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Abstract The Riemann problem for the discrete conservation law is classified using Whitham modulation theory, a quasi‐continuum approximation, and numerical simulations. A surprisingly elaborate set of solutions to this simple discrete regularization of the inviscid Burgers' equation is obtained. In addition to discrete analogs of well‐known dispersive hydrodynamic solutions—rarefaction waves (RWs) and dispersive shock waves (DSWs)—additional unsteady solution families and finite‐time blowup are observed. Two solution types exhibit no known conservative continuum correlates: (i) a counterpropagating DSW and RW solution separated by a symmetric, stationary shock and (ii) an unsteady shock emitting two counterpropagating periodic wavetrains with the same frequency connected to a partial DSW or an RW. Another class of solutions called traveling DSWs, (iii), consists of a partial DSW connected to a traveling wave comprised of a periodic wavetrain with a rapid transition to a constant. Portions of solutions (ii) and (iii) are interpreted as shock solutions of the Whitham modulation equations. 

Abstract Two-dimensional reductions of the Kadomtsev–Petviashvili(KP)–Whitham system, namely the overdetermined Whitham modulation system for five dependent variables that describe the periodic solutions of the KP equation, are studied and characterized. Three different reductions are considered corresponding to modulations that are independent ofx, independent ofy, and oft(i.e. stationary), respectively. Each of these reductions still describes dynamic, two-dimensional spatial configurations since the modulated cnoidal wave, generically, has a nonzero speed and a nonzero slope in thexyplane. In all three of these reductions, the integrability of the resulting systems of equations is proven, and various other properties are elucidated. Compatibility with conservation of waves yields a reduction in the number of dependent variables to two, three and four, respectively. As a byproduct of the stationary case, the Whitham modulation system for the classical Boussinesq equation is explicitly obtained. 

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 timetasymptotics 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 locationx, 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. 

The generalized nonlinear Schr\"odinger equation with full dispersion (FDNLS) is considered in the semiclassical regime. The Whitham modulation equations are obtained for the FDNLS equation with general linear dispersion and a generalized, local nonlinearity. Assuming the existence of a four-parameter family of two-phase solutions, a multiple-scales approach yields a system of four independent, first-order, quasi-linear conservation laws of hydrodynamic type that correspond to the slow evolution of the two wavenumbers, mass, and momentum of modulated periodic traveling waves. The modulation equations are further analyzed in the dispersionless and weakly nonlinear regimes. The ill-posedness of the dispersionless equations corresponds to the classical criterion for modulational instability (MI). For modulations of linear waves, ill-posedness coincides with the generalized MI criterion, recently identified by Amiranashvili and Tobisch [New J. Phys., 21 (2019), 033029]. A new instability index is identified by the transition from real to complex characteristics for the weakly nonlinear modulation equations. This instability is associated with long wavelength modulations of nonlinear two-phase wavetrains and can exist even when the corresponding one-phase wavetrain is stable according to the generalized MI criterion. Another interpretation is that while infinitesimal perturbations of a periodic wave may not grow, small but finite amplitude perturbations may grow, hence this index identifies a nonlinear instability mechanism for one-phase waves. Classifications of instability indices for multiple FDNLS equations with higher-order dispersion, including applications to finite-depth water waves and the discrete NLS equation, are presented and compared with direct numerical simulations.