Analysis of Malmquist-Takenaka-Christov rational approximations with applications to the nonlinear Benjamin equation

In the paper, we study approximation properties of the Malmquist-Takenaka-Christov (MTC) system. We show that the discrete MTC approximations converge rapidly under mild restrictions on functions asymptotic at infinity. This makes them particularly suitable for solving semi- and quasi-linear problems containing Fourier multipliers, whose symbols are not smooth at the origin. As a concrete application, we provide rigorous convergence and stability analyses of a collocation MTC scheme for solving the nonlinear Benjamin equation. We demonstrate that the method converges rapidly and admits an efficient implementation, comparable to the best spectral Fourier and hybrid spectral Fourier/finite-element methods described in the literature

Dynamics of solutions in the generalized Benjamin-Ono equation: a numerical study

We consider the generalized Benjamin-Ono (gBO) equation on the real line, $u_t + \partial_x (-\mathcal H u_{x} + \tfrac1{m} u^m) = 0, x \in \mathbb R, m = 2,3,4,5$, and perform numerical study of its solutions. We first compute the ground state solution to $-Q -\mathcal H Q^\prime +\frac1{m} Q^m = 0$ via Petviashvili's iteration method. We then investigate the behavior of solutions in the Benjamin-Ono ($m=2$) equation for initial data with different decay rates and show decoupling of the solution into a soliton and radiation, thus, providing confirmation to the soliton resolution conjecture in that equation. In the mBO equation ($m=3$), which is $L^2$-critical, we investigate solutions close to the ground state mass, and, in particular, we observe the formation of stable blow-up above it. Finally, we focus on the $L^2$-supercritical gBO equation with $m=4,5$. In that case we investigate the global vs finite time existence of solutions, and give numerical confirmation for the dichotomy conjecture, in particular, exhibiting blow-up phenomena in the supercritical setting