The stability of solitary traveling waves in a general class of conservative nonlinear dispersive equations is discussed

Akers, B

Milewski, Paul A

Communications in Mathematical Sciences

The stability of solitary traveling waves in a general class of conservative nonlinear dispersive equations is discussed. A necessary condition for the exchange of stability of traveling waves is presented; an unstable eigenmode may bifurcate from the neutral translational mode only at relative extrema of the wave energy. This paper extends a result from Hamiltonian systems, and from a few integrable partial differential equations, to a broader class of conservative differential equations, with particular application to gravity-capillary surface waves

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        Citation for published version:Akers, B & Milewski, PA 2008, 'A stability result for solitary waves in nonlinear dispersive equations',Communications in Mathematical Sciences, vol. 6, no. 3, pp. 791-797.Publication date:2008Link to publication© International Press.  First published in Communications in Mathematical Sciences in volume 6, number 3,2008, published by International Press.University of BathGeneral rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.Take down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Download date: 12. May. 2019COMMUN. MATH. SCI. c© 2008 International PressVol. 6, No. 3, pp. 791–797FAST COMMUNICATIONA STABILITY RESULT FOR SOLITARY WAVES IN NONLINEARDISPERSIVE EQUATIONS∗BENJAMIN AKERS† AND PAUL A. MILEWSKI‡Abstract. The stability of solitary traveling waves in a general class of conservative nonlineardispersive equations is discussed. A necessary condition for the exchange of stability of travelingwaves is presented; an unstable eigenmode may bifurcate from the neutral translational mode onlyat relative extrema of the wave energy. This paper extends a result from Hamiltonian systems, andfrom a few integrable partial differential equations, to a broader class of conservative differentialequations, with particular application to gravity-capillary surface waves.Key words. stability, solitary wave, gravity-capillary wave.AMS subject classifications. 76B45, 76B25, 76B151. IntroductionThis paper investigates the linear stability of nonlinear traveling waves in dis-persive evolution equations. It is well known that the derivative of a solution toa translation-invariant equation is a neutrally stable eigenfunction of the linearizedequation. The question addressed here is under what circumstances may the eigen-value for this eigenfunction, referred to here as the translational mode, bifurcate fromzero. This type of analysis originated in Hamiltonian systems, where Saffman provedthat the linear stability of traveling waves in a family of waves may only changeat local extrema of the speed-energy curve [1]. More recently, similar results havebeen shown for the the Nonlinear Schro¨dinger (NLS), Korteweg-de Vries (KdV), andKadomtsev-Petviashvilli (KP) equations [2, 3]. Here, we extend this result to classesof one-way and bidirectional conservative, nonlinear, dispersive equations. The resultapplies to models for gravity-capillary waves, where the dynamics and stability of soli-tary waves is of current interest [4, 5]. In the gravity-capillary case, the bifurcationstudied in this paper has been observed numerically [6]. The understanding of thisobserved bifurcation was the motivation for this paper.2. Model equations and solitary wavesConsider the first order in time evolution equation of the formηt − iΩη + 12Q∗(PQη)2x = 0. (2.1)Here, x is the preferred direction of propogation. In this variable, Ω has a real, oddFourier symbol; P has a real, even Fourier symbol, and it is self-adjoint. The operatorQ has an arbitrary Fourier symbol; Q∗ is the adjoint operator to Q whose Fouriermultiplier is the complex conjugate of Qˆ. Equation (2.1) conserves at least M =∫ηand E =∫ηPη, and is translation-invariant. For example, when Q = P = 1 and∗Received: July 9, 2008; accepted (in revised version): August 1, 2008. Communicated by DavidCai.†Department of Mathematics, University of Wisconsin, 480 Lincoln Dr., Madison, WI 53706(akers@math.wisc.edu).‡Department of Mathematics, University of Wisconsin, 480 Lincoln Dr., Madison, WI 53706(milewski@math.wisc.edu).791792 STABILITY OF DISPERSIVE SOLITARY WAVESΩˆ = k3 (where k is the Fourier dual variable to x) equation (2.1) is the KdV equation.When Ωˆ = sgn(k)k2, it is the (nonlocal) Benjamin-Ono equation. Equations in morethan one spatial dimension are also amenable to the analysis we present. For example,with Q = P = 1 and Ωˆ = k3 ∓ l2/k (where l is the Fourier dual variable to y), weget the KP-I and KP-II equations. For deep water gravity-capillary waves one mayderive models using the full dispersion relation Ωˆ = sign(k)√|k|(1 + k2); examples ofsuch models are derived in [6].We also consider the second order in time modelφtt + Ω2φ + Q∗(PQφt)2x = 0, (2.2)which is also translation-invariant, and conserves at least M2 =∫φt and E2 =∫φtPφt + φPΩ2φ. The operators Q,P and Ω satisfy the same conditions as in equa-tion (2.1). The operator Ω2 can thus correspond to any even, real, positive Fouriermultiplier. In the case of deep water gravity-capillary waves, Ωˆ2 = |k|(1 + |k|2).Equ. (2.2) can be factored symmetrically into left and right moving waves.2u = φt + iΩφ, 2v = φt − iΩφ,soφt = u + v, iΩφ = u− v.In terms of these new variables, equation (2.2) becomesut − iΩu + 12Q∗(PQ(u + v))2x = 0, (2.3a)vt + iΩv +12Q∗(PQ(u + v))2x = 0. (2.3b)If, at initial times, the waves are primarily propagating in one direction (for example,v(x, t) is small, then, depending on Q,P and Ω, these equations may decouple forlong times as in the gravity-capillary case [6]. Neglecting v, equation (2.3a) reducesto equation (2.1).The goal of this work is to investigate the stability of finite-amplitude, fully lo-calized solitary waves (also called lump solutions in higher dimensions). We areparticularly interested in wave-packet type waves: those that bifurcate from linearwavepackets at a finite wave-number. The equations that support such waves areusually non-local and non-integrable. Wave-packet solitary waves have been found,for example, as solutions to the free-surface Euler equations in deep water with surfacetension, and in a few model equations (for two dimensional examples, see [6, 3, 5, 7]).For Ωˆ = sign√|k|(1 + k2), both (2.1) and (2.2) have such solitary wave solutions formany choices of P and Q. Examples of these waves for one choice of P and Q thatconserves momentum in the fluid (see [6]) are plotted in Fig. 2.1. A diagnostic tool forthe existence of such waves at small amplitude in a particular evolution equation is theexistence of a focusing NLS equation governing wave-packets near the wavenumberwhere the group and phase speed are equal [6, 8, 7].3. Stability of traveling wavesThe stability of solitary waves has been of long standing interest [9, 10]. Inprevious work, the authors presented three simple one-dimensional one-way modelsfor infinite depth gravity-capillary waves, together with numerical computations oftheir stability and dynamics [6]. It was found that the stability of these waves canAKERS AND MILEWSKI 793−30 −20 −10 0 10 20 30−0.4−0.3−0.2−0.100.10.2Xη−30 −20 −10 0 10 20−0.4−0.3−0.2−0.100.10.20.3XηFig. 2.1. A depression (left) and an elevation (right) solitary wave solution to equation (2.1),with Ω = sign(k)p|k|(1 + k2), P = |Ω|, and Q = |Ω|−1.change as the amplitude increases. The purpose of this work is to find a criterion forthe stability change. The authors have proposed models for two dimensional waves[11], which support lump solutions, and the present analysis applies to those also.To begin, consider equation (2.1) in the frame of an exact solution moving withspeed c,ηt − cηx − iΩη + 12Q∗(PQη)2x = 0, (3.1)and suppose that η¯(x) is a steady solution to (3.1), that is, a traveling solution to(2.1). Consider perturbations of the form η = η¯+ ǫρ(x)eλt. The linearization of (3.1)isλρ− cρx − iΩρ + Q∗(PQη¯PQρ)x = 0. (3.2)Equ. (3.2) is of the form Lρ = λρ. This equation has the well known solution λ = 0,ρ(0) = η¯x, due to the translation-invariance of equation (3.1). The goal is now tounderstand how this solution behaves in a neighborhood of λ = 0. In particular, wewant to determine when eigenmodes with real nonzero eigenvalues bifurcate from thetranslational mode. First, we assume that |λ| ≪ 1 and express ρ as a power series inλ about ρ(0).ρ = ρ(0) + λρ(1) + λ2ρ(2) + ... (3.3)Collecting powers of λ transforms the linear eigenvalue problem into a series of linearproblems with the eigenvalue removed. The leading order equation is the originallinear problem for the zero eigenvalue, which has solution ρ(0) = η¯x. At O(λ), theleading order solution occurs as an inhomogeneous term in the equation for ρ(1):Lρ(1) = η¯x. (3.4)This equation has, as a particular solution, ρ(1) = −∂η¯∂c, which can be seen by differ-entiating equation (3.1) with respect to c. At order O(λ2) the first correction, ρ(1),forces the equation for ρ(2):Lρ(2) = −η¯c. (3.5)794 STABILITY OF DISPERSIVE SOLITARY WAVESUsing the Fredholm alternative, a solvability condition for equation (3.5) is that η¯c isorthogonal to solutions of the adjoint equationcvx + iΩv −Q∗P (PQη¯Qvx) = 0. (3.6)One solution to (3.6) is v = P η¯. The corresponding solvability condition is∫η¯cP η¯dx =12ddc∫ηPη =12dEdc= 0.We conclude that a necessary condition for the translational eigenmode to becomeunstable is that the energy is at a local extremum, as a function of the wave speed.The same technique can be used to derive stability conditions for second orderin time equations — by factoring into a system of first order in time equations,and proceeding as before. We consider the second order in time model (2.2) whichconserves M2 =∫φt and E2 =∫φtPφt + φPΩφ. Rewriting (2.2):φt = v (3.7a)vt = −Ω2φ + Q∗(PQv)2x. (3.7b)Changing to a traveling frame, traveling wave solutions (φ¯, v¯) to (3.7) are now steadysolutions to (3.8).φt − cφx = v (3.8a)vt − cvx = −Ω2φ + Q∗(PQv)2x (3.8b)Linearizing about these solutions, writing (φ, v) = (φ¯, v¯) + δ(Φ(x)e(λt), V (x)e(λt))yieldsλΦ− cΦx = V (3.9a)λV − cVx = −Ω2Φ + 2Q∗(PQv¯PQV )x. (3.9b)As before, equation (3.9) is of the formA(ΦV)= λ(ΦV).We write both Φ and V as a series in λ, about the translational solution (φ¯x, v¯x)Φ = φ¯x + λΦ1 + λ2Φ2 + ...V = v¯x + λV1 + λ2V2 + ...At O(λ) we get a forced linear system.A(Φ1V1)=(φ¯xv¯x)(3.10)Equ. (3.10) has solution (Φ1, V1) = −(∂φ¯∂c , ∂v¯∂c ). At O(λ2), the solutions (Φ1, V1) forcethe equation for (Φ2, V2)A(Φ2V2)= −(φ¯cv¯c). (3.11)AKERS AND MILEWSKI 7951.406 1.407 1.408 1.409 1.41 1.411 1.412 1.41300.511.522.533.5x 10−3Speedλ1.406 1.407 1.408 1.409 1.41 1.411 1.412 1.413 1.414 1.4150.330.340.350.360.370.38SpeedEnergyFig. 3.1. Left: The real positive eigenvalue for depression waves in equation (2.1), withΩ = sign(k)p|k|(1 + k2), P = (1− ∂2x) and Q = (1− ∂2x)−1, plotted as function of the wave speed.Right: The energy of depression waves in this equation plotted as a function of the wave speed.The solution bifurcates from a linear wavepacket at c =√2.For equation (3.11) to have solutions, a necessary condition is that the vector (∂φ¯∂c, ∂v¯∂c)is orthogonal to solutions to the adjoint equationA∗(ψu)=(c∂x Ω2−1 c∂x − 2Q∗P (PQv¯Q∂x))(ψu)= 0. (3.12)One solution to (3.12) is (ψ, u) = (PΩ2φ¯, P v¯). The corresponding solvability conditionis ∫∂v¯∂cP v¯ +∂φ¯∂cPΩ2φ¯ =12ddc(∫φtPφt + φPΩ2φ)=12dE2dc= 0.This stability condition allows one to understand the linear stability of travelingwaves via the graph of the speed versus the relevant conserved quantity, as in Fig. 3.1(right). The stability of traveling solitary waves of elevation and depression for twoequations of the form of (2.1) was investigated numerically in [6]. The linear spectrawere computed, and the instabilities were observed via numerical time evolution. Forequation (2.1) with Ω, P,Q as defined in Fig. 2.1 the waves of elevation were unstableat all amplitudes, and those of depression were stable. However, in an example withΩˆ = sign(k)√|k|(1 + k2), Pˆ = (1 + k2), and Qˆ = (1 + k2)−1, which corresponds toa model of gravity-capillary waves that conserves exactly the quadratic fluid energyE = ||u||2H1 =∫uPu =∫u2 + u2x, the stability is quite different. Depression waveshave a linear spectrum which is purely imaginary for both small and large amplitudewaves, but waves of intermediate amplitudes have a real positive eigenvalue whichbifurcates from zero at the local extrema of the energy. The unstable eigenvalue forthis example equation is plotted in Fig. 3.1, together with the speed-energy curve.The long time evolution of this instability is a time-periodic traveling solution, orbitingthe stable state at the energy level of the unstable wave [6].Note that neither derivation of the stability criterion relies on the whether φ or ηare functions of one or more space dimensions. In this case we assume that Ωˆ is oddin the vector k (for real η and φ). In two-dimensions, if Ωˆ is odd in k, then it must796 STABILITY OF DISPERSIVE SOLITARY WAVESbe even in l. If it does not vanish at k = 0 then extra conditions must be imposed onthe initial data’s mean in x, as in KP. There also may be additional conditions on Pand Q that we do not consider.Examples of two dimensional equations for which our result applies are the KP-Iequation,ut − 16uxxx +12∂−1x uyy +32uux = 0, (3.13)which is integrable and for which lumps are always stable, and the fifth-order KP-equation,ut +16uxxx +190uxxxxx +12∂−1x uyy +32uux = 0, (3.14)for which lumps seem to have more complicated stability characteristics [3]. Bothapply to a rather unphysical situation of gravity-capillary waves in shallow waterbased on the surface tension length scale: the former applies to a regime of B > 1/3and the latter for B ≈ 1/3, where B is the Bond number [12]. These equations governlong waves in x with even weaker variation in y, and their linear parts are obtainedby expanding the dispersion relation about k = (kx, ky) = 0. For gravity-capillarywaves on infinite depth, one can also derive a KP-like equation by expanding aboutk= (kx, ky) = (1, 0), and imposing some global conditions on the dispersion relation.The result isut +√24Hu−√24Huxx + ǫ(√22Huyy − 34√2uux)= 0. (3.15)The operator H is the Hilbert transform in x, whose Fourier symbol is Hˆ = −isign(k).The forgoing analysis predicts the change of stability of lump solutions to this equa-tion. An example lump solitary wave solution to equation (3.15) and its speed-energyplot are in Fig. 3.2. The authors present the derivation of equation (3.15), as well asstability and dynamics of solitary waves in [13].−20 −100 1020−40−2002040−0.5−0.4−0.3−0.2−0.100.10.2xyu0.65 0.66 0.67 0.68 0.69 0.71.71.81.922.12.22.32.42.52.62.7Speed||u|| 22Fig. 3.2. Left: An example of a lump depression wave solution to equation (3.15). Right:The speed-energy plot for depression lump solitary wave solutions to equation (3.15). The nonlinearsolution bifurcates from a linear wavepacket at c =√2/2.AKERS AND MILEWSKI 7974. ConclusionA necessary condition on the bifurcation of unstable eigenvalues for the transla-tional eigenfunction of traveling waves in one-way and bidirectional classes of disper-sive nonlinear equations was derived. Bifurcations can occur only at local extrema inthe graph of the speed versus the quadratic conserved quantity. Solitary waves arecomputed for an example equation, and the linear stability changes exactly at thelocal extremum in the speed-energy plot.This note focuses on equations with quadratic nonlinearities. One-way and bidi-rectional equations can be written with nonlinearities of higher degree. The resultcan be trivially extended to equations with higher power nonlinearities, for exampleηt + iΩη +∑nanQ∗(PQη)nx = 0 (4.1)with an arbitrary constants.REFERENCES[1] P.G. Saffman, The superharmonic instability of finite amplitude water waves, J. Fluid Mech.,159, 169–174, 1985.[2] D.E. Pelinovsky and R.H.J. Grimshaw, Asymptotic methods in soliton stability theory, Nonlin.Instab. Anal., 12, 245–312, 1997.[3] T.R. Akylas and Yeunwoo Cho, On the stability of lumps and wave collapse in water waves,Phil. Trans. A. Math. Phys. Eng. Sci., 2008.[4] F. Dias and C. Kharif, Nonlinear gravity and gravity-capillary waves, Ann. Rev. Fluid Mech.,31, 301–346, 1999.[5] E.I. Parau, Jean Mark Vanden-Broeck and M. J. Cooker, Nonlinear three-dimensional gravity-capillary solitary waves, J. Fluid Mech., 536, 99–105, 2005.[6] B. Akers and P.A. Milewski, Model equations for gravity-capillary waves in deep water, Stud.Appl. Math., 121, 49–69, 2008.[7] P.A. Milewski, Three-dimensional localized solitary gravity-capillary waves, Commun. Math.Sci., 3, 89–99, 2005.[8] B. Buffoni, M.D. Groves and J.F. Toland, A plethora of solitary gravity-capillary waves withnear critical bond and froude numbers, Phi. Trans. R. Soc. Lond. A, 354, 575–607, 1996.[9] T.B. Benjamin, On the stability of solitary waves, Proc. R. Soc. Lond. A, 328, 153–183, 1972.[10] B.B. Kadomtsev and V.I. Petviashvili, The stability of solitary waves in weakly dispersingmedia, Soviet Physics Doklady, 15, 539–541, December 1970.[11] B. Akers and P.A. Milewski, Stability and dynamics of three dimensional gravity-capillarywaves in deep water, preprint, 2008.[12] T.R. Akylas, Three-dimensional long water-wave phenomena, Ann. Rev. Fluid Mech., 26, 191–210, 1994.[13] B. Akers and P.A. Milewski, A new model equation governing three dimensional surface waves,preprint, 2008.

A stability result for solitary waves in nonlinear dispersive equations

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