Search for: All records

The question of well-posedness of the generalized focusing Ablowitz-Ladik (AL) and discrete nonlinear Schrödinger (DNLS) equations with nonzero boundary conditions on the infinite lattice is far less understood than in the case where the models are supplemented with vanishing boundary conditions. This question remains largely unexplored even in the standard case of cubic nonlinearities in which, in particular, the AL equation is completely integrable while the DNLS equation is not (in contrast with its continuous counterpart). We establish local well-posedness for both of these generalized nonlinear systems supplemented with a broad class of nonzero boundary conditions and, in addition, derive analytical upper bounds for the minimal guaranteed lifespan of their solutions. These bounds depend explicitly on the norm of the initial data, the background, and the nonlinearity exponents. In particular, they suggest the possibility of finite-time collapse (blow-up) of solutions. Furthermore, by comparing models with different nonlinearity exponents, we prove estimates for the distance between their respective solutions (measured in suitable metrics), valid up to their common minimal guaranteed lifespan. Highly accurate numerical studies illustrate that solutions of the generalized AL equation may collapse in finite time. Importantly, the numerically observed blow-up time is in excellent agreement with the theoretically predicted order of the minimal guaranteed lifespan. Furthermore, in the case of the DNLS equation on a finite lattice we prove global existence of solutions; this is consistent with our numerical observations of the phenomenon of quasi-collapse, manifested by narrow oscillatory spikes that nevertheless persist throughout time. Once again, the time of the emergence of this phenomenon is in excellent agreement with the theoretically established minimal guaranteed lifespan. Notably, our numerical simulations confirm our theoretical result on the proximity of the dynamics between the two models over time scales up to the common solution lifespan. Finally, for power nonlinearities, we prove the asymptotic equivalence between the two discrete models in the continuous limit. 

Abstract The asymptotic Dirichlet‐to‐Neumann (D‐N) map is constructed for a class of scalar, constant coefficient, linear, third‐order, dispersive equations with asymptotically time/periodic Dirichlet boundary data and zero initial data on the half‐line, modeling a wavemaker acting upon an initially quiescent medium. The large time asymptotics for the special cases of the linear Korteweg‐de Vries and linear Benjamin–Bona–Mahony (BBM) equations are obtained. The D‐N map is proven to be unique if and only if the radiation condition that selects the unique wave number branch of the dispersion relation for a sinusoidal, time‐dependent boundary condition holds: (i) for frequencies in a finite interval, the wave number is real and corresponds to positive group velocity, and (ii) for frequencies outside the interval, the wave number is complex with positive imaginary part. For fixed spatial location , the corresponding asymptotic solution is (i) a traveling wave or (ii) a spatially decaying, time‐periodic wave. The linearized BBM asymptotics are found to quantitatively agree with viscous core‐annular fluid experiments.