Modulational instability of a trapped Bose-Einstein condensate with two- and three-body interactions

Stability analysis on dark solitons in quasi-1D Bose-Einstein condensate with three-body interactions

The stability properties of dark solitons in quasi-one-dimensional Bose–Einstein condensate (BEC) loaded in a Jacobian elliptic sine potential with three-body interactions are investigated theoretically. The solitons are obtained by the Newton-Conjugate Gradient method. A stationary cubic-quintic nonlinear Schrödinger equation is derived to describe the profiles of solitons via the multi-scale technique. It is found that the three-body interaction has distinct effect on the stability properties of solitons. Especially, such a nonlinear system supports the so-called dark solitons (kink or bubble), which can be excited not only in the gap, but also in the band. The bubbles are always linearly and dynamically unstable, and they cannot be excited if the three-body interaction is absent. Both stable and unstable kinks, depending on the physical parameters, can be excited in the BEC system.

Effect of temperature fluctuation on the localized pattern of action potential in cardiac tissue

Based on the improved FitzHugh-Nagumo myocardial model driven by a constant external current, the effect of temperature fluctuation in a network of electrically coupled myocardial cells are investigated through analytical and numerical computations. Through the technique of multiple scale expansion, we successfully reduced the complex nonlinear system of equations to a more tractable and solvable nonlinear amplitude equation on which the analysis of linear stability is performed. Interestingly from this analysis, a plot of critical amplitude of action potential versus wave number revealed the growth rate of modulational instability (MI) is an increasing function of the thermoelectric couplings; [Formula: see text] and [Formula: see text], under fixed conditions of nonlinear electrical couplings. In order to verify our analytical predictions through the study the long-time evolution of the modulated cardiac impulses, numerical computation is finally carried out. Numerical experiment revealed the existence of localized coherent structures with some recognized features of synchronization. Through the mechanism of MI, changes in thermoelectrical couplings promote wave localization and mode transition in electrical activities in the cell lattice. Results could provide new insights in understanding the underlying mechanism of the manifestation of sudden heart disorder subjected to heavily temperature fluctuation.

Generation of localized patterns in anharmonic lattices with cubic-quintic nonlinearities and fourth-order dispersion via a variational approach

We use the time-dependent variational method to examine the formation of localized patterns in dynamically unstable anharmonic lattices with cubic-quintic nonlinearities and fourth-order dispersion. The governing equation is an extended nonlinear Schrödinger equation known for modified Frankel-Kontorova models of atomic lattices and here derived from an extended Bose-Hubbard model of bosonic lattices with local three-body interactions. In presence of modulated waves, we derive and investigate the ordinary differential equations for the time evolution of the amplitude and phase of dynamical perturbation. Through an effective potential, we find the modulationally unstable domains of the lattice and discuss the effect of the fourth-order dispersion in the dynamics. Direct numerical simulations are performed to support our analytical results, and a good agreement is found. Various types of localized patterns, including breathers and solitonic chirped-like pulses, form in the system as a result of interplay between the cubic-quintic nonlinearities and the second- and fourth-order dispersions.

Dynamical instability of a Bose-Einstein condensate with higher-order interactions in an optical potential through a variational approach

We investigate the dynamical instability of Bose-Einstein condensates (BECs) with higher-order interactions immersed in an optical lattice with weak driving harmonic potential. For this, we compute both analytically and numerically a modified Gross-Pitaevskii equation with higher-order nonlinearity and external potentials generated by magnetic and optical fields. Using the time-dependent variational approach, we derive the ordinary differential equations for the time evolution of the amplitude and phase of modulational perturbation. Through an effective potential, we obtain the modulational instability condition of BECs and discuss the effect of the higher-order interaction in the dynamics of the condensates in presence of optical potential. We perform direct numerical simulations to support our analytical results, and good agreement is found.

Modulated pressure waves in large elastic tubes

Modulational instability is the direct way for the emergence of wave patterns and localized structures in nonlinear systems. We show in this work that it can be explored in the framework of blood flow models. The whole modified Navier-Stokes equations are reduced to a difference-differential amplitude equation. The modulational instability criterion is therefore derived from the latter, and unstable patterns occurrence is discussed on the basis of the nonlinear parameter model of the vessel. It is found that the critical amplitude is an increasing function of α, whereas the region of instability expands. The subsequent modulated pressure waves are obtained through numerical simulations, in agreement with our analytical expectations. Different classes of modulated pressure waves are obtained, and their close relationship with Mayer waves is discussed.

Dynamics of kink-dark solitons in Bose-Einstein condensates with both two- and three-body interactions

The matter-wave solutions of Bose-Einstein condensates with three-body interaction are examined through the one-dimensional Gross-Pitaevskii equation. By using a modified lens-type transformation and a further extension of the tanh-function method we obtain the exact analytical solutions which describe the propagation of kink-shaped solitons, anti-kink-shaped solitons, and other families of solitary waves. We realize that the shape of a kink solitary wave depends on both the scattering length and the parameter of atomic exchange with the substrate. The stability of the solitary waves is examined using analytical and numerical methods. Our results can also be applied to nonlinear optics in the presence of cubic-quintic media.

Wave train generation of solitons in systems with higher-order nonlinearities

Considering the higher-order nonlinearities in a material can significantly change its behavior. We suggest the extended nonlinear Schrödinger equation to describe the propagation of ultrashort optical pulses through a dispersive medium with higher-order nonlinearities. Soliton trains are generated through the modulational instability and we point out the influence of the septic nonlinearity in the modulational instability gain. Experimental values are used for the numerical simulations and the input plane wave leads to the development of pulse trains, depending upon the sign of the septic nonlinearity.

Modulational instability in two-component discrete media with cubic-quintic nonlinearity

The effect of cubic-quintic nonlinearity and associated intercomponent couplings on the modulational instability (MI) of plane-wave solutions of the two-component discrete nonlinear Schrödinger (DNLS) equation is considered. Conditions for the onset of MI are revealed and the growth rate of small perturbations is analytically derived. For the same set of initial parameters as equal amplitudes of plane waves and intercomponent coupling coefficients, the effect of quintic nonlinearity on MI is found to be essentially stronger than the effect of cubic nonlinearity. Analytical predictions are supported by numerical simulations of the underlying coupled cubic-quintic DNLS equation. Relevance of obtained results to dense Bose-Einstein condensates (BECs) in deep optical lattices, when three-body processes are essential, is discussed. In particular, the phase separation under the effect of MI in a two-component repulsive BEC loaded in a deep optical lattice is predicted and found in numerical simulations. Bimodal light propagation in waveguide arrays fabricated from optical materials with non-Kerr nonlinearity is discussed as another possible physical realization for the considered model.