# Dark Solitons, Dispersive Shock Waves, and Transverse Instabilities

@article{Hoefer2012DarkSD, title={Dark Solitons, Dispersive Shock Waves, and Transverse Instabilities}, author={Mark A. Hoefer and Boaz Ilan}, journal={Multiscale Model. Simul.}, year={2012}, volume={10}, pages={306-341} }

The nature of transverse instabilities of dark solitons for the (2+1)-dimensional defocusing nonlinear Schrodinger/Gross–Pitaevskii˘ (NLS/GP) equation is considered. Special attention is given to the small (shallow) amplitude regime, which limits to the Kadomtsev–Petviashvili (KP) equation. We study analytically and numerically the eigenvalues of the linearized NLS/GP equation. The dispersion relation for shallow solitons is obtained asymptotically beyond the KP limit. This yields (1) the…

## 13 Citations

### Spectral transverse instabilities and soliton dynamics in the higher-order multidimensional nonlinear Schrödinger equation

- Physics, Mathematics
- 2015

### Rayleigh–Taylor instability in nonlinear Schrödinger flow

- Physics
- 2012

The Rayleigh–Taylor instability (RTI) is a fundamental fluid instability that occurs when a light fluid is accelerated into a heavier one. While techniques for observing the RTI in classical fluids…

### New Journal of Physics Rayleigh – Taylor instability in nonlinear Schrödinger flow

- Physics
- 2012

The Rayleigh–Taylor instability (RTI) is a fundamental fluid instability that occurs when a light fluid is accelerated into a heavier one. While techniques for observing the RTI in classical fluids…

### OBLIQUE SPATIAL DISPERSIVE SHOCK WAVES IN NONLINEAR

- Physics
- 2017

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…

### Oblique Spatial Dispersive Shock Waves in Nonlinear Schrödinger Flows

- PhysicsSIAM J. Appl. Math.
- 2017

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.

### Dynamics of shock waves in a superfluid unitary Fermi gas

- Physics
- 2014

We study the formation and dynamics of shock waves initiated by a repulsive potential in a superfluid unitary Fermi gas by using the order-parameter equation. In the theoretical framework, the…

### Oblique solitons generated by the flow of a polariton condensate past an obstacle

- Physics
- 2012

The formation of oblique solitons in a polariton condensate flowing past an obstacle is considered. Because of the finite lifetime of polaritons, the condensate flow is inhomogeneous, which leads to…

### On the Whitham system for the (2+1)-dimensional nonlinear Schr\"odinger equation

- Mathematics
- 2021

We derive the Whitham modulation equations for the nonlinear Schrödinger equation in the plane (2d NLS) with small dispersion. The modulation equations are derived in terms of both physical and…

## References

SHOWING 1-10 OF 65 REFERENCES

### Self-focusing of plane dark solitons in nonlinear defocusing media.

- PhysicsPhysical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics
- 1995

Dynamics of the transverse instability of the plane dark soliton of arbitrary amplitude is investigated analytically by means of the asymptotic technique, and numerically by direct integration of the two-dimensional NLS equation.

### Dispersive and classical shock waves in Bose-Einstein condensates and gas dynamics

- Physics
- 2006

A Bose-Einstein condensate (BEC) is a quantum fluid that gives rise to interesting shock-wave nonlinear dynamics. Experiments depict a BEC that exhibits behavior similar to that of a shock wave in a…

### Resolution of a shock in hyperbolic systems modified by weak dispersion.

- MathematicsChaos
- 2005

We present a way to deal with dispersion-dominated "shock-type" transition in the absence of completely integrable structure for the systems that one may characterize as strictly hyperbolic…

### Suppression of transverse instabilities of dark solitons and their dispersive shock waves

- Physics
- 2009

We investigate the impact of nonlocality, owing to diffusive behavior, on transverse instabilities of a dark stripe propagating in a defocusing cubic medium. The nonlocal response turns out to have a…

### On the transverse instability of solitary waves in the Kadomtsev-Petviashvili equation

- Physics, Mathematics
- 1997

### Eigenvalues, and instabilities of solitary waves

- MathematicsPhilosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences
- 1992

We study a type of ‘eigenvalue’ problem for systems of linear ordinary differential equations with asymptotically constant coefficients by using the analytic function D(λ) introduced by J. W. Evans…

### Dissipationless shock waves in media with positive dispersion

- Physics, Mathematics
- 2009

We consider nonstationary one-dimensional flows in dissipationless hydrodynamics with positive dispersion which is described by the nonlinear Schrodinger equation. We determine the conditions for the…

### Wave patterns generated by a supersonic moving body in a binary Bose-Einstein condensate

- Physics
- 2009

Generation of wave structures by a two-dimensional 2D object laser beam moving in a 2D twocomponent Bose-Einstein condensate with a velocity greater than the two sound velocities of the mixture is…