68 research outputs found
Nonlinear Schrödinger equations and the universal description of dispersive shock wave structure
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 non-integrable nonlinear dispersive equations. The proposed method extends DSW fitting theory that prescribes the motion of a DSW's edges into the DSW's interior, i.e., 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
Dynamic soliton-mean flow interaction with nonconvex flux
The interaction of localised solitary waves with large-scale, time-varying dispersive mean flows subject to non-convex flux is studied in the framework of the modified Kortewegâde Vries (mKdV) equation, a canonical model for internal gravity wave propagation and potential vorticity fronts in stratified fluids. The effect of large amplitude, dynamically evolving mean flows on the propagation of localised waves â essentially âsoliton steeringâ by the mean flow â is considered. A recent theoretical and experimental study of this new type of dynamic solitonâmean flow interaction for convex flux has revealed two scenarios where the soliton either transmits through the varying mean flow or remains trapped inside it. In this paper, it is demonstrated that the presence of a non-convex cubic hydrodynamic flux introduces significant modifications to the scenarios for transmission and trapping. A reduced set of Whitham modulation equations is used to formulate a general mathematical framework for solitonâmean flow interaction with non-convex flux. Solitary wave trapping is stated in terms of crossing modulation characteristics. Non-convexity and positive dispersion â common for stratified fluids â imply the existence of localised, sharp transition fronts (kinks). Kinks play dual roles as a mean flow and a wave, imparting polarity reversal to solitons and dispersive mean flows, respectively. Numerical simulations of the mKdV equation agree with modulation theory predictions. The mathematical framework developed is general, not restricted to completely integrable equations like mKdV, enabling application beyond the mKdV setting to other fluid dynamic contexts subject to non-convex flux such as strongly nonlinear internal wave propagation that is prevalent in the ocean
Similar oscillations on both sides of a shock. Part I. Even-odd alternative dispersions and general assignments of Fourier dispersions towards a unfication of dispersion models
We consider assigning different dispersions for different dynamical modes,
particularly with the distinguishment and alternation of opposite signs for
alternative Fourier components. The Korteweg-de Vries (KdV) equation with
periodic boundary condition and longest-wave sinusoidal initial field, as used
by N. Zabusky and M. D. Kruskal, is chosen for our case study with such
alternating-dispersion of the Fourier modes of (normalized) even and odd
wavenumbers. Numerical results verify the capability of our new model to
produce two-sided (around the shock) oscillations, as appear on both sides of
some ion-acoustic and quantum shocks, not admitted by models such as the
KdV(-Burgers) equation, but also indicate even more, including singular
zero-dispersion limit or non-convergence to the classical shock (described by
the entropy solution), non-thermalization (of the Galerkin-truncated models)
and applicability to other models (showcased by the modified KdV equation with
cubic nonlinearity). A unification of various dispersive models, keeping the
essential mathematical elegance (such as the variational principle and
Hamiltonian formulation) of each, for phenomena with complicated dispersion
relation is thus suggested with a further explicit example of two even-order
dispersions (from the Hilbert transforms) extending the Benjamin-Ono model. The
most general situation can be simply formulated by the introduction of the
dispersive derivative, the indicator function and the Fourier transform,
resulting in an integro-differential dispersion equation. Other issues such as
the real-number order dispersion model and the transition from
non-thermalization to thermalization and, correspondingly, from regularization
to non-regularization for untruncated models are also briefly remarked
Oblique spatial dispersive shock waves in nonlinear Schrodinger flows
In dispersive media, hydrodynamic singularities are resolved by coherent wavetrains known as dispersive shock waves (DSWs). Only dynamically expanding, temporal DSWs are possible in one-dimensional media. The additional degree of freedom inherent in two-dimensional media allows for the generation of time-independent DSWs that exhibit spatial expansion. Spatial oblique DSWs, dispersive analogs of oblique shocks in classical media, are constructed utilizing Whitham modulation theory for a class of nonlinear Schrodinger boundary value problems. Self-similar, simple wave solutions of the modulation equations yield relations between the DSWâs orientation and the
upstream/downstream flow fields. Time dependent numerical simulations demonstrate a convective or absolute instability of oblique DSWs in supersonic flow over obstacles. The convective instability results in an effective stabilization of the DSW
Stationary expansion shocks for a regularized Boussinesq system
Stationary expansion shocks have been recently identified as a new type of solution to hyperbolic conservation laws regularized by non-local dispersive terms that naturally arise in shallow-water theory. These expansion shocks were studied in [1] for the Benjamin-Bona-Mahony equation using matched asymptotic expansions. In this paper, we extend the analysis of [1] to the regularized Boussinesq system by using Riemann invariants of the underlying dispersionless shallow water equations. The extension for a system is non-trivial, requiring a combination of small amplitude, long-wave expansions with high order matched asymptotics. The constructed asymptotic solution is shown to be in excellent agreement with
accurate numerical simulations of the Boussinesq system for a range of appropriately smoothed Riemann data
Dispersive Riemann problems for the Benjamin-Bona-Mahony equation
Long time dynamics of the smoothed step initial value problem or dispersive Riemann problem for the BenjaminâBonaâMahony (BBM) equation u t + u u x = u xxt are studied using asymptotic methods and numerical simulations. The catalog of solutions of the dispersive Riemann problem for the BBM equation is much richer than for the related, integrable, Kortewegâde Vries equation u t + u u x + u xxx = 0 . The transition width of the initial smoothed step is found to significantly impact the dynamics. Narrow width gives rise to rarefaction and dispersive shock wave (DSW) solutions that are accompanied by the generation of twoâphase linear wavetrains, solitary wave shedding, and expansion shocks. Both narrow and broad initial widths give rise to twoâphase nonlinear wavetrains or DSW implosion and a new kind of dispersive Lax shock for symmetric data. The dispersive Lax shock is described by an approximate selfâsimilar solution of the BBM equation whose limit as t â â is a stationary, discontinuous weak solution. By introducing a slight asymmetry in the data for the dispersive Lax shock, the generation of an incoherent solitary wavetrain is observed. Further asymmetry leads to the DSW implosion regime that is effectively described by a pair of coupled nonlinear Schrödinger equations. The complex interplay between nonlocality, nonlinearity, and dispersion in the BBM equation underlies the rich variety of nonclassical dispersive hydrodynamic solutions to the dispersive Riemann problem
Dispersive shock waves and modulation theory
There is growing physical and mathematical interest in the hydrodynamics of dissipationless/dispersive media. Since G. B. Whithamâs seminal publication fifty years ago that ushered in the mathematical study of dispersive hydrodynamics, there has been a significant body of work in this area. However, there has been no comprehensive survey of the field of dispersive hydrodynamics. Utilizing Whithamâs averaging theory as the primary mathematical tool, we review the rich mathematical developments over the past fifty years with an emphasis on physical applications. The fundamental, large scale, coherent excitation in dispersive hydrodynamic systems is an expanding, oscillatory dispersive shock wave or DSW. Both the macroscopic and microscopic properties of DSWs are analyzed in detail within the context of the universal, integrable, and foundational models for uni-directional (Kortewegâde Vries equation) and bi-directional (Nonlinear Schrödinger equation) dispersive hydrodynamics. A DSW fitting procedure that does not rely upon integrable structure yet reveals important macroscopic DSW properties is described. DSW theory is then applied to a number of physical applications: superfluids, nonlinear optics, geophysics, and fluid dynamics. Finally, we survey some of the more recent developments including non-classical DSWs, DSW interactions, DSWs in perturbed and inhomogeneous environments, and two-dimensional, oblique DSWs
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