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Register a new userNew Hierarchies of Derivative nonlinear Schrodinger-Type Equation
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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.
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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-steepening 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.
We propose a general integrable lattice system involving some free parameters, which contains known integrable lattice systems such as the Ablowitz-Ladik discretization of the nonlinear Schrodinger (NLS) equation as special cases. With a suitable choice of the parameters, it provides a new integrable space-discretization of the derivative NLS equation known as the Chen-Lee-Liu equation. Analogously to the continuous case, the space-discrete Chen-Lee-Liu system possesses a Lax pair and admits a complex conjugation reduction between the two dependent variables. Thus, we obtain a proper space-discretization of the Chen-Lee-Liu equation defined on the three lattice sites $n-1$, $n$, $n+1$ for the first time. Considering a negative flow of the discrete Chen-Lee-Liu hierarchy, we obtain a proper discretization of the massive Thirring model in light-cone coordinates. Multicomponent generalizations of the obtained discrete equations are straightforward because the performed computations are valid for the general case where the dependent variables are vector- or matrix-valued.
We derive asymptotic formulas for the solution of the derivative nonlinear Schrodinger equation on the half-line under the assumption that the initial and boundary values lie in the Schwartz class. The formulas clearly show the effect of the boundary on the solution. The approach is based on a nonlinear steepest descent analysis of an associated Riemann-Hilbert problem.
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.
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.
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