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1. On the Whitham system for the radial nonlinear Schrödinger equation

   https://doi.org/10.1111/sapm.12254  

   Ablowitz, Mark J.; Cole, Justin T.; Rumanov, Igor (January 2019, Studies in Applied Mathematics)

   Abstract Dispersive shock waves (DSWs) of the defocusing radial nonlinear Schrödinger (rNLS) equation in two spatial dimensions are studied. This equation arises naturally in Bose‐Einstein condensates, water waves, and nonlinear optics. A unified nonlinear WKB approach, equally applicable to integrable or nonintegrable partial differential equations, is used to find the rNLS Whitham modulation equation system in both physical and hydrodynamic type variables. The description of DSWs obtained via Whitham theory is compared with direct rNLS numerics; the results demonstrate very good quantitative agreement. On the other hand, as expected, comparison with the corresponding DSW solutions of the one‐dimensional NLS equation exhibits significant qualitative and quantitative differences. 

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2. On the Whitham system for the (2+1)‐dimensional nonlinear Schrödinger equation

   https://doi.org/10.1111/sapm.12543  

   Ablowitz, Mark J.; Cole, Justin T.; Rumanov, Igor (December 2022, Studies in Applied Mathematics)

   Abstract Whitham modulation equations are derived for the nonlinear Schrödinger equation in the plane ((2+1)‐dimensional nonlinear Schrödinger [2d NLS]) with small dispersion. The modulation equations are obtained in terms of both physical and Riemann‐type variables; the latter yields equations of hydrodynamic type. The complete 2d NLS Whitham system consists of six dynamical equations in evolutionary form and two constraints. As an application, we determine the linear stability of one‐dimensional traveling waves. In both the elliptic and hyperbolic cases, the traveling waves are found to be unstable. This result is consistent with previous investigations of stability by other methods and is supported by direct numerical calculations. 

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3. Nonlinear Schrödinger equations and the universal description of dispersive shock wave structure

   https://doi.org/10.1111/sapm.12247  

   Congy, T.; El, G. A.; Hoefer, M. A.; Shearer, M. (November 2018, Studies in Applied Mathematics)

   Abstract The nonlinear Schrödinger (NLS) equation and the Whitham modulation equations both describe slowly varying, locally periodic nonlinear wavetrains, albeit in differing amplitude‐frequency domains. In this paper, we take advantage of the overlapping asymptotic regime that applies to both the NLS and Whitham modulation descriptions in order to develop a universal analytical description of dispersive shock waves (DSWs) generated in Riemann problems for a broad class of integrable and nonintegrable nonlinear dispersive equations. The proposed method extends DSW fitting theory that prescribes the motion of a DSW's edges into the DSW's interior, that is, this work reveals the DSW structure. Our approach also provides a natural framework in which to analyze DSW stability. We consider several representative, physically relevant examples that illustrate the efficacy of the developed general theory. Comparisons with direct numerical simulations show that inclusion of higher order terms in the NLS equation enables a remarkably accurate description of the DSW structure in a broad region that extends from the harmonic, small amplitude edge. 

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4. Two-dimensional reductions of the Whitham modulation system for the Kadomtsev–Petviashvili equation

   https://doi.org/10.1088/1361-6544/ad15b3  

   Biondini, Gino; Bivolcic, Alexander J; Hoefer, Mark A; Moro, Antonio (January 2024, Nonlinearity)

   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. 

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5. Whitham Modulation Theory and Two-Phase Instabilities for Generalized Nonlinear Schrödinger Equations with Full Dispersion

   https://doi.org/10.1137/23M1603078  

   Sprenger, Patrick; Hoefer, Mark A; Ilan, Boaz (August 2024, SIAM Journal on Applied Mathematics)

   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. 

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