# On the Whitham equations for the defocusing nonlinear Schrodinger equation with step initia.pdf

arXiv:nlin/0508036v1 [nlin.SI] 30 Aug 2005On the Whitham equations for the defocusing nonlinearSchr¨odinger equation with step initial dataGino Biondini∗ and Yuji Kodama†∗: State University of New York at Buffalo, Department of Mathematics, Buffalo, NY 14260†: Ohio State University, Department of Mathematics, Columbus, OH 43210February 8, 2008AbstractThe behavior of solutions of the finite-genus Whitham equations for the weak dispersion limit of thedefocusing nonlinear Schr¨odinger equation is investigated analytically and numerically for piecewise-constant initial data. In particular, the dynamics of constant-amplitude initial conditions with one ormore frequency jumps (i.e., piecewise linear phase) are considered. It is shown analytically and numer-ically that, for finite times, regions of arbitrarily high genus can be produced; asymptotically with time,however, the solution can be divided into expanding regions which are either of genus-zero, genus-oneor genus-two type, their precise arrangement depending on the specifics of the initial datum given. Thisbehavior should be compared to that of the Korteweg-de Vries equation, where the solution is devidedinto the regions which are either genus-zero or genus-one asymptotically. Finally, the potential appli-cation of these results to the generation of short optical pulses is discussed: the method proposed takesadvantage of nonlinear compression via appropriate frequency modulation, and allows control of boththe pulse amplitude and its width, as well as the distance along the fiber at which the pulse is produced.The weak dispersion limit of the defocusing nonlinear Schr¨odinger (NLS) equation has been extensivelystudied in recent years (see e.g., Refs. [13, 22, 27, 29, 31]), and in some sense it is well-characterizedmathematically. Not as much is known, however, about the detailed behavior of the solutions for specificchoices of initial datum[9, 10, 14, 25]. The purpose of this work is to present analytical and numerical resultsregarding the behavior of a special class of solutions of the NLS equation in the weak dispersion limit. Moreprecisely, we consider the initial value problem for the NLS equation in the weak dispersion limit with theinitial data having a constant-amplitude and piecewise constant frequency. We believe that, on one hand,this behavior is interesting mathematically, and that, on the other hand, it could have potential applicationsin the generation of intense, ultra-short optical pulses. As such, it is worthy of further study.The structure of this document is as follows: in section 1 we introduce the problem, and in section 2 wereview some well-known results regarding the weak dispersion limit of the NLS equation. In section 3 wediscuss the behavior of solutions corresponding to “single-jump” initial conditions, which are the startingpoint for our investigation. Then, in sections 4 and 5 we present the analytical calculations which are themain results of this work. In section 6 we demonstrate these results through numerical simulations of theNLS equation and obtain further information about the solution behavior, and in section 7 we discuss theapplication of our results to the generation of intense, ultra-short optical pulses. Appendix A.1 describes1our nondimensionalizations and our choice of units, Appendices A.2 and A.3 review some known resultsregarding genus-one (i.e., periodic) solutions of the NLS equation, the Whitham averaging method andthe NLS-Whitham equations, and Appendix A.4 gives the details of some calculations whose results arepresented in sections 3 and 4.1 The NLS equation with small dispersionIn this section we recall some basic results regarding the behavior of solutions of the nonlinear Schr¨odinger(NLS) equation with small dispersion. This will establish the background and the notation necessary toextend these results in the following sections.The semiclassical limit of the NLS equation. We start from the defocusing NLS equation (that is, theNLS in the normal dispersion regime in optical fibers) in dimensionless form:iε∂q∂t − 12ε2 ∂2q∂x2 +|q|2q = 0, (1.1)where we assume 0 0, and if ε≪1,Eqs. (1.4) are approximated to leading order by the following reduced hydrodynamical system[26, 34]∂∂tparenleftbiggρuparenrightbigg=parenleftbiggu ρ1 uparenrightbigg ∂∂xparenleftbiggρuparenrightbigg. (1.5)Equation (1.5) is called the dispersionless NLS equaton, and is used to describe a surface wave motion inshallow water. In the hydrodynamical setting, ρ and −u represent respectively the depth and velocity ofwater, and x and t are dimensionless space and time. For ρ 0, the eigenvalues u±√ρ of the coefficientmatrix are real, and the system (1.5) is strictly hyperbolic. This system, which is known as the shallow waterwave equations and has been intensively studied (see e.g. Ref. [34]), can be rewritten in Riemann invariant(i.e., diagonal) form∂rk∂t = sk∂rk∂x , k = 1,2, (1.6)where the Riemann invariants r1,2(x,t) are given byr1 = u−2√ρ, r2 = u+2√ρ, (1.7)and the characteristic speeds s1,2(x,t) ares1 = 14(3r1 +r2) = u−√ρ, s2 = 14(r1 +3r2) = u+√ρ.Note that sk 0 implies a left-moving wave (i.e., dx/dt =−sk), and that Eqs. (1.7) are equivalent toρ = 116(r2 −r1)2 , u = 12(r1 +r2). (1.8)Since the system of PDEs described by Eq. (1.6) is strictly hyperbolic, it is possible to show that, for“rarefaction” initial data, namely, when r1,2(x,0) are both monotonically decreasing functions of x, a globalsolution exists for all t 0. In many cases of interest, however, the initial data do not satisfy this propertyand as a consequence they develop a shock, as we shall see in the following.Square-wave initial conditions. Consider first the initial datum given by the following rectangular pulseof width 2L:ρ(x,0) =braceleftbiggq20 |x| L, (1.9a)u(x,0) = 0, (1.9b)with q0 0, corresponding in the optics framework to an NRZ pulse. The system of equations (1.5) withinitial conditions (1.9) is known in the literature as the “dam-breaking” problem. (In the hydrodynamicanalogy, the problem describes the behavior of a mass of water which is initially confined in a uniform,3spatially localized state by two dams located at x = ±L and both of which are removed at t = 0.) TheRiemann invariants for this situation are shown as the dashed lines in Fig. 2.1a. Note that the initial datafor the Riemann invariants r1, r2 corresponding to the initial condition (1.9) is not of rarefaction type. Itis possible however to obtain initial data of rarefaction-type by properly redefining the initial value of theinvariants r1 and r2 for |x| L, as shown in Fig. 2.1b later. (This procedure is a special case of the processknown as regularization, see next section.) Then the system has the following solution up to the timet0 = L/q0: for 0 L +2q0 t, ρ(x,t) = u(x,t) = 0, with ρ(−x,t) = ρ(x,t) and u(−x,t) = −u(x,t) (cf. Ref. [25]).The full solution of the NLS equation (1.1) with initial condition (1.9), as obtained from numerical simu-lations (described in section 6) is depicted in Fig. 1.1a. (This kind of solution is usually called a fan in thecontext of hydrodynamics.) The speeds of the boundaries of the top (ρ(x,t) = q20) and bottom (ρ(x,t) = 0)regions are easily obtained from Eqs. (1.10) (see also Fig. 2.1b later); these two speeds are respectivelys−2 = q0 and s+2 =−2q0.Equations (1.10) cease to be valid beyond the time t0 = L/q0 when the boundaries of the top regionmeet at x = 0 (i.e., the time at which the left-moving characteristic emanating from x = L meets with theright-moving characteristic from x = −L). An analytical expression for q(x,t) when t L/q0 however canbe obtained using the dispersionless limit of the scattering transform for the NLS equation[31]. The corre-sponding behavior of the full solution of the NLS equation is shown in Fig. 1.1b. The appearance of smalloscillations in the numerical solution in Figs. 1.1a,b was discussed in Ref. [14], and is the consequence of ap-proximating the discontinuous initial data (1.9) with a continuous initial datum in the numerical simulations(as described in section 6). It should also be noted that some care must be taken regarding the regulariza-tion of the discontinuous initial datum (1.9) for Eq. (1.5), since a weak solution with a discontinuity is notunique. In fact, one can construct a different solution of (1.4) with a discontinuity. In our case, however,the regularization described in section 2 enforces the continuity of ρ(x,t), thus removing the ambiguity andproducing a unique solution.Frequency jumps and high-frequency oscillations. In terms of optical pulses, the above results implythat the initial condition in Eq. (1.9) rapidly spreads out, as shown in Figs. 1.1a,b. This behavior can bepartly prevented (or, alternatively, reinforced) by employing initial conditions with nontrivial phase. Forexample, consider the following:u(x,0) =braceleftbigg−u0 x 0, (1.9b′)with ρ(x,0) still given by Eq. (1.9a). Hereafter, we will use r01(x) and r02(x) to refer to the value of theRiemann invariants (1.7) at t = 0. If ρ(x,0) is given by Eq. (1.9a) and u(x,0) by Eq. (1.9b′), for |x| 0, (1.11a)r02(x) =braceleftbigg−u0 +2q0 x 0, (1.11b)4as shown by dashed lines in Figs. 3.2a,b. Note that the value of the invariants can be redefined for |x| L,since ρ(x,0) = 0 there.In terms of optical pulses, Eq. (1.9b′) amounts to imposing a frequency jump at the center of the pulse(x = 0). If u0 0, the right half of the pulse acquires a positive frequency and the left half a negativefrequency. Thus, owing to the normal dispersion, the two halves of the pulse will tend to move towards eachother.[26] In terms of the hydrodynamical problem, this corresponds to assigning an inward initial velocityto the mass of water, as if two pistons were acting on each side of it. (For this reason, this case is oftenreferred to as the “piston” problem.) Note that if u0 0 the initial data r01,2(x) are increasing. Thus, anotherconsequence of the initial frequency jump is that if u0 0 a shock develops at x = 0, and the solutiondevelops high-frequency oscillations, as shown in Fig. 3.1a. (This type of shock is called collisionless, ordispersive, to distinguish it from the usual type of shock, which is dissipative; e.g., see Ref. [34].) Thecharacteristic frequency of these oscillations is O(1/ε) (i.e., one period of the oscillation shown in Fig. 3.1ais of order ε). If u0 0, (2.4a)rk 0, i.e., it is regular.In general, the initial conditions for the two Riemann invariants obtained from the dispersionless sys-tem (1.4) do not satisfy monotonicity and the separability condition (2.5). In other words, Eqs. (1.7) (thatis, the system (2.3) with g = 0) in general do not have a global solution. The regularization process thenconsists in enlarging the set of Riemann invariants so that the resulting NLS-Whitham equations have aglobal solution. This is done by representing the initial data for the NLS Eq. (1.1) in such a way that allthe Riemann invariants r1,.,r2g+2 are monotonically decreasing functions of x at t = 0. Note that it isalways possible to do so for piecewise-constant initial data, since in this case there is some ambiguity inhow the spectrum of the Lax operator is represented in terms of the Riemann invariants. This ambiguitycan then be exploited to redefine the initial datum for the Whitham Eqs. (2.3) by adding degenerate gaps(e.g., see Figs. 3.2 and 4.1). All of the solutions discussed in sections 3, 4 and 5 fall within the framework ofpiecewise-constant initial conditions. In this case the data at t = 0 are always genus-0, but highly degenerate,in the sense that they can be described in terms of a higher genus with degenerate gaps.It should be noted that the adiabatic approximation (the use of the Whitham equations for the slowevolution) implies that the local genus of the solution is preserved. A separate issue is how the genus of thesolution changes from one region to the next. This is a bifurcation problem through a critical point, and canbe approached by regularization, i.e., by trying to patch two Whitham systems with different genus at thispoint. Our regularization acts as a “globalization”, in the sense that the Whitham equations with a proper (ingeneral larger) genus now describes a global behavior beyond the perturbation range. For fixed x, a regularscheme must solve a connection problem in t because of the change of genus, and for fixed t one also needsto solve a connection problem to match regions with different genus.It is also important to realize that for piecewise-constant initial data there is more than one way toxrr1r2r4--r2+r1--r1+s4--s1+2q0-2q0r3r4s2+s3--r3--xrr1r2r2--r2+r1--r1+s2--s1+2q0-2q0Figure 2.1: Evolution of the Riemann invariants in two equivalent cases. Dashed lines: the Riemann invariants att = 0; solid lines: the invariants at t negationslash= 0. Figure 2.1a (left) corresponds to the square-wave initial datum in Eqs. (1.9)and Fig. 1.1, regularized by genus-1 data; Figure 2.1b (right) shows an equivalent diagram of the left one, and it isgiven by genus-0 data. Hereafter, the subscripts “−” and “+” refer to the value of the invariants respectively to theleft and to the right of their initial discontinuity. The ellipses in Fig. 2.1a indicate “locking” points, i.e., points x∗ forwhich r1(x∗,t)= r2(x∗,t)= r3(x∗,t) (on the left) and r2(x∗,t)= r3(x∗,t)= r4(x∗,t) (on the right) for all t. This impliesthat the regularization at those points is trivial. Finally, note that the initial conditions r01,2(x) in the equivalent figure(Fig. 2.1b) have been redefined whenever ρ(x,0) = 0 (i.e., u(x) itself is not defined in the NLS-Whitham equationwhen ρ(x) = 0, see Eqs. (A.9)).8regularize the initial datum (e.g., see Figs. 2.1a,b). The minimum number of Riemann invariants that arenecessary so that the system becomes regular is related to the genus of the solution of the NLS-Whithamequations. That is, 2g + 2 invariants correspond to a genus-g solution of Eqs. (2.3). The local genus ofthe solution of the NLS equation is roughly speaking the number of distinct frequencies which are locallypresent in the solution over regions of order one in (x,t). Since the Riemann invariants are the branch pointsof the spectrum of the finite-genus solution of NLS which locally approximates the full solution, it is thenclear that the local genus is equal to the number of gaps determined by the local value of the Riemanninvariants (e.g., see Figs. 5.1 and 5.2). The opening or closing of one of the gaps for some values of (x,t)corresponds to a local change of genus in the solution of the NLS equation. Note however that all finite-genus solutions of the NLS equation with non-zero genus become singular in the limit ε →0+. In this sense,the solution of the Whitham equations represents a weak limit, since when g negationslash= 0 the solutions of the NLSequation only converge in an average sense (i.e., weak convergence).Finally, with regards to Fig. 2.1 we should note that genus-1 data is necessary in o