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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. 

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 Using the Darboux transformation for the Korteweg–de Vries equation, we construct and analyze exact solutions describing the interaction of a solitary wave and a traveling cnoidal wave. Due to their unsteady, wavepacket-like character, these wave patterns are referred to as breathers. Both elevation (bright) and depression (dark) breather solutions are obtained. The nonlinear dispersion relations demonstrate that the bright (dark) breathers propagate faster (slower) than the background cnoidal wave. Two-soliton solutions are obtained in the limit of degeneration of the cnoidal wave. In the small amplitude regime, the dark breathers are accurately approximated by dark soliton solutions of the nonlinear Schrödinger equation. These results provide insight into recent experiments on soliton-dispersive shock wave interactions and soliton gases. 

Conduits generated by the buoyant dynamics between two miscible Stokes fluids with high viscosity contrast, a type of core–annular flow, exhibit a rich nonlinear wave dynamics. However, little is known about the fundamental wave dispersion properties of the medium. In the present work, a pump is used to inject a time-periodic flow that results in the excitation of propagating small- and large-amplitude periodic travelling waves along the conduit interface. This wavemaker problem is used as a means to measure the linear and nonlinear dispersion relations and corresponding periodic travelling wave profiles. Measurements are favourably compared with predictions from a fully nonlinear, long-wave model (the conduit equation) and the analytically computed linear dispersion relation for two-Stokes flow. A critical frequency is observed, marking the threshold between propagating and non-propagating (spatially decaying) waves. Measurements of wave profiles and the wavenumber–frequency dispersion relation quantitatively agree with wave solutions of the conduit equation. An upshift from the conduit equation's predicted critical frequency is observed and is explained by incorporating a weak recirculating flow into the full two-Stokes flow model. When the boundary condition corresponds to the temporal profile of a nonlinear periodic travelling wave solution of the conduit equation, weakly nonlinear and strongly nonlinear, cnoidal-type waves are observed that quantitatively agree with the conduit nonlinear dispersion relation and wave profiles. This wavemaker problem is an important precursor to the experimental investigation of more general boundary value problems in viscous fluid conduit nonlinear wave dynamics. 

Abstract The Whitham modulation equations for the defocusing nonlinear Schrödinger (NLS) equation in two, three and higher spatial dimensions are derived using a two-phase ansatz for the periodic traveling wave solutions and by period-averaging the conservation laws of the NLS equation. The resulting Whitham modulation equations are written in vector form, which allows one to show that they preserve the rotational invariance of the NLS equation, as well as the invariance with respect to scaling and Galilean transformations, and to immediately generalize the calculations from two spatial dimensions to three. The transformation to Riemann-type variables is described in detail; the harmonic and soliton limits of the Whitham modulation equations are explicitly written down; and the reduction of the Whitham equations to those for the radial NLS equation is explicitly carried out. Finally, the extension of the theory to higher spatial dimensions is briefly outlined. The multidimensional NLS-Whitham equations obtained here may be used to study large amplitude wavetrains in a variety of applications including nonlinear photonics and matter waves. 

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. 

Resonant Y-shaped soliton solutions to the Kadomtsev–Petviashvili II (KPII) equation are modelled as shock solutions to an infinite family of modulation conservation laws. The fully two-dimensional soliton modulation equations, valid in the zero dispersion limit of the KPII equation, are demonstrated to reduce to a one-dimensional system. In this same limit, the rapid transition from the larger Y soliton stem to the two smaller legs limits to a travelling discontinuity. This discontinuity is a multivalued, weak solution satisfying modified Rankine–Hugoniot jump conditions for the one-dimensional modulation equations. These results are applied to analytically describe the dynamics of the Mach reflection problem, V-shaped initial conditions that correspond to a soliton incident upon an inward oblique corner. Modulation theory results show excellent agreement with direct KPII numerical simulation.