The problem of nonlinear adjustment of localized front-like perturbations to a state of geostrophic equilibrium (balanced state) is studied in the framework of rotating shallow-water equations with no dependence on the along-front coordinate. We work in Lagrangian coordinates, which turns out to be conceptually and technically advantageous. First, a perturbation approach in the cross-front Rossby number is developed and splitting of the motion into slow and fast components is demonstrated for non-negative potential vorticities. We then give a non-perturbative proof of existence and uniqueness of the adjusted state, again for configurations with nonnegative initial potential vorticities. We prove that wave trapping is impossible within this adjusted state and, hence, adjustment is always complete for small enough departures from balance. However, we show that retarded adjustment occurs if the adjusted state admits quasi-stationary states decaying via tunnelling across a potential barrier. A description of finite-amplitude periodic nonlinear waves known to exist in configurations with constant potential vorticity in this model is given in terms of Lagrangian variables. Finally, shock formation is analysed and semi-quantitative criteria based on the values of initial gradients and the relative vorticity of initial states are established for wave breaking showing, again, essential differences between the regions of positive and negative vorticity.

We consider a Leith model of turbulence (Leith C 1967 Phys. Fluids 10 1409) in which the energy spectrum obeys a nonlinear diffusion equation. We analytically prove the existence of a self-similar solution with a power-law asymptotic on the low-wavenumber end and a sharp boundary on the high-wavenumber end, which propagates to infinite wavenumbers in a finite-time t*. We prove that this solution has a power-law asymptotic with an anomalous exponent x*, which is less than the Kolmogorov value, x* > 5/3. This is a result that was previously discovered by numerical simulations in Connaughton and Nazarenko (2004 Phys. Rev. Lett. 92 044501). We also prove the convergence to this self-similar solution of the spectrum evolving from an arbitrary finitely supported initial data as t → t*.

Abstract. Differential models for hydrodynamic, passive-scalar and wave turbulence given by nonlinear first-and second-order evolution equations for the energy spectrum in the k-space were analysed. Both types of models predict formation an anomalous transient power-law spectra. The second-order models were analysed in terms of self-similar solutions of the second kind, and a phenomenological formula for the anomalous spectrum exponent was constructed using numerics for a broad range of parameters covering all known physical examples. The first-order models were examined analytically, including finding an analytical prediction for the anomalous exponent of the transient spectrum and description of formation of the Kolmogorovtype spectrum as a reflection wave from the dissipative scale back into the inertial range. The latter behaviour was linked to pre-shock/shock singularities similar to the ones arising in the Burgers equation. Existence of the transient anomalous scaling and the reflection-wave scenario are argued to be a robust feature common to the finitecapacity turbulence systems. The anomalous exponent is independent of the initial conditions but varies for for different models of the same physical system. arXiv:1502.06750v2 [physics.flu-dyn]

We review the main physical and mathematical properties of dispersion-managed (DM) optical solitons. Theory of DM solitons can be presented at two levels of accuracy: first, simple, but nevertheless, quantitative models based on ordinary differential equations governing evolution of the soliton width and phase parameter (the so-called chirp); and second, a comprehensive path-average theory that is capable of describing in detail both the fine structure of DM soliton form and its evolution along the fiber line. An analogy between DM soliton and a macroscopic nonlinear quantum oscillator model is also discussed.

Using the cubic Ginzburg-Landau equation as an example, we demonstrate how the inverse scattering transform can be applied to characterize coherent structures in dissipative nonlinear systems. Using this approach one can reduce the number of the effective degrees of freedom in the system when the dynamic is dominated by the coherent structures, even if they are embedded in the dispersive waves and demonstrate unstable behavior.