اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدOscillatory dynamics of the classical Nonlinear Schrodinger equation
146
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We study numerically the statistical properties of the modulation instability (MI) developing from condensate solution seeded by weak, statistically homogeneous in space noise, in the framework of the classical (integrable) one-dimensional Nonlinear Schrodinger (NLS) equation. We demonstrate that in the nonlinear stage of the MI the moments of the solutions amplitudes oscillate with time around their asymptotic values very similar to sinusoidal law. The amplitudes of these oscillations decay with time $t$ as $t^{-3/2}$, the phases contain the nonlinear phase shift that decays as $t^{-1/2}$, and the period of the oscillations is equal to $pi$. The asymptotic values of the moments correspond to Rayleigh probability density function (PDF) of waves amplitudes appearance. We show that such behavior of the moments is governed by oscillatory-like, decaying with time, fluctuations of the PDF around the Rayleigh PDF; the time dependence of the PDF turns out to be very similar to that of the moments. We study how the oscillations that we observe depend on the initial noise properties and demonstrate that they should be visible for a very wide variety of statistical distributions of noise.
قيم البحث
اقرأ أيضاً
We consider an extension of the methodology of the modified method of simplest equation to the case of use of two simplest equations. The extended methodology is applied for obtaining exact solutions of model nonlinear partial differential equations
for deep water waves: the nonlinear Schrodinger equation. It is shown that the methodology works also for other equations of the nonlinear Schrodinger kind.
We generate hierarchies of derivative nonlinear Schrodinger-type equations and their nonlocal extensions from Lie algebra splittings and automorphisms. This provides an algebraic explanation of some known reductions and newly established nonlocal reductions in integrable systems.
We consider an integrable generalization of the nonlinear Schrodinger (NLS) equation that was recently derived by one of the authors using bi-Hamiltonian methods. This equation is related to the NLS equation in the same way that the Camassa Holm equa
tion is related to the KdV equation. In this paper we: (a) Use the bi-Hamiltonian structure to write down the first few conservation laws. (b) Derive a Lax pair. (c) Use the Lax pair to solve the initial value problem. (d) Analyze solitons.
The second-type derivative nonlinear Schrodinger (DNLSII) equation was introduced as an integrable model in 1979. Very recently, the DNLSII equation has been shown by an experiment to be a model of the evolution of optical pulses involving self-steep
ening without concomitant self-phase-modulation. In this paper the $n$-fold Darboux transformation (DT) $T_n$ of the coupled DNLSII equations is constructed in terms of determinants. Comparing with the usual DT of the soliton equations, this kind of DT is unusual because $T_n$ includes complicated integrals of seed solutions in the process of iteration. By a tedious analysis, these integrals are eliminated in $T_n$ except the integral of the seed solution. Moreover, this $T_n$ is reduced to the DT of the DNLSII equation under a reduction condition. As applications of $T_n$, the explicit expressions of soliton, rational soliton, breather, rogue wave and multi-rogue wave solutions for the DNLSII equation are displayed.
By using the Darboux transformation, we obtain two new types of exponential-and-rational mixed soliton solutions for the defocusing nonlocal nonlinear Schrodinger equation. We reveal that the first type of solution can display a large variety of inte
ractions among two exponential solitons and two rational solitons, in which the standard elastic interaction properties are preserved and each soliton could be either the dark or antidark type. By developing the asymptotic analysis technique, we also find that the second type of solution can exhibit the elastic interactions among four mixed asymptotic solitons. But in sharp contrast to the common solitons, the asymptotic mixed solitons have the t-dependent velocities and their phase shifts before and after interaction also grow with |t| in the logarithmical manner. In addition, we discuss the degenerate cases for such two types of mixed soliton solutions when the four-soliton interaction reduces to a three-soliton or two-soliton interaction.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
