This paper introduces
analytical and numerical solutions of the nonlinear Langevin’s equation under
square nonlinearity with stochastic non-homogeneity. The solution is obtained by using the Wiener-Hermite expansion with
perturbation (WHEP) technique, and the
results are compared with those of Picard iterations and the homotopy
perturbation method (HPM). The WHEP technique is used to obtain up to fourth
order approximation for different number of corrections. The mean and variance
of the solution are obtained and compared among the different methods, and some parametric studies are done by using Matlab.

In this paper, quadratic nonlinear oscillators under stochastic
excitation are considered. The Wiener-Hermite expansion with perturbation
(WHEP) method and the homotopy perturbation method (HPM) are used and compared.
Different approximation orders are considered and statistical moments are
computed in the two methods. The two methods show efficiency in estimating the
stochastic response of the nonlinear differential equations.

Abstract:
This work deals with soliton solutions of the nonlinear Schroedinger equation with cubic and quintic nonlinearities. We extend the procedure put forward in a recent Letter and we solve the equation in the presence of linear background, and cubic and quintic interactions which are modulated in space and time. As a result, we show how a simple parameter can be used to generate brightlike or darklike localized nonlinear waves which oscillate in several distinct ways, driven by the space and time dependence of the parameters that control the trapping potential, and the cubic and quintic nonlinearities.

Abstract:
We consider the existence of a dynamically stable soliton in the one-dimensional cubic-quintic nonlinear Schr\"odinger model with strong cubic nonlinearity management for periodic and random modulations. We show that the predictions of the averaged cubic-quintic NLS equation and modified variational approach for the arrest of collapse coincide. The analytical results are confirmed by numerical simulations of one-dimensional cubic-quintic NLS equation with rapidly and strongly varying cubic nonlinearity coefficient.

Abstract:
The interaction of both scalar and counter-rotating polarized steady state pulses (SSP) is studied numerically for a medium characterized by nonlinear susceptibilities of the third and the fifth order (a cubic-quintic medium with,$\chi_{3}>0,\chi_{5}<0$). The collision of two plateau-shaped solitons proved to be essentially inelastic, as a number of secondary elliptically polarized solitary waves arise as a result of interaction of steady-state pulses.

Abstract:
The influence of an externally applied magnetic field upon classic cubic quintic dissipative solitons is investigated using both exact simulations and a Lagrangian technique. The basic approach is to use a spatially inhomogeneous magnetic field and to consider two important geometries, namely the Voigt and the Faraday effects. A layered structure is selected for the Voigt case with the principal aim being to demonstrate non-reciprocal behaviour for various classes of spatial solitons that are known to exist as solutions of the complex Ginzburg-Landau cubic-quintic envelope equation under dissipative conditions. The system is viewed as dynamical and an opportunity is taken to display the behaviour patterns of the spatial solitons in terms of two-dimensional dynamical plots involving the total energy and the peak amplitude of the spatial solitons. This action this leads to limit cycle plots that beautifully reveal the behaviour of the solitons solutions at all points along the propagation axis. The closed contour that exists in the absence of a magnetic field is opened up and a limit point is exposed. The onset of chaos is revealed in a dramatic way and it is clear that detailed control by the external magnetic field can be exercised. The Lagrangian approach is adjusted to deal with dissipative systems and through the choice of particular trial functions, aspects of the dynamic behaviour of the spatial are predicted by this approach. Finally, some vortex dynamics in the Faraday configuration are investigated.

Abstract:
Using theoretical arguments, we prove the numerically well-known fact that the eigenvalues of all localized stationary solutions of the cubic-quintic 2D+1 nonlinear Schrodinger equation exhibit an upper cut-off value. The existence of the cut-off is inferred using Gagliardo-Nirenberg and Holder inequalities together with Pohozaev identities. We also show that, in the limit of eigenvalues close to zero, the eigenstates of the cubic-quintic nonlinear Schrodinger equation behave similarly to those of the cubic nonlinear Schrodinger equation.

This paper presents a
comprehensive stability analysis of the dynamics of the damped cubic-quintic
Duffing oscillator. We employ the derivative expansion method to investigate
the slightly damped cubic-quintic Duffing oscillator obtaining a uniformly
valid solution. We obtain a uniformly valid solution of the un-damped
cubic-quintic Duffing oscillator as a special case of our solution. A phase
plane analysis of the damped cubic-quintic Duffing oscillator is undertaken
showing some chaotic dynamics which sends a signal that the oscillator may be
useful as model for prediction of earth- quake occurrence.

Abstract:
In the present work, we examine the combined effects of cubic and quintic terms of the long range type in the dynamics of a double well potential. Employing a two-mode approximation, we systematically develop two cubic-quintic ordinary differential equations and assess the contributions of the long-range interactions in each of the relevant prefactors, gauging how to simplify the ensuing dynamical system. Finally, we obtain a reduced canonical description for the conjugate variables of relative population imbalance and relative phase between the two wells and proceed to a dynamical systems analysis of the resulting pair of ordinary differential equations. While in the case of cubic and quintic interactions of the same kind (e.g. both attractive or both repulsive), only a symmetry breaking bifurcation can be identified, a remarkable effect that emerges e.g. in the setting of repulsive cubic but attractive quintic interactions is a "symmetry restoring" bifurcation. Namely, in addition to the supercritical pitchfork that leads to a spontaneous symmetry breaking of the anti-symmetric state, there is a subcritical pitchfork that eventually reunites the asymmetric daughter branch with the anti-symmetric parent one. The relevant bifurcations, the stability of the branches and their dynamical implications are examined both in the reduced (ODE) and in the full (PDE) setting.

Abstract:
We have investigated modulation instability in metamaterials (MM) with both cubic and quintic nonlinearities, based on a model appropriate for pulse propagation in MMs with cubic-quintic nonlinearities and higher order dispersive effects. We have included loss into account in our analysis and found that loss distorts the sidebands of the MI gain spectrum. We find that the combined effect of cubic-quintic nonlinearity increases the MI gain. The role of higher order nonlinear dispersive effects on MI has been also discussed.