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We propose an adaptive approach in picking the wave-number parameter of absorbing boundary conditions for Schr\"{o}dinger-type equations. Based on the Gabor transform which captures local frequency information in the vicinity of artificial boundaries, the parameter is determined by an energy-weighted method and yields a quasi-optimal absorbing boundary conditions. It is shown that this approach can minimize reflected waves even when the wave function is composed of waves with different group velocities. We also extend the split local absorbing boundary (SLAB) method [Z. Xu and H. Han, {\it Phys. Rev. E}, 74(2006), pp. 037704] to problems in multidimensional nonlinear cases by coupling the adaptive approach. Numerical examples of nonlinear Schr\"{o}dinger equations in one- and two dimensions are presented to demonstrate the properties of the discussed absorbing boundary conditions., Comment: 18 pages; 12 figures. A short movie for the 2D NLS equation with absorbing boundary conditions can be downloaded at http://home.ustc.edu.cn/~xuzl/movie.avi. To appear in Journal of Computational Physics

We propose a numerical method for finding solitary wave solutions of generalized Korteweg-de Vries equations by solving the nonlinear eigenvalue problem on an unbounded domain. The artificial boundary conditions are obtained to make the domain finite. We specially discuss the soliton solutions of the K(m, n) equation and KdV-K(m,n) equation. Furthermore for the mixed models of linear and nonlinear dispersion, the collision behaviors of soliton-soliton and soliton-antisoliton are observed., Comment: 9 pages, 4 figures