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The Lambert W Function © Copyright, Princeton University Press. No part of this book may be distributed, posted, or reproduced in any form by digital or mechanical means without prior written permission of the publisher. III.17. The Lambert W Function 151 faster than shorter ones, Zabusky and Kruskal simu- quantities, infinitely many higher-order symmetries, lated the collision of two waves in a nonlinear crys- and an infinite number of soliton solutions. tal lattice and observed that each retains its shape and As well as being applicable to shallow-water waves, speed after collision. Interacting solitary waves merely the KdV equation is ubiquitous in applied science. It experience a phase shift, advancing the faster wave and describes, for example, ion-acoustic waves in a plasma, retarding the slower one. In analogy with colliding par- elastic waves in a rod, and internal waves in the atmo- ticles, they coined the word “solitons” to describe these sphere or ocean. The KP equation models, for exam- elastically colliding waves. ple, water waves, acoustic waves, and magnetoelastic To model water waves that are weakly nonlinear, waves in anti-ferromagnetic materials. The nonlinear weakly dispersive, and weakly two-dimensional, with Schrödinger equation describes weakly nonlinear and all three effects being comparable, Kadomtsev and dispersive wave packets in physical systems, e.g., light Petviashvili (KP) derived a two-dimensional version of pulses in optical fibers, surface waves in deep water, (2) in 1970: Langmuir waves in a plasma, and high-frequency vibra- 2 tions in a crystal lattice. Equation (7) with an extra lin- (ut + 6uux + uxxx)x + 3σ uyy = 0, (6) ear term V(x)u to account for the external potential 2 =± where σ 1 and the y-axis is perpendicular to the V(x) also arises in the study of Bose–Einstein conden- direction of propagation of the wave (along the x-axis). sates, where it is referred to as the time-dependent The KdV and KP equations, and the nonlinear Gross–Pitaevskii equation. schrödinger equation [III.26] 2 Further Reading iut + uxx + κ|u| u = 0 (7) (where κ is a constant and u(x, t) is a complex-valued Ablowitz, M. J. 2011. Nonlinear Dispersive Waves: Asymp- function), are famous examples of so-called completely totic Analysis and Solitons. Cambridge: Cambridge Univer- sity Press. integrable nonlinear PDEs. This means that they can Ablowitz, M. J., and P. A. Clarkson. 1991. Solitons, Nonlinear be solved with the inverse scattering transform, a Evolution Equations and Inverse Scattering. Cambridge: nonlinear analogue of the Fourier transform. Cambridge University Press. The inverse scattering transform is not applied to (2) Kasman, A. 2010. Glimpses of Soliton Theory. Providence, directly but to an auxiliary system of linear PDEs, RI: American Mathematical Society. Osborne, A. R. 2010. Nonlinear Ocean Waves and the Inverse + + 1 = ψxx (λ 6 αu)ψ 0, (8) Scattering Transform. Burlington, MA: Academic Press. + 1 + + = ψt 2 αuxψ αuψx 4ψxxx 0, (9) which is called the Lax pair for the KdV equation. Equa- III.17 The Lambert W Function tion (8) is a linear Schrödinger equation for an eigen- Robert M. Corless and David J. Jeffrey function ψ, a constant eigenvalue λ, and a potential (−αu)/6. Equation (9) governs the time evolution of ψ. 1 Definition and Basic Properties The two equations are compatible, i.e., ψxxt = ψtxx, if and only if u(x, t) satisfies (2). For given u(x, 0) For a given complex number z, the equation decaying sufficiently fast as |x|→∞, the inverse scat- wew = z tering transform solves (8) and (9) and finally deter- has a countably infinite number of solutions, which are mines u(x, t). denoted by Wk(z) for integers k. Each choice of k speci- fies a branch of the Lambert W function. By convention, 4 Properties and Applications only the branches k = 0 (called the principal branch) =− Scientists remain intrigued by the rich mathemati- and k 1 are real-valued for any z; the range of every cal structure of completely integrable nonlinear PDEs. other branch excludes the real axis, although the range −∞ − These PDEs can be written as infinite-dimensional bi- of W1(z) includes ( , 1/e] in its closure. Only W0(z) Hamiltonian systems and have additional, remarkable contains positive values in its range (see figure 1). When =− features. For example, they have an associated Lax pair, z 1/e (the only nonzero branch point), there is a =− w = they can be written in Hirota’s bilinear form, they admit double root w 1 of the basic equation we z. Bäcklund transformations, and they have the Painlevé The conventional choice of branches assigns property. They have an infinite number of conserved W0(−1/e) = W−1(−1/e) =−1 For general queries, contact [email protected] © Copyright, Princeton University Press. No part of this book may be distributed, posted, or reproduced in any form by digital or mechanical means without prior written permission of the publisher. 152 III. Equations, Laws, and Functions of Applied Mathematics W W 1 0 3 2 –e–1 z –1 1 2 3 1 ) ) z ( –1 k 0 W ( Im –1 –2 –2 –3 –3 W–1 –4 –3 –2 –1 3210 Re (Wk (z ) ) Figure 1 Real branches of the Lambert W function. The solid line is the principal branch W0; the dashed line is W−1, Figure 2 Images of circles and rays in the z-plane under the −1 which is the only other branch that takes real values. The maps z → Wk(z). The circle with radius e maps to a curve small filled circle at the branch point corresponds to the that goes through the branch point, as does the ray along one in figure 2. the negative real axis. This graph was produced in Maple x+iy by numerical evaluation of ω(x + iy) = WK(iy)(e ) first for a selection of fixed x and varying y, and then for a 2 and implies that W1(−1/e−iε ) =−1+O(ε) is arbitrar- selection of fixed y and varying x. These two sets produce ily close to −1 because the conventional branch choice orthogonal curves as images of horizontal and vertical lines means that the point −1 is on the border between these in x and y under ω or, equivalently, images of circles with = x = three branches. Each branch is a single-valued com- constant r e and rays with constant θ y under W . plex function, analytic away from the branch point and branch cuts. ln z is the principal branch of the logarithm, having The set of all branches is often referred to, loosely, as −π<Im(ln z) π. the Lambert W “function”; but of course W is multival- ued. Depending on context, the symbol W(z) can refer The Wright ω function helps to solve the equation + = = ± − to the principal branch (k = 0) or to some unspecified y ln y z. We have that, if z t iπ for t< 1, = = − − branch. Numerical computation of any branch of W is then y ω(z).Ifz t iπ for t< 1, then there is = + − typically carried out by Newton’s method or a variant no solution to the equation; if z t iπ for t< 1, iθ then there are two solutions: ω(z) and ω(z − 2πi). thereof. Images of Wk(r e ) for various k, r , and θ are shown in figure 2. In contrast to more commonly encountered multi- 1.1 Derivatives branched functions, such as the inverse sine or cosine, the branches of W are not linearly related. However, by Implicit differentiation yields rephrasing things slightly, in terms of the unwinding = −W(z) + number W (z) e /(1 W(z)) z − ln(ez) K(z) := =− 2πi as long as W(z) 1. The derivative can be simplified and the related single-valued function to the rational differential equation = z dW W ω(z) : WK(z)(e ), = dz z(1 + W) which is called the Wright ω function, we do have the somewhat simple relationship between branches that if, in addition, z = 0. Higher derivatives follow natu- Wk(z) = ω(lnk z), where lnk z denotes ln z + 2πik and rally. For general queries, contact [email protected] © Copyright, Princeton University Press. No part of this book may be distributed, posted, or reproduced in any form by digital or mechanical means without prior written permission of the publisher. III.17. The Lambert W Function 153 1.2 Integrals is split at the maximum of the integrand (t = n), and each integral is transformed using the substitu- Integrals containing W(x)can often be performed ana- 2 tions t =−nW (−e−1−z /2), where k = 0 is used for = k lytically by the change of variable w W(x), used in =− w t n and k 1 otherwise. The integrands then an inverse fashion: x = we . Thus, 2 simplify to tne−t = nne−ne−nz /2 and the differentials sin W(x)dx = (1 + w)ew sin w dw, dt are obtained as series from the above expansions. Term-by-term integration leads to and integration using usual methods gives n+1 k+1/2 1 1 n 2 1 (1 + w)ew sin w − wew cos w, n! ∼ (2k + 1)a2k+1 Γ(k+ ), 2 2 en n 2 k0 which eventually gives where Γ is the gamma function. x →∞ 2 sin W(x)dx = x + sin W(x) Asymptotic series for z have been known since W(x) de Bruijn’s work in the 1960s. He also proved that − x cos W(x)+ C.
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