@article{akrivis_dougalis_karakashian_mckinney_2003, title={Numerical approximation of blow-up of radially symmetric solutions of the nonlinear Schrodinger equation}, volume={25}, ISSN={["1095-7197"]}, DOI={10.1137/S1064827597332041}, abstractNote={We consider the initial-value problem for the radially symmetric nonlinear Schrodin\-ger equation with cubic nonlinearity (NLS) in d=2 and 3 space dimensions. To approximate smooth solutions of this problem, we construct and analyze a numerical method based on a standard Galerkin finite element spatial discretization with piecewise linear, continuous functions and on an implicit Crank--Nicolson type time-stepping procedure. We then equip this scheme with an adaptive spatial and temporal mesh refinement mechanism that enables the numerical technique to approximate well singular solutions of the NLS equation that blow up at the origin as the temporal variable t tends from below to a finite value $t^\star$. For the blow-up of the amplitude of the solution we recover numerically the well-known rate $(t^\star - t)^{-1/2}$ for d=3. For d=2 our numerical evidence supports the validity of the $\log \log$ law $[\ln\ln \frac {1}{t^\star -t} /(t^\star-t)]^{1/2}$ for t extremely close to $t^\star.$ The scheme also approximates well the details of the blow-up of the phase of the solution at the origin as $t\to t^\star.$}, number={1}, journal={SIAM JOURNAL ON SCIENTIFIC COMPUTING}, author={Akrivis, GD and Dougalis, VA and Karakashian, OA and McKinney, WR}, year={2003}, pages={186–212} }
 @article{bona_mckinney_restrepo_2000, title={Stable and unstable solitary-wave solutions of the generalized regularized long-wave equation}, volume={10}, ISSN={["1432-1467"]}, DOI={10.1007/s003320010003}, abstractNote={Investigated here are interesting aspects of the solitary-wave solutions of the generalized Regularized Long-Wave equation $$u_t + u_x + \alpha \left( {u^p } \right)_x - \beta u_{xxt} = 0.$$ For p>5, the equation has both stable and unstable solitary-wave solutions, according to the theory of Souganidis and Strauss. Using a high-order accurate numerical scheme for the approximation of solutions of the equation, the dynamics of suitably perturbed solitary waves are examined. Among other conclusions, we find that unstable solitary waves may evolve into several, stable solitary waves and that positive initial data need not feature solitary waves at all in its long-time asymptotics.}, number={6}, journal={JOURNAL OF NONLINEAR SCIENCE}, author={Bona, JL and McKinney, WR and Restrepo, JM}, year={2000}, pages={603–638} }