Nonlinear waves often exhibit critical interior physics, such as shock formation, self-focusing, and soliton evolution, demanding high-fidelity numerical schemes to simulate. Existing pseudo-spectral methods have structural limitations: Fourier methods are restricted to periodic domains, while Chebyshev methods suffer from resolution mismatch and stringent stability constraints. To address these issues, we propose a flexible B-spline-Fourier (BSPF) pseudo-spectral method that accommodates non-periodic boundaries and adaptive meshes. The method decomposes the solution into a B-spline expansion that matches boundary constraints and a periodic residual, without the need to extend the domain as in Fourier continuation methods. The B-spline derivative is analytically computed, while the periodic residual is spectrally differentiated, preserving near-spectral convergence within the domain. We validate BSPF using canonical differentiation tests and four nonlinear PDE benchmarks. In the viscous Burgers’ equation, BSPF outperforms Chebyshev by resolving traveling shock waves with superior convergence and relaxed stability constraints. For the KdV equation, a soliton simulation demonstrates low dispersion errors compared to finite difference schemes. For the nonlinear Schrödinger equation, two canonical solutions (a traveling bright soliton and a Kuznetsov–Ma breather) further demonstrate BSPF’s spectral-like accuracy for complex-valued dispersive waves under different boundary treatments.Finally, a 2D shallow-water simulation demonstrates the method’s scalability to multi-dimensional problems and its ability to resolve steep wave fronts, with results validated against finite-difference solutions.Beyond these benchmarks, the BSPF framework is broadly applicable to a wide range of multi-scale nonlinear wave and transport problems in science and engineering.