Nonlinear Waves in Integrable and Nonintegrable Systems

The front matter includes the title page, series page, copyright page, TOC, list of figures, and preface

In this chapter, we show how nonlinear wave equations can be derived from physical systems under certain asymptotic limits.

1.1 The Nonlinear Schrödinger Equation for Weakly Nonlinear Wave Packets

The nonlinear Schrödinger (NLS) equation governs weakly nonlinear and dispersive wave packets in one-dimensional (1D) physical systems. It was first derived by Benney and Newell (1967) in a general setting. Zakharov (1968) also derived this equation in his study of modulational stability of deep water waves. Then Hasegawa and Tappert (1973) showed that the same equation governs light pulse propagation in optical fibers. Below, we illustrate the derivation of the NLS equation for wave packets in a general KdV-type model equation: ψt+ψxxx+ƒ(ψ)ψx=0,1.1 where ψ is a real variable, and ƒ (ψ) is a general real function of ψ. For the convenience of analysis, we expand ƒ (ψ) into a Taylor series: ƒ(ψ)=c1ψ+c2ψ2+c3ψ3+⋯.1.2 Here the constant term c0 in this Taylor expansion is omitted, because this term can be eliminated from Eq. (1.1) by a trivial coordinate transform. We also assume that c1 and c2 are not zero simultaneously, which is the generic case.

Now we consider the evolution of a low-amplitude slowly varying wave packet in Eq. (1.1). This wave packet is illustrated in Fig. 1.1. The leading-order approximation for this wave packet is ψ∼ε[U(x,t)ei(k0x−ω0t)+c.c.].1.3

Integrable equations are an important class of nonlinear wave equations. Notable examples include the KdV equation, the NLS equation, the sine-Gordon equation, the Kadomtsev—Petviashvili (KP) equation, and many others. These equations have the remarkable property that their solitary waves collide elastically; i.e., they pass through each other without change of shape or speed. In addition, these equations can be integrated exactly by the inverse scattering transform method. This method was developed first by Gardner et al. (1967) for the KdV equation. Lax (1968) realized that if a nonlinear wave equation is the compatibility condition between two linear operators (now called the Lax pair), then this equation would be integrable by the inverse scattering transform method. Based on this idea, Zakharov and Shabat (1971) showed that the NLS equation is also integrable. Subsequently, Ablowitz et al. (1974) developed a method to derive a large class of integrable equations such as an integrable hierarchy. From that time on, numerous advances have been made on integrable equations, and this field is still very active today.

In this and the next chapters, we cover the inverse scattering transform method and related theories on integrable equations. The original inverse scattering transform method was based on the Gel'fand—Levitan—Marchenko integral equations (Gardner et al. (1967), Ablowitz and Segur (1981)). Later on, a Riemann—Hilbert formulation was developed which streamlines and simplifies the inverse scattering transform method considerably (Zakharov et al. (1984)). Our presentation will be based on the Riemann—Hilbert formulation. In this chapter, we develop the theory for the (focusing) NLS equation and its integrable hierarchy. In the next chapter, theories for other higher-order integrable equations will be developed.

The NLS equation and the AKNS hierarchy in the previous chapter are associated with the second-order Zakharov—Shabat scattering problem. In this chapter, we consider integrable equations which are associated with higher-order scattering problems. Our treatments will closely resemble those in the previous chapter (with only minor modifications). Thus these treatments are universal for a large class of integrable systems.

First, we consider a few examples. One is the Manakov system iut+uxx+(|u|2+|v|2)u=0,3.1 ivt+vxx+(|u|2+|v|2)v=0,3.2 which was shown to be integrable by Manakov (1973). The scattering operator of the Lax pair for this system is Yx=−iζ0u0−iζv−u*−v*iζY,3.3 which is a third-order system. Another equation is the Sasa—Satsuma equation ut+uxxx+6|u|2ux+3u(|u|2)x=0,3.4 which was shown to be integrable by Sasa and Satsuma (1991).

Integrable equations as considered in the previous two chapters can describe certain physical wave systems at the lowest order of approximation. For instance, pulse transmission in optical fibers is governed by the NLS equation at the leading order. Integrable equations can support soliton solutions which travel without change of shape. When perturbations such as damping, higher-order dispersion, and higher-order nonlinearity are brought into consideration, a physical system is then modeled by a perturbed integrable equation (see Eq. (1.50) for optical pulses in fibers for instance). In a perturbed system, solitons may not propagate stationarily anymore. Their shapes may be distorted over time. In addition, energy radiation can be excited which can affect the soliton's evolution in nontrivial ways. In order to describe soliton evolution under perturbations, a soliton perturbation theory is required.

Perturbation theories for single solitons have been developed for many integrable equations. Examples include the KdV equation (Karpman and Maslov (1977), Kaup and Newell (1978a), Kodama (1985), Herman (1990), Grimshaw and Mitsudera (1993), Yan and Tang (1996)), the NLS equation (Kaup (1976a), Keener and Mclaughlin (1977), Kaup and Newell (1978a), Karpman and Maslov (1978), Kaup (1990), Hasegawa and Kodama (1995)), the sine-Gordon equation (Fogel et al. (1977)), the Benjamin—Ono equation (Kaup et al. (1999)), the derivative NLS equation (and the related modified NLS equation) (Shch-esnovich and Doktorov ( 1999), Chen and Yang (2002)), the Manakov equations (Lakoba and Kaup (1997), Shchesnovich and Doktorov (1997)), the massive Thirring model (Kaup and Lakoba (1996)), the fifth-order KdV equation (Yang (2001a)), the complex modified KdV equation (Yang (2003), Hoseini and Marchant (2009)), and many others (see also the review article by Kivshar and Malomed (1989) and the references therein).

Most nonlinear wave phenomena in science and engineering are governed by nonintegrable equations, and these equations are not near integrable systems either. In such cases, the inverse scattering theory and the soliton perturbation theory as presented in the previous three chapters are not applicable. Theories for nonintegrable equations started to appear forty years ago. By now, great advances have been made in this area, and many new phenomena unique to nonintegrable equations have been reported and theoretically explained. for instance, it has been found that solitary waves in nonintegrable equations can be unstable; collisions and weak interactions of solitary waves in nonintegrable equations can exhibit fractal-type dependence on initial conditions; and so on. These behaviors differ significantly from those in integrable systems. In this chapter, we present various theories for nonintegrable equations.

5.1 Solitary Waves in Nonintegrable Equations

Nonintegrable equations often admit solitary waves which are localized in space and propagate stationarily with time. These solitary waves are the counterpart of single solitons in integrable equations. These solitary waves, when exist, will play an important role in the solution dynamics. In this section, we use a generalized NLS equation to introduce these solitary waves.

The generalized NLS equation we consider is iUt+Uxx+F(|U|2)U=0.5.1 Here F(⋅) is a real function, and we assume F(0)=0. This equation describes nonlinear light propagation in a non-Kerr medium (see Chapter 1). It is nonintegrable in general (except the special case of F(|U|2)=|U|2, where it reduces to the NLS equation).

In previous chapters, we have shown nonlinear wave phenomena occurring in homogeneous media, where the corresponding model equations all have space-independent coefficients. In this chapter, we discuss nonlinear wave phenomena in periodic media, where the corresponding model equations have spatially periodic coefficients now. The periodic media fundamentally change the dispersion relations, and thus induce a wide variety of novel nonlinear phenomena which have no counterparts in homogeneous media. These new phenomena attracted a lot of interest in nonlinear optics and Bose—Einstein condensates and were heavily studied in the past decade.

First we describe the physical situations where nonlinear wave propagation in periodic media arises. In Sec. 1.2, we have shown that when a light beam travels in a Kerr medium, if the refractive index of the medium exhibits periodic transverse variation, then under certain approximations, propagation of this light beam is described by the NLS equation with an external potential (1.43). Physical examples include photonic crystal fibers with weak index variation (Argyros et al. (2005)), laser-written waveguide arrays (Szameit et al. (2006)), and planar etched waveguide arrays on AlGaAs substrates (Eisenberg et al. (1998), Iwanow et al. (2004)). After variable normalizations, this model equation reduces to iUz+Uxx+Uyy−V(x,y)U+σ|U|2U=0,6.1 where U is the envelope function of the light beam, z is the distance of propagation, (x, y) is the transverse plane, V (x, y) is a periodic potential function, and σ is the sign of nonlinearity. In most Kerr media, the nonlinearity is self-focusing, thus σ=1. Equation (6.1) is a model equation for nonlinear wave propagation in periodic media which exhibits Kerr (cubic) nonlinearity.

In photorefractive crystals, the nonlinearity is of saturable type (see Sec. 1.2).

Numerical computations play an important role in the study of nonlinear waves. Many important phenomena, such as elastic collision of KdV solitons and fractal scattering in solitary wave interactions, were discovered first by numerical computations (see Zabusky and Kruskal (1965) and Chapter 5). Such numerical discoveries motivated further analytical investigations, which then led to deeper understanding of those phenomena. Numerical computations are especially important for the study of nonintegrable equations. Indeed, in Chapters 5 and 6 on nonintegrable equations, many numerical results were presented. A dynamic interplay between numerics and analysis was a distinctive feature for the topics in those chapters. In recent years, great advances have been made on numerical methods for nonlinear wave equations, and many highly accurate and efficient numerical techniques have been developed. In addition, some convenient commercial software for scientific computing (such as MATLAB) have also become available, which greatly facilitated the coding and implementation of those numerical algorithms. Due to these advances, nontrivial numerical computations of nonlinear wave equations can often be done with several-inch-long MATLAB codes and hence are accessible to anybody with or without prior programming experience.

In this chapter, we describe numerical methods for various aspects of nonlinear wave computations such as evolution simulation, computation of solitary wave solutions, and computation of linear-stability spectra of solitary waves. Fundamental properties of these methods, such as their accuracy, numerical stability, convergence conditions, and convergence speeds, will be elaborated. We will show that all these methods we present have spectral spatial accuracy; i.e., the spatial error decays exponentially with the grid spacing (for smooth functions). Thus these methods yield highly accurate numerical results.

The back matter includes bibliography and index