Stability analysis of the D-dimensional nonlinear Schrodinger equation with trap and two- and three-body interactions

Considering the static solutions of the D-dimensional nonlinear Schrodinger equation with trap and attractive two-body interactions, the existence of stable solutions is limited to a maximum critical number of particles, when D greater than or equal to 2. In case D = 2, we compare the variational approach with the exact numerical calculations. We show that, the addition of a positive three-body interaction allows stable solutions beyond the critical number. In this case, we also introduce a dynamical analysis of the conditions for the collapse. (C) 2000 Published by Elsevier B.V. B.V. All rights reserved.

Asymptotic soliton train solutions of the defocusing nonlinear Schrodinger equation

Asymptotic behavior of initially large and smooth pulses is investigated at two typical stages of their evolution governed by the defocusing nonlinear Schrodinger equation. At first, wave breaking phenomenon is studied in the limit of small dispersion. A solution of the Whitham modulational equations is found for the case of dissipationless shock wave arising after the wave breaking point. Then, asymptotic soliton trains arising eventually from a large and smooth initial pulse are studied by means of a semiclassical method. The parameter varying along the soliton train is calculated from the generalized Bohr-Sommerfeld quantization rule, so that the distribution of eigenvalues depends on two functions-intensity rho(0)(x) of the initial pulse and its initial chirp v(0)(x). The influence of the initial chirp on the asymptotic state is investigated. Excellent agreement of the numerical solution of the defocusing NLS equation with predictions of the asymptotic theory is found.

GALACTIC NONLINEAR DYNAMIC MODEL

We develop a model for spiral galaxies based on a nonlinear realization of the Newtonian dynamics starting from the momentum and mass conservations in the phase space. The radial solution exhibits a rotation curve in qualitative accordance with the observational data.

Nonlinear Schrodinger equation for a superfluid Bose gas from weak coupling to unitarity: Study of vortices

We introduce a nonlinear Schrodinger equation to describe the dynamics of a superfluid Bose gas in the crossover from the weak-coupling regime, where an(1/3)<<1 with a the interatomic s-wave scattering length and n the bosonic density, to the unitarity limit, where a ->+infinity. We call this equation the unitarity Schrodinger equation (USE). The zero-temperature bulk equation of state of this USE is parametrized by the Lee-Yang-Huang low-density expansion and Jastrow calculations at unitarity. With the help of the USE we study the profiles of quantized vortices and vortex-core radius in a uniform Bose gas. We also consider quantized vortices in a Bose gas under cylindrically symmetric harmonic confinement and study their profile and chemical potential using the USE and compare the results with those obtained from the Gross-Pitaevskii-type equations valid in the weak-coupling limit. Finally, the USE is applied to calculate the breathing modes of the confined Bose gas as a function of the scattering length.

Effective nonlinear Schrodinger equations for cigar-shaped and disc-shaped Fermi superfluids at unitarity

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Bound state solution of the grassmannian nonlinear sigma model with fermions

We construct the S-matrix for bound state (gauge-invariant) scattering for nonlinear sigma models defined on the manifold SU(n) S(U(p)⊗U(n-p)) with fermions. It is not possible to compute gauge non-singlet matrix elements. In the present language, constraints from higher conservation laws determine the bound state solution. An alternative derivation is also presented. © 1988.

Nonlinear dynamic model and simulation of morphing wing profile actuated by shape memory alloys

Morphing aircraft have the ability to actively adapt and change their shape to achieve different missions efficiently. The development of morphing structures is deeply related with the ability to model precisely different designs in order to evaluate its characteristics. This paper addresses the dynamic modeling of a sectioned wing profile (morphing airfoil) connected by rotational joints (hinges). In this proposal, a pair of shape memory alloy (SMA) wires are connected to subsequent sections providing torque by reducing its length (changing airfoil camber). The dynamic model of the structure is presented for one pair of sections considering the system with one degree of freedom. The motion equations are solved using numerical techniques due the nonlinearities of the model. The numerical results are compared with experimental data and a discussion of how good this approach captures the physical phenomena associated with this problem. © The Society for Experimental Mechanics, Inc. 2012.

Radial sign-changing solutions to biharmonic nonlinear Schrodinger equations

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Wide localized solutions of the parity-time-symmetric nonautonomous nonlinear Schrodinger equation

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Hirota method for oblique solitons in two-dimensional supersonic nonlinear Schrodinger flow

In a previous work El et al. (2006) [1] exact stable oblique soliton solutions were revealed in two-dimensional nonlinear Schrodinger flow. In this work we show that single soliton solution can be expressed within the Hirota bilinear formalism. An attempt to build two-soliton solutions shows that the system is "close" to integrability provided that the angle between the solitons is small and/or we are in the hypersonic limit. (C) 2012 Elsevier B.V. All rights reserved.

The concept of quasi-integrability for modified non-linear Schrodinger models

We consider modifications of the nonlinear Schrodinger model (NLS) to look at the recently introduced concept of quasi-integrability. We show that such models possess an in finite number of quasi-conserved charges which present intriguing properties in relation to very specific space-time parity transformations. For the case of two-soliton solutions where the fields are eigenstates of this parity, those charges are asymptotically conserved in the scattering process of the solitons. Even though the charges vary in time their values in the far past and the far future are the same. Such results are obtained through analytical and numerical methods, and employ adaptations of algebraic techniques used in integrable field theories. Our findings may have important consequences on the applications of these models in several areas of non-linear science. We make a detailed numerical study of the modified NLS potential of the form V similar to (vertical bar psi vertical bar(2))(2+epsilon), with epsilon being a perturbation parameter. We perform numerical simulations of the scattering of solitons for this model and find a good agreement with the results predicted by the analytical considerations. Our paper shows that the quasi-integrability concepts recently proposed in the context of modifications of the sine-Gordon model remain valid for perturbations of the NLS model.

Experimental validation of a nonlinear μ FE model based on cohesive-frictional plasticity for trabecular bone

Trabecular bone is a porous mineralized tissue playing a major load bearing role in the human body. Prediction of age-related and disease-related fractures and the behavior of bone implant systems needs a thorough understanding of its structure-mechanical property relationships, which can be obtained using microcomputed tomography-based finite element modeling. In this study, a nonlinear model for trabecular bone as a cohesive-frictional material was implemented in a large-scale computational framework and validated by comparison of μFE simulations with experimental tests in uniaxial tension and compression. A good correspondence of stiffness and yield points between simulations and experiments was found for a wide range of bone volume fraction and degree of anisotropy in both tension and compression using a non-calibrated, average set of material parameters. These results demonstrate the ability of the model to capture the effects leading to failure of bone for three anatomical sites and several donors, which may be used to determine the apparent behavior of trabecular bone and its evolution with age, disease, and treatment in the future.

The development and stability analysis of a nonlinear growth model for microorganisms

A nonlinear dynamic model of microbial growth is established based on the theories of the diffusion response of thermodynamics and the chemotactic response of biology. Except for the two traditional variables, i.e. the density of bacteria and the concentration of attractant, the pH value, a crucial influencing factor to the microbial growth, is also considered in this model. The pH effect on the microbial growth is taken as a Gaussian function G0e-(f- fc)2/G1, where G0, G1 and fc are constants, f represents the pH value and fc represents the critical pH value that best fits for microbial growth. To study the effects of the reproduction rate of the bacteria and the pH value on the stability of the system, three parameters a, G0 and G1 are studied in detail, where a denotes the reproduction rate of the bacteria, G0 denotes the impacting intensity of the pH value to microbial growth and G1 denotes the bacterial adaptability to the pH value. When the effect of the pH value of the solution which microorganisms live in is ignored in the governing equations of the model, the microbial system is more stable with larger a. When the effect of the bacterial chemotaxis is ignored, the microbial system is more stable with the larger G1 and more unstable with the larger G0 for f0 > fc. However, the stability of the microbial system is almost unaffected by the variation G0 and G1 and it is always stable for f0 < fc under the assumed conditions in this paper. In the whole system model, it is more unstable with larger G1 and more stable with larger G0 for f0 < fc. The system is more stable with larger G1 and more unstable with larger G0 for f0 > fc. However, the system is more unstable with larger a for f0 < fc and the stability of the system is almost unaffected by a for f0 > fc. The results obtained in this study provide a biophysical insight into the understanding of the growth and stability behavior of microorganisms.

Type-I intermittency with discontinuous reinjection probability density in a truncation model of the derivative nonlinear Schrödinger equation

In previous papers, the type-I intermittent phenomenon with continuous reinjection probability density (RPD) has been extensively studied. However, in this paper type-I intermittency considering discontinuous RPD function in one-dimensional maps is analyzed. To carry out the present study the analytic approximation presented by del Río and Elaskar (Int. J. Bifurc. Chaos 20:1185-1191, 2010) and Elaskar et al. (Physica A. 390:2759-2768, 2011) is extended to consider discontinuous RPD functions. The results of this analysis show that the characteristic relation only depends on the position of the lower bound of reinjection (LBR), therefore for the LBR below the tangent point the relation {Mathematical expression}, where {Mathematical expression} is the control parameter, remains robust regardless the form of the RPD, although the average of the laminar phases {Mathematical expression} can change. Finally, the study of discontinuous RPD for type-I intermittency which occurs in a three-wave truncation model for the derivative nonlinear Schrodinger equation is presented. In all tests the theoretical results properly verify the numerical data

Channel model and lower capacity bound for the transmission based on nonlinear Fourier transform

The integrability of the nonlinear Schräodinger equation (NLSE) by the inverse scattering transform shown in a seminal work [1] gave an interesting opportunity to treat the corresponding nonlinear channel similar to a linear one by using the nonlinear Fourier transform. Integrability of the NLSE is in the background of the old idea of eigenvalue communications [2] that was resurrected in recent works [3{7]. In [6, 7] the new method for the coherent optical transmission employing the continuous nonlinear spectral data | nonlinear inverse synthesis was introduced. It assumes the modulation and detection of data using directly the continuous part of nonlinear spectrum associated with an integrable transmission channel (the NLSE in the case considered). Although such a transmission method is inherently free from nonlinear impairments, the noisy signal corruptions, arising due to the ampli¯er spontaneous emission, inevitably degrade the optical system performance. We study properties of the noise-corrupted channel model in the nonlinear spectral domain attributed to NLSE. We derive the general stochastic equations governing the signal evolution inside the nonlinear spectral domain and elucidate the properties of the emerging nonlinear spectral noise using well-established methods of perturbation theory based on inverse scattering transform [8]. It is shown that in the presence of small noise the communication channel in the nonlinear domain is the additive Gaussian channel with memory and signal-dependent correlation matrix. We demonstrate that the effective spectral noise acquires colouring", its autocorrelation function becomes slow decaying and non-diagonal as a function of \frequencies", and the noise loses its circular symmetry, becoming elliptically polarized. Then we derive a low bound for the spectral effiency for such a channel. Our main result is that by using the nonlinear spectral techniques one can significantly increase the achievable spectral effiency compared to the currently available methods [9]. REFERENCES 1. Zakharov, V. E. and A. B. Shabat, Sov. Phys. JETP, Vol. 34, 62{69, 1972. 2. Hasegawa, A. and T. Nyu, J. Lightwave Technol., Vol. 11, 395{399, 1993. 3. Yousefi, M. I. and F. R. Kschischang, IEEE Trans. Inf. Theory, Vol. 60, 4312{4328, 2014. 4. Yousefi, M. I. and F. R. Kschischang, IEEE Trans. Inf. Theory, Vol. 60, 4329{4345 2014. 5. Yousefi, M. I. and F. R. Kschischang, IEEE Trans. Inf. Theory, Vol. 60, 4346{4369, 2014. 6. Prilepsky, J. E., S. A. Derevyanko, K. J. Blow, I. Gabitov, and S. K. Turitsyn, Phys. Rev. Lett., Vol. 113, 013901, 2014. 7. Le, S. T., J. E. Prilepsky, and S. K. Turitsyn, Opt. Express, Vol. 22, 26720{26741, 2014. 8. Kaup, D. J. and A. C. Newell, Proc. R. Soc. Lond. A, Vol. 361, 413{446, 1978. 9. Essiambre, R.-J., G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, J. Lightwave Technol., Vol. 28, 662{701, 2010.