Wave Instability Under Short-Wave Amplitude Modulations

Physics Letters A 377 (2013) 1260–1263 Contents lists available at SciVerse ScienceDirect Physics Letters A www.elsevier.com/locate/pla Wave instability under short-wave amplitude modulations ∗ Benno Rumpf a, ,AlanC.Newellb a Department of Mathematics, Southern Methodist University, Dallas, TX 75275, USA b Department of Mathematics, University of Arizona, Tucson, AZ 85721, USA article info abstract Article history: The instabilities of nonlinear waves with a square-root dispersion ω ∼ |k| arestudied.Wepresenta Received 21 December 2012 new type of instability that affects wavelengths of the order of the carrier wave. This instability can Received in revised form 8 March 2013 initiate the formation of collapses and of narrow pulses. Accepted 9 March 2013 © 2013 Elsevier B.V. All rights reserved. Available online 15 March 2013 Communicated by C.R. Doering Keywords: Benjamin–Feir instability Nonlinear waves Majda–McLaughlin–Tabak equation 1. Introduction iφ˙ + ∂2φ/∂x2 − σ φ|φ|2 = 0. This equation is canonical in the sense that it can be obtained from many wave equations by ex- Modulational instabilities of monochromatic waves in weakly panding the linear dispersion about a carrier wavenumber, and nonlinear and strongly dispersive systems [1] are not only ubiq- by expanding the nonlinearity in powers of the small amplitude. uitous in nature (in water waves [2], plasma waves [3],optical In this context the nonlinear Schrödinger equation describes long waves [4], elastic waves [5]), but are responsible for much of the modulations of the carrier wave. But here we take the nonlin- striking behaviors seen in light filamentation [6] and freak waves ear Schrödinger equation as an equation of its own right that is in fluid mechanics [7] and in optics [8]. This instability causes an satisfied by monochromatic waves φ = A exp(ikx − i(k2 + σ A2)t) increase of initially small amplitude variations of a monochromatic where the amplitude A is considered to be real and positive for wave, which can ultimately destroy this wave. Such modulations simplicity. A modulation of this carrier wave is achieved if the affect a long spatial scale that usually scales as one over the small amplitude is initially not constant, but varies slightly in space amplitude of the carrier wave. The new finding of this Letter is an with a wavenumber q as A + a cos(qx) + b sin(qx).Thisleads instability at a length scale that is shorter than the carrier wave. to a system of coupled modes, namely two at k and at k ± q. We investigate this kind of instability for a class of wave equa- The four corresponding frequencies are almost phase matched as tions ω(k + q) + ω(k − q) − 2ω(k) = 2q2 (Fig. 1(a)). Still, these modes can interact resonantly when the nonlinearity has the appropriate ∂φ 2 sign to offset the detuning [9].Forσ =−1 (the focusing nonlinear i = Lαφ + σ φ|φ| (1) ∂t Schrödinger equation) waves are unstable under long-wave modu- ∼ in one dimension that describe the dynamics of a complex ampli- lations with q A. Majda, McLaughlin and Tabak [10] have studied an equation of tude φ(x,t) in one spatial dimension. The operator Lα is defined α the form (1) in which the dispersion relation ω = |k| is not an- via its eigenfunctions and eigenvalues Lα exp(ikx) =|k| exp(ikx). alytic and whose graph is concave down (Figs. 1(b)–(d)), in order Eq. (1) can be derived from a Hamiltonian H = H2 + H4 with 2 α 4 to mimic the dispersion of gravity waves in deep water. This MMT H2 = ω(k)|ak| dk, ω(k) =|k| and H4 = σ |φ| /2 dx, where ak is the Fourier transform of φ. We consider the two important cases equation was originally designed to check predictions of wave tur- α = 2 and α = 1/2. bulence theory in one dimension. Numerical investigations show α = 2, ω = k2 yields the second order differential operator that coherent structures, namely short-wave pulses [11,12] for 2 2 σ = 1 and wave collapses [13–16] for σ =−1emergefromdis- L2 =−∂ /∂x so that (1) is the nonlinear Schrödinger equation integrating waves in this system. In order to gain some understanding of these findings, we now = * Corresponding author. consider the instability of monochromatic waves of (1) with ω E-mail address: [email protected] (B. Rumpf). |k|. Fig. 1(b) shows that the four corresponding frequencies at k, 0375-9601/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2013.03.015 B. Rumpf, A.C. Newell / Physics Letters A 377 (2013) 1260–1263 1261 M = |k + q|α −|k − q|α /2 0. (4) The characteristic equation is 2 λ4 + 2λ2 M2 − Q + M2 + Q = 0(5) with Q =−D(D + 2σ A2). The eigenvalues are λ1,2,3,4 =±iM ± Q . (6) Q > 0 signals an instability Re(λ) > 0. Note that the matrix B is non-normal and also defective when the eigenvalues are degener- ate either by M = 0, or by D =−2σ A2 and therefore Q = 0, or by Q =−M2. In these cases the eigenvectors don’t span the full vector space. This can lead to algebraic growth of perturbations. It is well known that the focusing nonlinear Schrödinger equation (σ =−1) is modulationally unstable with Re(λ) = ± q2(2A2 − q2) for 2A2 > q2, whereas waves of the defocussing (σ = 1) nonlinear Schrödinger equation are not modulationally unstable. To study the square-root dispersion α = 1/2weintroduce the parameter s = q/k and the rescaled function Fig. 1. The dispersion ω = k2 of the nonlinear Schrödinger equation (a), and the = | | dispersion ω k of the MMT equation (b)–(d). The nonlinear Schrödinger equa- 1/2 tion (a) and the MMT equation (b) admit long-wave modulational instabilities (with R(s) = D/|k| | | q k) for opposite signs of the nonlinearity. The instability to short-wave am- = | + |1/2 +| − |1/2 − plitude modulations q ≈ 5k/4 is possible for either sign of the nonlinearity in the 1 s 1 s 2 /2. (7) MMT equation (c). Solitons in the MMT equation lose power by radiation of reso- = = = nantly excited modes on the second branch of the dispersion (d). This function is depicted in Fig. 2. It has zeros R(s 0) R(s 5/4) = 0, and R(s)<0for0<√ s < 5/4 with a cusp-shaped minimum at s = 1, R(s = 1) = 1/ 2 − 1. R(s) is positive for s > 5/4 k ± q almost match as ω(k + q) + ω(k − q) − 2ω(k) ≈−q2/(4k3/2) √ with R(s) ∼ s for large s. for |q||k|. The negative detuning can be offset by a nonlinearity with σ = 1. This indicates the possibility of a long-wave modulational instability with q ∼ A. Note that this instability occurs for a 2.2. Domains of the instability positive sign of σ , while the nonlinear Schrödinger equation admits modulational instabilities only for σ negative. 2.2.1. Zero eigenvalues and maximum eigenvalues The fact that no such instability is possible for α = 1/2, σ =−1 To determine the√ domain of instability we first consider its = = = appears to contradict ample numerical evidence for the existence boundaries Re(λ) Q 0. This is satisfied by D 0 and for =− 2 = of wave collapses [13–16] in this equation: How can a wave col- D 2σ A . D 0orequivalently lapse be initiated without a Benjamin–Feir instability? Fig. 1(c) = suggests that the concave down dispersion ω ∼ |k| admits an R(s) 0(8) additional instability: The resonance condition can be fulfilled for is solved by s = 0 and s = 5/4 and yields the two lines q = 0 and − some q > k > 0, so that k q is at the left branch of the disper- q = 5k/4 where Re(λ) = 0. The condition D =−2σ A2 can be writ- + sion while k and k q are at the right branch, a case that has not ten as been recognized in the literature on this subject. We investigate ˜ −1/2 this instability in detail and discuss its effects on the formation of R(s) =−2σ |k| (9) coherent structures. ˜ with the rescaled dimensionless wavenumber k = k/A4 (k and A4 2. Long- and short-wave instabilities have the same dimensions). To determine the maximum of Re(λ) we note that ∂ Q /∂k = 0 2 2.1. Linear stability analysis and ∂ Q /∂q = 0 are both satisfied at D =−σ A . In rescaled vari- ablesthisreads We perform a stability analysis by perturbing a monochromatic =− |˜|−1/2 wave with small modulations |a|, |b|A to obtain R(s) σ k . (10) − We discuss the domains with Re(λ) > 0 separately for σ = 1 and φ(x,t) = ei(kx Ωt) A + a(t) cos(qx) + b(t) sin(qx) . (2) σ =−1. The unmodulated wave (a = b = 0) has the nonlinear dispersion relation Ω =|k|α + σ A2. It is assumed with no loss of gener- 2.2.2. Instabilities for σ = 1 ality that k > 0, q > 0. The linearized dynamics of the vector Condition (9) corresponds to intersection points of the lines ˜ −1/2 v = (Re(a), Im(a), Re(b), Im(b))T is given by v˙ = Bv with the ma- R(s) and −2|k| in Fig. 2. There are two such intersection points ˜ −1/2 trix at s1 and s2 if the minimum of R(s) is below the line −2|k| . ⎛ ⎞ In the intervals 0 < s < s and s < s < 5/4weobtainQ > 0 and 0 D −M 0 1 2 therefore Re(λ) > 0.

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