We present a method for determination of the blow up point for a complex nonautonomous ordinary differential equation with a cubic nonlinearity. This equation describes stationary nonlinear modes for two important NLS-based models: (i) the Gross–Pitaevskii equation with complex potential and repulsive nonlinearity and (ii) the Lugiato–Lefever equation in the case of normal dispersion. We derive and justify an asymptotic expansion in the vicinity of the blow up point. This expansion is employed for the construction of a numeric procedure for the computation of the coordinate of the blow up point. We illustrate applications of the proposed procedure by two examples, where exact analytical solutions are available. It is shown that it allows one to find the blow up point with high accuracy. The method may be efficiently used for the search of soliton solutions for vector and scalar NLS-type equations within the strategy of ’filtering out’ of blow up solutions (Alfimov et al., Physica D, 394, 39 (2019)).
start from the derivation of the Abrikosov-Ryzhkin model for the 1D random potential problem. In its framework I find closed functional representations for various physical quantities. The representation uses number-valued fields only. These functional integrals are calculated exactly without the use of any perturbative expansions. Expressions for the multipoint densities correlators are obtained. These correlators allow to compute the distribution function of inverse sizes of localized wave functions valid both for an infinite sample and for a sample with a finite length.
Internal tidal wave entering shallow waters transforms into an undular bore and this process can be described in the framework of the Gardner equation (extended version of the Korteweg-de Vries equation with both quadratic and cubic nonlinear terms). Our numerical computations demonstrate the features of undular bore developing for different signs of the cubic nonlinear term. If cubic nonlinear term is negative, and initial wave amplitude is large enough, two undular bores are generated from the two breaking points formed on both crest slopes (within dispersionless Gardner equation). Undular bore consists of one table-top soliton and a group of small soliton-like waves passing through the table-top soliton. If the cubic nonlinear term is positive and again the wave amplitude is large enough, the breaking points appear on crest and trough generating groups of positive and negative soliton-like pulses. This is the main difference with respect to the classic Korteweg-de Vries equation, where the breaking point is single. It is shown also that nonlinear interaction of waves happens similarly to one of scenarios of two-soliton interaction of "exchange" or "overtake" types with a phase shift. If small-amplitude pulses interact with large-amplitude soliton-like pulses, their speed in average is negative in the case when "free" velocity is positive. Nonlinear interaction leads to the generation of higher harmonics and spectrum width increases with amplitude increase independently of the sign of cubic nonlinear term. The breaking asymptotic k4/3 predicted within the dispersionless Gardner equation emerges during the process of undular bore development. The formation of soliton-like perturbations leads to appearance of several spectral peaks which are downshifting with time. © 2015 Elsevier B.V.
A one-parameter family of time-reversible systems on three-dimensional torus is considered. It is shown that the dynamics is not conservative, namely the attractor and repeller intersect but not coincide. We explain this as the manifestation of the so-called mixed dynamics phenomenon which corresponds to a persistent intersection of the closure of the stable periodic orbits and the closure of the completely unstable periodic orbits. We search for the stable and unstable periodic orbits indirectly, by finding non-conservative saddle periodic orbits and heteroclinic connections between them. In this way, we are able to claim the existence of mixed dynamics for a large range of parameter values. We investigate local and global bifurcations that can be used for the detection of mixed dynamics.
We develop a theory of turbulence based on the Navier–Stokes equation, without using dimensional or phenomenological considerations. A small scale vortex filament is the main element of the theory. The theory allows to obtain the scaling law and to calculate the scaling exponents of Lagrangian and Eulerian velocity structure functions in the inertial range. The obtained results are shown to be in very good agreement with numerical simulations and experimental data. The introduction of stochasticity into the equations and derivation of scaling exponents are discussed in details. A weak dependence on statistical propositions is demonstrated. The relation of the theory to the multifractal model is discussed.
Recent advances in the applications of the Kuramoto model to a wide range of real-life processes require the reconstruction of processes' parameters from observations. This paper explores the inverse problem for the Kuramoto model of two nonlinear oscillators with slowly varying coupling in the form of a single-step function, sine-wave, and auto-regressive process with a view to deriving the basic properties of the reconstruction procedure, that is the connection of the reconstruction efficiency with the coupling strength and estimates of the time it takes for a system to phase-lock. By investigating the de-synchronization of the solar foculae series, which represent signals coming from the northern and southern solar hemispheres, we relate the de-synchronization of the series, which occurred in the early'1960s to the changes in the coupling of the underlying real oscillators.
The goal of this paper is to identify the universal Whitham hierarchy of genus zero with a dispersionless limit of the multi-component KP hierarchy. To this end, the multi-component KP hierarchy is (re)formulated to depend on several discrete variables called ``charges''. These discrete variables play the role of lattice coordinates in underlying Toda field equations. A multi-component version of the so called differential Fay identity are derived from the Hirota equations of the tau-function of this ``charged'' multi-component KP hierarchy. These multi-component differential Fay identities have a well-defined dispersionless limit (the dispersionless Hirota equations). The dispersionless Hirota equations turn out to be equivalent to the Hamilton-Jacobi equations for the S-functions of the universal Whitham hierarchy. The differential Fay identities themselves are shown to be a generating functional expression of auxiliary linear equations for scalar-valued wave functions of the multi-component KP hierarchy.
We consider the damped and driven Navier–Stokes system with stress free boundary conditions and the damped Euler system in a bounded domain Ω ⊂ R2.We show that the damped Euler system has a (strong) global attractor in H1(Ω). We also show that in the vanishing viscosity limit the global attractors of the Navier–Stokes system converge in the non-symmetric Hausdorff distance in H1(Ω) to the strong global attractor of the limiting damped Euler system (whose solutions are not necessarily unique).