonlinearly self-adjoint and then its conservation laws are calculated.