High-Frequency Instabilities of a Boussinesq-Whitham System: A Perturbative Approach
aa r X i v : . [ m a t h . A P ] F e b HIGH-FREQUENCY INSTABILITIES OF A BOUSSINESQ-WHITHAM SYSTEM:A PERTURBATIVE APPROACH
RYAN CREEDON , BERNARD DECONINCK , OLGA TRICHTCHENKO DEPT. OF APPLIED MATHEMATICS, U. OF WASHINGTON, SEATTLE, WA, 98105, USA (
[email protected] ) DEPT. OF APPLIED MATHEMATICS, U. OF WASHINGTON, SEATTLE, WA, 98105, USA (
[email protected] ) DEPT. OF PHYSICS AND ASTRONOMY, U. OF WESTERN ONTARIO, LONDON, ON, N6A 3K7, CA (
[email protected] ) FEBRUARY 9, 2021
Abstract.
We analyze the spectral stability of small-amplitude, periodic, traveling-wave solutionsof a Boussinesq-Whitham system. These solutions are shown numerically to exhibit high-frequencyinstabilities when subject to bounded perturbations on the real line. We use a formal perturbationmethod to estimate the asymptotic behavior of these instabilities in the small-amplitude regime.We compare these asymptotic results with direct numerical computations. Introduction
We investigate small-amplitude, π/κ -periodic, traveling waves of a Boussinesq-Whitham systemproposed by Hur and Pandey [15] and Hur and Tao [16]: η t = − h u x − ( ηu ) x ,u t = − g K [ η x ] − uu x . (1.1)In this model, η ( x, t ) represents the displacement of a wave profile from its equilibrium depth h , u ( x, t ) is the horizontal velocity along η , and K is a Fourier multiplier operator defined so that thelinearized dispersion relation of (1.1) matches that of the Euler water wave problem (WWP) [27].For functions f ∈ L per ( − π/κ, π/κ ) , K is defined as d K [ f ]( κn ) = tanh( κnh ) κnh b f ( κn ) , n ∈ Z , (1.2)where b · denotes the Fourier transform of f : b f ( k ) = κ π Z π/κ − π/κ f ( x ) e − ikx dx. (1.3)Alternatively, K can be defined in physical variables as the pseudo-differential operator K [ f ] = (cid:18) tanh( h D ) h D (cid:19) f, (1.4)where D = − i∂ x . For the remainder of this manuscript, we refer to (1.1) as the Hur-Pandey-Tao–Boussinesq-Whitham system, or HPT–BW for short.The HPT–BW system is Hamiltonian [15] with H = 12 Z π/κ − π/κ (cid:0) h u + gη K [ η ] + ηu (cid:1) dx. (1.5)and non-canonical Poisson structure J = − (cid:18) ∂ x ∂ x (cid:19) . (1.6)The system has a one-parameter family of small-amplitude, π/κ -periodic, traveling-wave solutionsfor each κ > . We call these solutions the Stokes waves of HPT–BW by analogy with solutions of the Figure 1. (Top, left) A zero-amplitude ( ε = 0 ) Stokes wave (solid blue). (Bottom,left) The stability spectrum of the zero-amplitude Stokes wave (solid black). Colli-sions of nonzero spectral elements are denoted by red crosses, while the collision ofzero spectral elements is denoted by the green polygon. (Top, right) A small ampli-tude ( | ε | ≪ ) Stokes wave (solid blue). (Bottom, right) The stability spectrum ofthe small-amplitude Stokes wave. Stable elements are in solid black. High-frequencyisolas are in solid red. The modulational instability figure-eight pattern is in dashedgreen and is present only if κh is sufficiently large.WWP of the same name [22, 24, 25]. In Section 2, we derive a power series expansion for HPT–BWStokes waves in a small parameter ε that scales with the amplitude of the waves.Perturbing Stokes waves yields a spectral problem after linearizing the governing equations of theperturbations. The spectral elements of this problem define the stability spectrum of Stokes waves;see Section 3. The stability spectrum is purely continuous [7, 18, 23], but Floquet theory decomposesthe spectrum into an uncountable union of point spectra. Each of these point spectra is indexed bythe Floquet exponent [11, 14, 17].The stability spectrum inherits quadrafold symmetry from the Hamiltonian structure of (1.1), i.e.,the spectrum is invariant under conjugation and negation [14, 18]. Because of quadrafold symmetry,all elements of the stability spectrum have non-positive real component only if the stability spectrumis a subset of the imaginary axis. Therefore, HPT–BW Stokes waves are spectrally stable only if thespectrum is on the imaginary axis. Otherwise, the Stokes waves are spectrally unstable.If the aspect ratio κh is sufficiently large, both HPT–BW [15] and WWP Stokes waves [3, 4, 5]have stability spectra near the origin that leave the imaginary axis for < | ε | ≪ , resulting inmodulational instability. Using the Floquet-Fourier-Hill (FFH) method [11], recent numerical workby [8] and [12] shows, respectively, that HPT–BW and WWP Stokes waves also have stability spectraaway from the origin that leave the imaginary axis, regardless of κh . These spectra give rise to theso-called high-frequency instabilities [13], shown schematically in Figure 1.High-frequency instabilities arise from the collision of nonzero stability eigenvalues of zero-amplitude( ε = 0 ) Stokes waves. At these collided spectral elements, a Hamiltonian-Hopf bifurcation occurs,resulting in a locus of spectral elements bounded away from the origin that leave the imaginary axisas | ε | increases. We refer to this locus of spectral elements as a high-frequency isola.High-frequency isolas are difficult to find using numerical methods like FFH. To capture the isolaclosest to the origin, for example, the interval of Floquet exponents that parameterizes the isola haswidth O (cid:0) ε (cid:1) (Figure 2). For isolas further from the origin, this width appears to decay geometricallyin ε . To compound these difficulties, the isolas drift away from their initial collision sites (Figure 2),meaning that the Floquet exponent that gives rise to the collided spectral elements at ε = 0 isnot contained in the interval parameterizing the corresponding isola for | ε | sufficiently large. Tocircumvent these difficulties, one must supply the numerical method with asymptotic expressions forthe interval of Floquet exponents corresponding to the desired isolas. We discover these expressionsfor high-frequency isolas of HPT–BW. IGH-FREQUENCY INSTABILITIES OF A BOUSSINESQ-WHITHAM SYSTEM: A PERTURBATIVE APPROACH 3
Figure 2. (Left) The high-frequency isola closest to the origin of HPT–BW Stokeswaves with κ = g = h = 1 and ε = 2 . × − (red), × − (burgundy), . × − (purple), and − (blue). The collision that generates the isola when ε = 0 is indicated by the red star. The imaginary axis is recentered to show themagnitude of the drift of the isola from the collision site λ . (Right) Interval ofFloquet exponents that parameterize the isola on the left as a function of ε . Thesolid black lines indicate the boundaries of the interval, while the dashed black linegives the Floquet exponent of the most unstable spectral element of the isola. Thecolored dots provide the Floquet data for the correspondingly colored isola in theleft plot. The red star indicates the Floquet exponent µ that generates the isolawhen ε = 0 . The Floquet axis is recentered to show the magnitude of the drift ofthe Floquet exponents from µ .Our motivation for studying the HPT–BW system, apart from its inherent interest, is that itretains the full dispersion relation (both branches) of the more complicated WWP. Our goal is theapplication of the perturbation method developed herein to the WWP. The first step towards thisgoal was the investigation of the stability spectra of Stokes waves of the Kawahara equation [9]. Theinvestigations presented here constitute our second step, before proceeding to the finite-depth WWPnext [10].For a given high-frequency isola of an HPT–BW Stokes wave, we obtain (i) an asymptotic rangeof Floquet exponents that parameterize the isola, (ii) an asymptotic estimate for the most unstablespectral element of the isola, (iii) expressions of curves that are asymptotic to the isola, (iv) wavenum-bers for which the given isola is not present. Our approach is inspired by a perturbation methodoutlined in [2], but modified appropriately for higher-order calculations. We compare all asymptoticresults with numerical results computed by the FFH method.2. Small-Amplitude Stokes Waves
In a traveling frame moving with velocity c , x → x − ct and (1.1) becomes η t = cη x − h u x − ( ηu ) x ,u t = cu x − g K ( η x ) − uu x . (2.1)Non-dimensionalizing (2.1) according to η → h η , u → u √ gh , x → α − h x , t → t p h /g , and c → c √ gh yields the following system: α − η t = cη x − u x − ( ηu ) x ,α − u t = cu x − K α ( η x ) − uu x . (2.2)The parameter α is chosen to map π/κ -periodic solutions of (2.1) to π -periodic solutions of (2.2).Consequently, α = κh > , the aspect ratio of the solutions, and IGH-FREQUENCY INSTABILITIES OF A BOUSSINESQ-WHITHAM SYSTEM: A PERTURBATIVE APPROACH 4 \ K α [ f ]( n ) = tanh( αn ) αn b f ( n ) , n ∈ Z , (2.3)or, alternatively, K α [ f ] = (cid:18) tanh( αD ) αD (cid:19) f, (2.4)for f ∈ L per ( − π, π ) with the Fourier transform (1.3) redefined over ( − π, π ) .Stokes wave solutions of (2.2) are independent of time. Equating time derivatives in (2.2) to zeroand integrating in x , we find cη = u + ηu + I cu = K α ( η ) + 12 u + I , (2.5)where I j are integration constants. For each α > , there exists a three-parameter family of infinitelydifferentiable, even, small-amplitude, π -periodic solutions of (2.5), provided I j are sufficiently small[15]. We call these solutions the HPT–BW Stokes waves, denoted ( η S ( x ; ε, I j ) , u S ( x ; ε, I j )) T , where ε is a small-amplitude parameter defined implicitly in terms of the first Fourier mode of η S ( x ; ε, I j ) : ε := 2 c η S (1) = 1 π Z π − π η S ( x ; ε, I j ) e − ix dx. (2.6) Remark . Redefining c → c − I and u → u − I in (2.5) implies I = 0 without loss of generality. Ifwe also equate I = 0 , our Stokes waves reduce to a one-parameter family of solutions to (2.5) suchthat (2.6) ensures that η S ( x ; ε ) ∼ ε cos( x ) as ε → . We restrict to this case for simplicitly, but themethodology in Sections 4 and 5 are unchanged if I = 0 . For series representations of Stokes wavesthat include I j , see [15].The Stokes waves and their velocity may be expanded as power series in ε : η S = η S ( x ; ε ) = ε cos( x ) + ∞ X j =2 η j ( x ) ε j , (2.7a) u S = u S ( x ; ε ) = c ε cos( x ) + ∞ X j =2 u j ( x ) ε j , (2.7b) c = c ( ε ) = c + ∞ X j =1 c j ε j , c = tanh( α ) α , (2.7c)where η j ( x ) and u j ( x ) are analytic, even, and π -periodic for each j . Substituting these expansionsinto (2.5) (with I j = 0 ) and following a Poincaré-Lindstedt perturbation method [27], one determines η j ( x ) , u j ( x ) , and c j order by order. In Appendix A, we report expansions of η S , u S , and c up tofourth order in ε ; this is sufficient for our asymptotic calculations of high-frequency isolas discussedin Sections 4 and 5. 3. The Stability Spectrum of Stokes Waves
Consider perturbations to ( η S , u S ) T of the form (cid:18) η ( x, t ; ε ) u ( x, t ; ε ) (cid:19) = (cid:18) η S u S (cid:19) + ρ (cid:18) H ( x, t ; ε ) U ( x, t ; ε ) (cid:19) + O (cid:0) ρ (cid:1) , (3.1)where | ρ | ≪ is a parameter independent of ε and H and U are sufficiently smooth, boundedfunctions of x on R for all t ≥ . When (3.1) is substituted into (2.2), terms of O (cid:0) ρ (cid:1) cancel by (2.5)(with I j = 0 ). Equating terms of O ( ρ ) , the perturbation ( H, U ) T solves the linear system ∂∂t (cid:18) HU (cid:19) = α (cid:18) − u ′ S + ( c − u S ) ∂ x − η ′ S − (1 + η S ) ∂ x − i tanh( αD ) α − u ′ S + ( c − u S ) ∂ x (cid:19) (cid:18) HU (cid:19) , (3.2) IGH-FREQUENCY INSTABILITIES OF A BOUSSINESQ-WHITHAM SYSTEM: A PERTURBATIVE APPROACH 5 where primes denote differentiation with respect to x . Formally separating variables, (cid:18) H ( x, t ; ε ) U ( x, t ; ε ) (cid:19) = e λt (cid:18) H ( x, t ; ε ) U ( x, t ; ε ) (cid:19) , (3.3)where ( H , U ) T solves the spectral problem λ (cid:18) HU (cid:19) = α (cid:18) − u ′ S + ( c − u S ) ∂ x − η ′ S − (1 + η S ) ∂ x − i tanh( αD ) α − u ′ S + ( c − u S ) ∂ x (cid:19) (cid:18) HU (cid:19) . (3.4)Since the entries of the matrix operator above are π -periodic, one can use Floquet theory to solve(3.4) for ( H , U ) T . These solutions take the form (cid:18) H ( x ; ε ) U ( x ; ε ) (cid:19) = e iµx (cid:18) h ( x ; ε ) u ( x ; ε ) (cid:19) , (3.5)where µ ∈ [ − / , / is called the Floquet exponent and h , u ∈ H per ( − π, π ) . Substituting (3.5) into(3.4) results in a spectral problem for w = ( h , u ) T : λ ε,µ w = L ε,µ w , (3.6)with L ε,µ = α (cid:18) − u ′ S + ( c − u S )( iµ + ∂ x ) − η ′ S − (1 + η S )( iµ + ∂ x ) − i tanh( α ( µ + D )) α − u ′ S + ( c − u S ) ∂ x (cid:19) . (3.7)For sufficiently small ε , (3.6) has a countable collection of eigenvalues λ ε,µ for each Floquet exponent µ [17]. The union of these eigenvalues over µ ∈ [ − / , / recovers the purely continuous spectrumof (3.4) for fixed ε ; this is the stability spectrum of HPT–BW Stokes waves. We use the Floquet-Fourier-Hill method [11] to compute the stability spectrum numerically.If there exists a µ such that there is a λ ε,µ with Re ( λ ε,µ ) > , then there exists a perturbation(3.3) that grows exponentially in time, and Stokes waves of amplitude ε are spectrally unstable. Ifno such µ is found, then the Stokes waves are spectrally stable. Because of the quadrafold symmetrymentioned in the introduction, Stokes waves are spectrally stable if and only if their stability spectrumis a subset of the imaginary axis.When ε = 0 , L ,µ has constant coefficients, and its spectral elements are given exactly by λ ( σ )0 ,µ,n = − i Ω σ ( n + µ ) , n ∈ Z , σ = ± , (3.8)where Ω σ are the two branches of the linear dispersion relation of (2.2) with c → c ( c is given in(2.7)). Explicitly, Ω σ ( k ) = − αc k + σω α ( k ) , (3.9)where ω α ( k ) = sgn ( k ) p αk tanh( αk ) . (3.10)As expected, λ ( σ )0 ,µ,n is a countable collection of eigenvalues for each µ , and the resulting stabilityspectrum has quadrafold symmetry. In addition, the stability spectrum coincides with the imaginaryaxis, implying that zero-amplitude Stokes waves are spectrally stable.For some µ = µ , nonzero eigenvalues of L ,µ with double multiplicity may give rise to Hamiltonian-Hopf bifurcations and, thus, to high-frequency instabilities for < | ε | ≪ . These eigenvalues existprovided there exists µ , m , and n such that λ ( σ )0 ,µ ,n = λ ( σ )0 ,µ ,m = 0 . (3.11) Strictly speaking, Floquet theory applies only to linear, local operators. Work by [6] extends this theory to nonlocaloperators.
IGH-FREQUENCY INSTABILITIES OF A BOUSSINESQ-WHITHAM SYSTEM: A PERTURBATIVE APPROACH 6
We view (3.11) as a collision of two simple, nonzero eigenvalues. It can be shown that such a collisionoccurs only if σ = σ [1, 13, 15]. Theorem 4 in Appendix B shows that, for any p ∈ Z \ { , ± } , thereexist unique µ , m , and n that satisfy (3.11) with m − n = p . Thus, there are a countably infinitenumber of nonzero eigenvalue collisions in the zero-amplitude stability spectrum; each of which haspotential to develop a high-frequency instability in the small-amplitude stability spectrum. Remark . Using results in Appendix B, it can be shown that the Krein signatures [20] of the col-liding eigenvalues have opposite signs. This is a second necessary criterion for the occurance ofhigh-frequency instabilities [13, 21].
Remark . The WWP shares the same collided eigenvalues with HPT–BW, since (3.9) is also thedispersion relation of the WWP.4.
High-Frequency Instabilities: p = 2 We use perturbation methods to investigate the high-frequency instability that develops from thecollision of λ (1)0 ,µ ,n and λ ( − ,µ ,m , where µ ∈ [ − / , / is the unique Floquet exponent for which(3.11) is satisfied and m − n = 2 . This instability corresponds to the high-frequency isola closest tothe origin; see Theorem 4 in Appendix B. For sufficiently small ε , this is also the isola with largestreal component.4.1. The O (cid:0) ε (cid:1) Problem
The p = 2 isola develops from the spectral data λ = λ (1)0 ,µ ,n = − i Ω ( µ + n ) = − i Ω − ( µ + m ) = λ ( − ,µ ,m = 0 , (4.1a) w ( x ) = (cid:18) h ( x ) u ( x ) (cid:19) = γ − ω α ( m + µ ) α ( m + µ ) ! e imx + γ ω α ( n + µ ) α ( n + µ ) ! e inx , (4.1b)where γ j are arbitrary, nonzero constants. As | ε | increases, we assume the spectral data vary analyt-ically [1] with ε : λ = λ + ελ + ε λ + O (cid:0) ε (cid:1) , (4.2a) w = w + ε w + ε w + O (cid:0) ε (cid:1) (4.2b) = (cid:18) h u (cid:19) + ε (cid:18) h u (cid:19) + ε (cid:18) h u (cid:19) + O (cid:0) ε (cid:1) , (4.2c)where we suppress functional dependencies for ease of notation. We normalize w so that b h ( n ) = 12 π Z π − π h e − inx dx = 1 , (4.3)or, alternatively, so that b h ( n ) = 1 , b h j ( n ) = 0 , ∀ j ∈ N . (4.4)This normalization ensures that h fully resolves the n th Fourier mode of h , a convenient choice forthe perturbation calculations that follow. With this normalization, w ( x ) = (cid:18) h ( x ) u ( x ) (cid:19) = γ − ω α ( m + µ ) α ( m + µ ) ! e imx + ω α ( n + µ ) α ( n + µ ) ! e inx . (4.5) Because the spectrum (3.9) has the symmetry λ ( σ )0 , − µ , − n = λ ( σ )0 ,µ ,n , where the overbar denotes complex conju-gation, choosing p = − gives the isola conjugate to that for p = 2 . Thus, we may choose p = 2 without loss ofgenerality. IGH-FREQUENCY INSTABILITIES OF A BOUSSINESQ-WHITHAM SYSTEM: A PERTURBATIVE APPROACH 7
The arbitrary constant γ will be determined at higher order, leading to a unique expression for w . Remark . The eigenvalue corrections λ j derived below are independent of the normalization cho-sen for w .If λ is a semi-simple, isolated eigenvalue of L ,µ , we may justify (4.2) using analytic perturba-tion theory [19], provided the Floquet exponent is fixed. For ε sufficiently small, this method of proofgives two spectral elements on the isola. Numerical and asymptotic calculations show that thesespectral elements quickly leave the isola as a result of the change in its Floquet parameterizationwith ε (Figure 2, Figure 5, Figure 10). To account for this variation, we allow the Floquet exponentto depend on ε as well: µ = µ + εµ + ε µ + O (cid:0) ε (cid:1) . (4.6) Remark . In the calculations that follow, explicit expressions of select quantities are suppressed forease of readability. The interested reader may consult the supplemental Mathematica file hptbw_isolap2.nb for these expressions.4.2.
The O ( ε ) Problem
Substituting the expansions of the Stokes wave (2.7), spectral data (4.2), and Floquet exponent(4.6) into the spectral problem (3.6) and collecting terms of O ( ε ) , we find ( L ,µ − λ ) w = λ w − L w , (4.7)with L = α (cid:18) − u ′ + ic µ − u ( iµ + ∂ x ) − η ′ − iµ − η ( iµ + ∂ x ) − iµ sech ( α ( µ + D )) − u ′ + ic µ − u ( iµ + ∂ x ) (cid:19) . (4.8)The inhomogeneous terms on the RHS of (4.7) can be evaluated using expressions for η , u , and w . Each of these quantities are finite linear combinations of π -periodic sinusoids. As a result, theinhomogeneous terms can be rewritten as a finite Fourier series, and (4.7) becomes ( L ,µ − λ ) w = m +1 X j = n − T , j e ijx , (4.9)where T , j depend on µ , α , and γ ; see the Mathematica file for details. Remark . Since m − n = 2 , the index j ∈ { n − , n, n, m, m + 1 } . When evaluating the in-homogeneous terms, one finds vector multiples of exp ( i (1 + n ) x ) and exp ( i ( m − x ) . These vectorsare combined to give T , + n .For (4.9) to have a solution w , the inhomogeneous terms must be orthogonal (in the L per ( − π, π ) × L per ( − π, π ) sense) to the nullspace of the hermitian adjoint of L ,µ − λ by the Fredholm alternative.The hermitian adjoint of L ,µ − λ is ( L ,µ − λ ) † = (cid:18) − αc ( iµ + ∂ x ) − λ tan( α ( iµ + ∂ x )) α ( iµ + ∂ x ) − αc ( iµ + ∂ x ) − λ (cid:19) , (4.10)where overbars denote complex conjugation. Its nullspace isNull h ( L ,µ − λ ) † i = Span " α ( µ + n ) ω α ( µ + n ) ! e inx , − α ( µ + m ) ω α ( µ + m ) ! e imx . (4.11)Thus, according to the Fredholm alternative, there exists a solution w to (4.9) if IGH-FREQUENCY INSTABILITIES OF A BOUSSINESQ-WHITHAM SYSTEM: A PERTURBATIVE APPROACH 8 * α ( µ + n ) ω α ( µ + n ) ! e inx , T , n e inx + = 0 , * − α ( µ + m ) ω α ( µ + m ) ! e imx , T , m e imx + = 0 , (4.12)where h· , ·i is the standard inner product on L per ( − π, π ) × L per ( − π, π ) . Substituting expressions for T , n and T , m gives solvability conditions λ + iµ c g ( µ + n ) = 0 , (4.13a) γ (cid:0) λ + iµ c g − ( µ + m ) (cid:1) = 0 , (4.13b)where c g σ ( k ) = Ω ′ σ ( k ) is the group velocity of Ω σ . Lemma 3 in Appendix B shows that c g ( µ + n ) = c g − ( µ + m ) . Since γ is nonzero, λ = 0 = µ . (4.14)Consequently, T , n = = T , m , simplifying the inhomogeneous terms in (4.9).With the solvability conditions satisfied, we solve for the particular solution of w in (4.9). Com-bining with the nullspace of L ,µ − λ , w = m +1 X j = n − j = n,m W , j e ijx + β ,m − ω α ( m + µ ) α ( m + µ ) e imx + β ,n ω α ( n + µ ) α ( n + µ ) e inx , (4.15)where β ,j are arbitrary constants and W , j are found in the Mathematica file. Enforcing the nor-malization condition (4.4), one finds β ,n = 0 . For ease of notation, let β ,m → γ so that w = m +1 X j = n − j = n,m W , j e ijx + γ − ω α ( m + µ ) α ( m + µ ) e imx . (4.16)4.3. The O (cid:0) ε (cid:1) Problem
Using (4.14), the spectral problem (3.6) at O (cid:0) ε (cid:1) is ( L ,µ − λ ) w = λ w − L | µ =0 w − L | µ =0 w , (4.17)where L | µ =0 is the same as above, but evaluated at µ = 0 , and L | µ =0 = α (cid:18) − u ′ + ( c − u )( iµ + ∂ x ) + iµ c − η ′ − iµ − η ( iµ + ∂ x ) − iµ sech ( α ( µ + D )) − u ′ + ( c − u )( iµ + ∂ x ) + iµ c (cid:19) . (4.18)One can evaluate the inhomogeneous terms of (4.18) using η j , u j , and w j − for j ∈ { , } . Theseinhomogeneous terms can be expressed as a finite Fourier series, giving ( L ,µ − λ ) w = m +2 X j = n − j = n − T , j e ijx . (4.19)It can be shown that T , n − = .Proceeding similarly to the previous order, solvability conditions for (4.19) are λ + i C ,n ) + iγ S ,n = 0 , (4.20a) γ ( λ + i C − ,m ) + i S ,m = 0 , (4.20b)where IGH-FREQUENCY INSTABILITIES OF A BOUSSINESQ-WHITHAM SYSTEM: A PERTURBATIVE APPROACH 9
Figure 3.
A plot of S vs. α . No roots of S are found for α > . It is likely thatHPT–BW Stokes waves of all wavenumbers experience a p = 2 instability. C ,n = µ c g ( µ + n ) − P ,n , (4.21a) C − ,m = µ c g − ( µ + m ) − P ,m . (4.21b)Expressions for S ,j and P ,j have no dependence on γ , γ , µ , or λ ; see the attached Mathematicafile for details.Conditions (4.20a) and (4.20b) form a nonlinear system for γ and λ . Solving for λ yields λ = − i (cid:18) C − ,m + C ,n (cid:19) ± s − (cid:18) C − ,m − C ,n (cid:19) − S ,n S ,m . (4.22)A direct calculation shows that S ,n S ,m = − S ω α ( µ + m ) ω α ( µ + n ) , (4.23)where S is given in the attached Mathematica file. Then, λ = − i (cid:18) C − ,m + C ,n (cid:19) ± s − (cid:18) C − ,m − C ,n (cid:19) + S ω α ( µ + m ) ω α ( µ + n ) . (4.24)A corollary of Lemma 3 in Appendix B shows that ω α ( µ + m ) ω α ( µ + n ) is positive . Provided S = 0 and c g − ( µ + m ) = c g ( µ + n ) , λ has nonzero real part for µ ∈ ( M , − , M , + ) , where M , ± = µ , ∗ ± |S || c g − ( µ + m ) − c g ( µ + n ) | p ω α ( µ + m ) ω α ( µ + n ) , (4.25)and µ , ∗ = P ,m − P ,n c g − ( µ + m ) − c g ( µ + n ) . (4.26)That c g − ( µ + m ) = c g ( µ + n ) follows from Lemma 2 in Appendix B. A plot of S as a function of α suggests that S > for all values of α > (Figure 3). We conjecture that HPT–BW Stokeswaves of any wavenumer experience a p = 2 high-frequency instability at O (cid:0) ε (cid:1) .For µ ∈ ( M , − , M , + ) , a quick calculation shows that (4.24) parameterizes an ellipse asymptoticto the numerically observed p = 2 high-frequency isola (Figure 4). The ellipse has semi-major and-minor axes that scale with ε , and the center of the ellipse drifts along the imaginary axis like ε from λ , the collision point at ε = 0 .The midpoint of ( M , − , M , + ) maximizes the real part of λ . Thus, the most unstable spectralelement of the isola has Floquet exponent This corollary is equivalent to satisfying the Krein signature condition mentioned in Section 3.
IGH-FREQUENCY INSTABILITIES OF A BOUSSINESQ-WHITHAM SYSTEM: A PERTURBATIVE APPROACH 10 -4 -2 0 2 410 -8 -8 -10.0237-9.2681-8.5125-7.7570-7.0014-6.2459-5.4903-4.7348-3.9792-3.223710 -8 -2 -1 0 1 210 -8 -10-9-8-7-6-5-4-3 10 -8 Figure 4. (Left) The p = 2 isola with α = 1 and ε = 5 × − (zero-order imaginarycorrection removed for better visibility). The solid red curve is the ellipse obtainedby our perturbation calculations. Blue circles are a subset of spectral elements fromthe numerically computed isola using FFH. (Right) The Floquet parameterizationof the real (blue) and imaginary (red) components of the isola (zero- and second-order Floquet corrections and zero-order imaginary correction removed for bettervisibility). Solid curves illustrate perturbation results. Circles indicate FFH results. µ ∗ = µ + µ , ∗ ε + O (cid:0) ε (cid:1) , (4.27)and its real and imaginary components are λ r, ∗ = |S | p ω α ( µ + m ) ω α ( µ + n ) ! ε + O (cid:0) ε (cid:1) , (4.28a) λ i, ∗ = − Ω ( µ + n ) − C ,n ε + O (cid:0) ε (cid:1) , (4.28b)respectively. These expansions agree well with the FFH results, see Figure 5.5. High-Frequency Instabilities: p = 3 According to Theorem 4 in Appendix B, the p = 3 high-frequency instability is the second-closestto the origin. As will be seen, this instability arises at O (cid:0) ε (cid:1) . Let µ correspond to the uniqueFloquet exponent in [ − / , / that satisfies the collision condition (3.11) with m − n = 3 . Then,the spectral data (4.1) give rise to the p = 3 high-frequency instability. We assume these data andthe Floquet exponent vary analytically with ε . For uniqueness, we normalize the eigenfunction w according to (4.4) so that w is given by (4.5). We proceed as in the p = 2 case. Remark . In the calculations that follow, explicit expressions of select quantities are suppressed forease of readability. The interested reader may consult the supplemental Mathematica file hptbw_isolap3.nb for these expressions.5.1.
The O ( ε ) Problem
Substituting expansions (2.7), (4.2), and (4.6) into the spectral problem (3.6), equating terms of O ( ε ) , and using expression for η , u , and w to simplify, we find ( L ,µ − λ ) w = m +1 X j = n − T , j e ijx . (5.1)Expressions for T , j depend on µ , α , and γ ; see the Mathematica file attached. Since m − n = 3 , j ∈ { n − , n, n, m − , m, m + 1 } . The functional expressions for T , n − and T , m + are identical IGH-FREQUENCY INSTABILITIES OF A BOUSSINESQ-WHITHAM SYSTEM: A PERTURBATIVE APPROACH 11
Figure 5. (Left) The interval of Floquet exponents that parameterize the p = 2 isola as a function of ε with α = 1 (zero- and second-order Floquet correctionsremoved for better visibility). Up to recentering the Floquet axis and accountingfor a wider range of ε , this plot is identical to the right panel of Figure 2. Solidblue curves indicate the boundaries of this interval according to our perturbationcalculations. Blue circles indicate the boundaries computed numerically by FFH.The solid red curve gives the Floquet exponent corresponding to the most unstablespectral element of the isola according to our perturbation calculations. Red circlesindicate the same but computed numerically using FFH. (Right) The real (blue) andimaginary (red) components of the most unstable spectral element of the isola as afunction of ε (zero-order imaginary correction removed for better visibility). Solidcurves illustrate perturbation calculations. Circles illustrate FFH results.to those in the p = 2 case .Solvability conditions for (5.1) simplify to µ = 0 = λ . Together with the normalization (4.4),these conditions guarantee a solution to (5.1) of the form w = m +1 X j = n − j = n,m W , j e ijx + γ − ω α ( m + µ ) α ( m + µ ) e imx , (5.2)where γ is arbitrary and expressions for W , j are found in the supplemental Mathematica file. Be-cause T , n − and T , m + are identical to their p = 2 counterparts, W , n − and W , m + are as well.5.2. The O (cid:0) ε (cid:1) Problem
The O (cid:0) ε (cid:1) problem takes the same form as (4.17). Evaluating at η j , u j , and w j − for j ∈ { , } , wefind ( L ,µ − λ ) w = m +2 X j = n − j = n − T , j e ijx , (5.3)For the same reasons as in the p = 2 case, T , n − = , and expressions for T , n − and T , m + areidentical to their p = 2 counterparts.Since γ = 0 , the solvability conditions for (5.3) simplify to λ + iµ c g ( µ + n ) − i P ,n = 0 , (5.4a) λ + iµ c g − ( µ + m ) − i P ,m = 0 , (5.4b) They do not evaluate to the same vectors, however, as µ is different for p = 2 and p = 3 in general. IGH-FREQUENCY INSTABILITIES OF A BOUSSINESQ-WHITHAM SYSTEM: A PERTURBATIVE APPROACH 12 where P ,j are independent of λ , µ , γ , and γ ; see supplemental Mathematica file. Note that theseterms are distinct from those introduced in (4.21).Solving (5.4a) and (5.4b) for λ and µ yields λ = − i (cid:18) P ,m c g ( µ + n ) − P ,n c g − ( µ + m ) c g − ( µ + m ) − c g ( µ + n ) (cid:19) , (5.5a) µ = P ,m − P ,n c g − ( µ + m ) − c g ( µ + n ) . (5.5b)Thus the spectral elements and Floquet parameterization of the p = 3 isola have nontrivial correctionsat O (cid:0) ε (cid:1) . However, since Re ( λ ) = 0 , we have yet to determine the leading-order behavior of theisola. We find this at the next order.Imposing solvability conditions (5.4a) and (5.4b) as well as the normalization condition on w , thesolution of (5.3) is w = m +1 X j = n − j = n − W , j e ijx + γ − ω α ( m + µ ) α ( m + µ ) e imx , (5.6)where γ is an arbitrary constant. Since T , n − = , W , n − = .5.3. The O (cid:0) ε (cid:1) Problem At O (cid:0) ε (cid:1) , the spectral problem (3.6) takes the form ( L ,µ − λ ) w = X j =2 λ j w − j − X j =1 L j | µ =0 w − j , (5.7)where L j | µ =0 for j ∈ { , } are as before and L | µ =0 = α (cid:18) − u ′ − iµ u − u ( iµ + ∂ x ) + iµ c − η ′ − iµ − iη µ − η ( iµ + ∂ x ) − µ sech ( α ( µ + D )) u ′ − iµ u − u ( iµ + ∂ x ) + iµ c (cid:19) . (5.8)Evaluating (5.7) at η j , u j , and w j − for j ∈ { , , } , one finds ( L ,µ − λ ) w = m +3 X j = n − j = n − T , j e ijx , (5.9)where T , n − = .The solvability conditions for (5.9) are λ + iµ c g ( µ + n )) + iγ S ,n = 0 , (5.10a) γ ( λ + iµ c g − ( µ + m )) + i S ,m + iγ T ,m = 0 , (5.10b)where S ,j and T ,m have no dependence on γ , γ , µ , or λ ; see supplemental Mathematica file.Using (5.4a) and (5.4b) from the previous order as well as (3.11), one can show that T ,m ≡ . Inaddition, similar to (4.23) for the p = 2 isola, we have S ,n S ,m = − S ω α ( µ + m ) ω α ( µ + n ) , (5.11)where S is given in the supplemental Mathematica file. As a result, (5.10a) and (5.10b) form anonlinear system for λ and γ . Solving for λ , one finds IGH-FREQUENCY INSTABILITIES OF A BOUSSINESQ-WHITHAM SYSTEM: A PERTURBATIVE APPROACH 13 -7 Figure 6. (Left) A plot of S vs. α . The quantity S has a root α = 1 . ... (red star), implying HPBT–BW Stokes waves of this aspect ratio do not have a p = 3 instability at O (cid:0) ε (cid:1) . (Right) A plot of the maximum real component of thenumerical p = 3 isola (computed by FFH) as a function of α for ε = 10 − (solidblue), ε = 7 . × − (dot-dashed purple), ε = 5 × − (dashed light blue), and ε = 2 . × − (dotted cyan). The p = 3 isola vanishes when α = 1 . ... (red star). λ = − iµ (cid:18) c g − ( µ + m ) + c g ( µ + n )2 (cid:19) (5.12) ± s − µ (cid:18) c g − ( µ + m ) − c g ( µ + n )2 (cid:19) + S ω α ( µ + m ) ω α ( µ + n ) . As in the p = 2 case, ω α ( µ + m ) ω α ( µ + n ) > and c g − ( µ + m ) = c g ( µ + n ) . Provided S = 0 , λ has nonzero real part if µ ∈ ( − M , M ) , where M = |S || c g − ( µ + m ) − c g ( µ + n ) | p ω α ( µ + m ) ω α ( µ + n ) . (5.13)A plot of S vs. α reveals that S = 0 only at α = 1 . ... (Figure 6). For this wave aspectratio, the p = 3 instability does not occur at O (cid:0) ε (cid:1) . In fact, Figure 6 shows that, if α approaches1.1862.... for fixed ε , the numerically computed p = 3 isola shrinks to a point on the imaginary axis. We conjecture that HPT–BW Stokes waves with α = 1 . ... are not succeptible to the p = 3 instability , even beyond O (cid:0) ε (cid:1) . Indeed, in the next subsection, we find that λ is purelyimaginary, so Stokes waves with aspect ratio α = 1 . ... do not exhibit p = 3 instabilities to O (cid:0) ε (cid:1) .Assuming α = 1 . ... , µ ∈ ( − M , M ) parameterizes an ellipse asymptotic to the p = 3 high-frequeny isola; see Figure 7. The ellipse has semi-major and -minor axes that scale with ε . Thecenter of this ellipse drifts along the imaginary axis like ε due to the purely imaginary correctionfound at O (cid:0) ε (cid:1) .The interval of Floquet exponents that parameterizes the p = 3 isola is µ ∈ (cid:0) µ + µ ε − M ε , µ + µ ε + M ε (cid:1) + O (cid:0) ε (cid:1) . (5.14)The width of this interval is an order of magnitude smaller than that of the p = 2 isola. Consequently,the p = 3 isola is more challenging to find numerically than the p = 2 isola, at least for methodssimilar to FFH (Table 1).For α = 1 and | ε | < × − , (5.14) provides an excellent approximation to the numericallycomputed interval of Floquet exponents (Figure 8). Fourth-order corrections are necessary to improveagreement between (5.14) and numerical computations for larger ε , see Section 5.4 below.Choosing µ = 0 maximizes the real part of λ . Thus, the most unstable spectral element of the p = 3 isola has Floquet exponent IGH-FREQUENCY INSTABILITIES OF A BOUSSINESQ-WHITHAM SYSTEM: A PERTURBATIVE APPROACH 14
Table 1.
Intervals of Floquet exponents that parameterize the p = 2 and p = 3 high-frequency isolas with ε = 10 − and α = 1 / , , and . The first digit for whichthe boundary values disagree is underlined and colored red. If a uniform mesh ofFloquet exponents in [ − / , / is used for numerical methods like FFH, the spacingof the mesh must be finer than ε to capture the p = 2 instability and ε to capturethe p = 3 instability. The intervals vary with α as well, making it difficult to adaptand refine a uniform mesh to find high-frequency isolas. p = 2 α = (-0.106478 α = 1 (-0.26090 α = 2 (-0.330352 p = 3 α = (-0.37544887 α = 1 (0.257196721 α = 2 (0.0440583313 µ ∗ = µ + µ ε + O (cid:0) ε (cid:1) , (5.15)where µ is as in (5.5b), and its real and imaginary components are λ r, ∗ = |S | p ω α ( µ + m ) ω α ( µ + n ) ! ε + O (cid:0) ε (cid:1) , (5.16a) λ i, ∗ = − Ω ( µ + n ) − (cid:18) P ,m c g ( µ + n ) − P ,n c g − ( µ + m ) c g − ( µ + m ) − c g ( µ + n ) (cid:19) ε + O (cid:0) ε (cid:1) , (5.16b) -1 -0.5 0 0.5 110 -11 -12 -1-0.8-0.6-0.4-0.200.20.40.60.8110 -11 -4 -2 0 2 410 -12 -1-0.8-0.6-0.4-0.200.20.40.60.81 10 -11 Figure 7. (Left) p = 3 isola with α = 1 and ε = 5 × − (zero- and second-order imaginary corrections removed for better visibility). The solid red curve is theellipse obtained by our perturbation calculations. The blue circles are a subset ofspectral elements from the numerically computed isola using FFH. (Right) Floquetparameterization of the real (blue) and imaginary (red) components of the isola (zero-and second-order imaginary and Floquet corrections removed for better visibility).Solid curves illustrate perturbation results. Circles indicate FFH results. IGH-FREQUENCY INSTABILITIES OF A BOUSSINESQ-WHITHAM SYSTEM: A PERTURBATIVE APPROACH 15
Figure 8. (Left) The interval of Floquet exponents that parameterize the p = 3 isolaas a function of ε with α = 1 (zero- and second-order Floquet corrections removed forbetter visibility). Solid blue curves indicate the boundaries of this interval accordingto our perturbation calculations. Blue circles indicate the boundaries computed nu-merically by FFH. The solid red curve gives the Floquet exponent corresponding tothe most unstable spectral element of the isola according to our perturbation calcu-lations. Red circles indicate the same but computed numerically using FFH. (Right)The real (blue) and imaginary (red) components of the most unstable spectral ele-ment of the isola as a function of ε (zero- and second-order imaginary and Floquetcorrections removed for better visibility). Solid curves illustrate perturbation calcu-lations. Circles illustrate FFH results.respectively. The expansion for λ r, ∗ is in excellent agreement with numerical results using the FFHmethod (Figure 8). As with (5.14), corrections to µ ∗ and λ i, ∗ at O (cid:0) ε (cid:1) improve the agreementbetween numerical and asymptotic results for these quantities.Before proceeding to O (cid:0) ε (cid:1) , we solve (5.9) for w , assuming solvability conditions (5.4a) and(5.4b) and normalization condition (4.4) are satisfied. We find w = m +3 X j = n − j = n − W , j e ijx + γ − ω α ( m + µ ) α ( m + µ ) e imx ! , (5.17)where γ is arbitrary and W , n − = (since T , n − = ).5.4. The O (cid:0) ε (cid:1) Problem
The spectral problem (3.6) is ( L ,µ − λ ) w = X j =2 λ j w − j − X j =1 L j | µ =0 w − j , (5.18)where L j | µ =0 are as before and L | µ =0 = α L (1 , L (1 , L (2 , L (1 , ! , (5.19) IGH-FREQUENCY INSTABILITIES OF A BOUSSINESQ-WHITHAM SYSTEM: A PERTURBATIVE APPROACH 16 with L (1 , = ic µ − iµ u + iµ ( c − u ) + ( c − u )( iµ + ∂ x ) − u ′ , (5.20a) L (1 , = − iµ − iµ η − iµ η − η ( iµ + ∂ x ) − η ′ , (5.20b) L (2 , = − iµ sech ( α ( µ + D )) + iαµ sech( α ( µ + D )) tanh( α ( µ + D )) . (5.20c)Substituting η j , u j , and w j − for j ∈ { , , } into (5.18), we find ( L ,µ − λ ) w = m +4 X j = n − j = n − T , j e ijx , (5.21)where T , n − = (since W , n − = ).The solvability conditions for (5.21) can be expressed as (cid:18) i S ,n γ λ + iµ c g − ( µ + m )) (cid:19) (cid:18) λ γ (cid:19) + iγ (cid:18) T ,m (cid:19) = − i (cid:18) µ c g ( µ + n ) − P ,n γ (cid:0) µ c g − ( µ + m ) − P ,m (cid:1)(cid:19) . (5.22)Expressions for P ,j are in the supplemental Mathematica file. Using the solvability condition (5.4b)together with the collision condition (3.11) shows that T ,m ≡ . What remains is a linear system for λ and γ .If α = 1 . ... , then an application of the third-order solvability condition (5.10a) shows that, for µ ∈ ( − M , M ) , det (cid:18) i S ,n γ λ + iµ c g − ( µ + m )) (cid:19) = 8 λ ,r , (5.23)where λ ,r = Re ( λ ) . For µ in this interval, λ ,r = 0 by construction; thus, (5.22) is an invertiblelinear system.We solve (5.22) for λ by Cramer’s rule, using (5.10a) to eliminate the dependence on γ . Then, λ = i (cid:20) ( λ + iµ c g − ( µ + m ))( c g ( µ + n ) − P ,n )2 λ ,r + ( λ + iµ c g ( µ + n ))( c g − ( µ + m ) − P ,m )2 λ ,r (cid:21) . (5.24)To simplify further, we separate the real and imaginary components of (5.24). Since λ (5.5a) ispurely imaginary, P ,j are real-valued, and µ ∈ ( − M , M ) , we have λ ,i = Im ( λ ) = − iµ (cid:18) c g − ( µ + m ) + c g ( µ + n )2 (cid:19) , (5.25)according to (5.12). Equation (5.24) decomposes into λ = λ ,r + iλ ,i , where λ ,r = µ (cid:2) ( c g − ( µ + m ) − c g ( µ + n )) (cid:0) µ ( c g − ( µ + m ) − c g ( µ + n )) + P ,n − P ,m (cid:1)(cid:3) , (5.26a) λ ,i = − (cid:2) µ ( c g − ( µ + m ) + c g ( µ + n )) − ( P ,m − P ,n ) (cid:3) . (5.26b)As | µ | → M , λ ,r → . If λ ,r is to remain bounded, the numerator of (5.26a) must vanish in thislimit. Since c g − ( µ + m ) = c g ( µ + n ) , we must have µ = P ,m − P ,n c g − ( µ + m ) − c g ( µ + n ) . (5.27)We refer to this equality as the regular curve condition : it ensures that the curve asymptotic tothe p = 3 isola is continuous near its intersections with the imaginary axis. From the regular curvecondition , we get IGH-FREQUENCY INSTABILITIES OF A BOUSSINESQ-WHITHAM SYSTEM: A PERTURBATIVE APPROACH 17 λ = − i (cid:18) P ,m c g ( µ + n ) − P ,n c g − ( µ + m ) c g − ( µ + m ) − c g ( µ + n ) (cid:19) . (5.28)As expected, the Floquet parameterization and imaginary component of the p = 3 isola have a nonzerocorrection at O (cid:0) ε (cid:1) . These corrections improve the agreement between numerical and asymptoticresults observed at the previous order (Figure 9, Figure 10). No corrections to the real component ofthe isola are found at fourth order. Remark . If α = 1 . ... , one can show that λ = 0 = µ and S ,n = 0 . Applying the Fred-holm alternative to (5.22) gives (5.27). Then, λ is given by (5.28), and γ = 1 . The constant γ remains arbitrary at this order for this value of α only.6. Conclusions
We have extended a formal perturbation method, first introduced in [2], to obtain asymptoticbehavior of the largest ( p = 2 , ) high-frequency instabilities of small-amplitude, HPT–BW Stokeswaves. In particular, we have computed explicit expressions for (i) the interval of Floquet exponentsthat asymptotically parameterize the p th isola, (ii) the leading-order behavior of its most unstablespectral elements, (iii) the leading-order curve asymptotic to the isola, and (iv) wavenumbers that donot have a p th isola. Items (i)-(iii) can be extended to higher-order if necessary using the regular curvecondition . In all instances, our perturbation calculations are in excellent agreement with numericalresults computed by the FFH method [11].We restrict to p = 2 and p = 3 in this work, but our method can provide asymptotic expressions for p > isolas. We conjecture that this method yields the first real-component correction ofthe isola at O ( ε p ) , similar to the cases p = 2 and p = 3 . If correct, this conjecture highlights the maindifficulty of computing higher-order high-frequency instabilities, both numerically and perturbatively.The asymptotic expressions derived in this paper are intimidating and cumbersome. Although itis satisfying to have asymptotic expressions for the results previously obtained only numerically, thisis not the main point of our work. Rather, (i) the perturbation method demonstrated allows one toapproximate an entire isola at once, going beyond standard eigenvalue perturbation theory [19], (ii)the results obtained constitute a first step toward a proof of the presence of the high-frequency insta-bilities, and (iii) the asymptotic expressions for the range of Floquet exponents allow for a far more -1 -0.5 0 0.5 110 -11 -12 -1-0.8-0.6-0.4-0.200.20.40.60.8110 -11 -4 -2 0 2 410 -12 -1-0.8-0.6-0.4-0.200.20.40.60.81 10 -11 Figure 9. (Left) The p = 3 isola with α = 1 and ε = 5 × − (zero- and second-order imaginary corrections removed for better visibility). Solid and dashed redcurves are given by perturbation calculations to O (cid:0) ε (cid:1) and O (cid:0) ε (cid:1) , respectively.Blue circles are a subset of spectral elements from the numerically computed isolausing FFH. (Right) The Floquet parameterization of the real (blue) and imaginary(red) components of the isola (zero- and second-order imaginary and Floquet correc-tions removed for better visibility). Solid and dashed curves illustrate perturbationcalculations to O (cid:0) ε (cid:1) and O (cid:0) ε (cid:1) , respectively. Circles indicate FFH results. IGH-FREQUENCY INSTABILITIES OF A BOUSSINESQ-WHITHAM SYSTEM: A PERTURBATIVE APPROACH 18
Figure 10. (Left) The interval of Floquet exponents that parameterize the p = 3 isola as a function of ε with α = 1 (zero- and second-order Floquet correctionsremoved for better visibility). Solid and dashed blue curves indicate the boundaries ofthis interval according to perturbation calculations to O (cid:0) ε (cid:1) and O (cid:0) ε (cid:1) , respectively.Blue circles indicate the boundaries computed numerically by FFH. The solid redcurve gives the Floquet exponent corresponding to the most unstable spectral elementof the isola according to our perturbation calculations. Red circles indicate the samebut computed numerically using FFH. (Right) The real (blue) and imaginary (red)components of the most unstable spectral element of the isola as a function of ε (zero-order imaginary correction removed for better visibility). Solid and dashedcurves illustrate perturbation calculations to O (cid:0) ε (cid:1) and O (cid:0) ε (cid:1) , respectively. Circlesillustrate FFH results.efficient numerical computation of the high-frequency isolas, which are difficult to track numericallyas the amplitude of the solution increases. Acknowledgements : This research was funded partially by the ARCS Foundation Fellowship.7.
Appendix A. Stokes Wave Expansions
Below are the Stokes wave expansions of (2.5) (with I j = 0 ) to fourth order in the small-amplitudeparameter ε . In what follows, C k = tanh( αk ) αk and D z = 1 c − z , where c = C . (7.1)For the surface displacement η S ( x ; ε ) , η S ( x ; ε ) = εη ( x ) + ε η ( x ) + ε η ( x ) + ε η ( x ) + O (cid:0) ε (cid:1) = ε cos( x ) + (cid:16) N , + 2 N , cos(2 x ) (cid:17) ε + 2 N , ε cos(3 x )+ (cid:16) N , + 2 N , cos(2 x ) + 2 N , cos(4 x ) (cid:17) ε + O (cid:0) ε (cid:1) , (7.2) IGH-FREQUENCY INSTABILITIES OF A BOUSSINESQ-WHITHAM SYSTEM: A PERTURBATIVE APPROACH 19 with N , = 3 c D , (7.3a) N , = 3 c D C , (7.3b) N , = c D C D C (cid:16) c + 4 C (cid:17) , (7.3c) N , = − c D D C (cid:16) c + 4 C + 2 c C ( − C ) − c (2 + 11 C ) (7.3d) + c (1 + 22 C + 10 C ) (cid:17) ,N , = c D D C D C (cid:16) − c + 4 C C + c (2 + 31 C + 50 C ) (7.3e) + c C ( C + 2 C (10 + 7 C )) − c (38 C + 32 C + C ( − C )) (cid:17) ,N , = c D C D C D C (cid:16) c − C C + 5 c (4 C + 5 C ) − c (7 C + 8 C C ) (cid:17) . (7.3f)For the horizontal velocity u S ( x ; ε ) along η S ( x ; ε ) , u S ( x ; ε ) = εu ( x ) + ε u ( x ) + ε u ( x ) + ε u ( x ) + O (cid:0) ε (cid:1) = c ε cos( x ) + (cid:16) U , + 2 U , cos(2 x ) (cid:17) ε + (cid:16) U , cos( x ) + 2 U , cos(3 x ) (cid:17) ε + (cid:16) U , + 2 U , cos(2 x ) + 2 U , cos(4 x ) (cid:17) ε + O (cid:0) ε (cid:1) , (7.4)with U , = c D (cid:16) c (cid:17) , (7.5a) U , = c D C (cid:16) C + c (cid:17) , (7.5b) U , = 3 c D D C (cid:16) − c + 2 C (cid:17) , (7.5c) U , = c D C D C (cid:16) c + 2 C C + 2 c ( C + 2 C ) (cid:17) , (7.5d) U , = − c D D C (2 + c )64 (cid:16) c ( − c ) + C ( − c − c ) + 2 C (1 + 2 c ) (cid:17) , (7.5e) U , = c D D C D C (cid:16) − c + 8 C C + c (7 − C + 45 C ) + c (cid:0) C + 32 C (7.5f) + C ( −
27 + 37 C ) (cid:1) − c (60 C + 37 C C + C ( −
22 + 56 C )) (cid:17) ,U , = c D C D C D C (cid:16) c − C C C + 2 c ( − C + 5 C C + 6 C ( − C + C )) (7.5g) + c (8 C + 15( C + 2 C )) − c (5 C C C + 3 C ( C + 2 C )) (cid:17) . For the velocity of the Stokes waves c ( ε ) , c ( ε ) = c + c ε + c ε + O (cid:0) ε (cid:1) , (7.6) IGH-FREQUENCY INSTABILITIES OF A BOUSSINESQ-WHITHAM SYSTEM: A PERTURBATIVE APPROACH 20 with c = 3 c D D C (cid:16) c + 5 c − C (2 + c ) (cid:17) , (7.7a) c = 3 c D D C D C (cid:16) c (cid:0) c − c + 15 c − c + c C (11 + 3 c − c + 205 c ) (7.7b) − c C ( − c + 3 c + 38 c ) − C ( − c − c + c ) (cid:1) + C (cid:0) c ( −
103 + 309 c − c + 355 c ) − c C (31 − c + 57 c + 185 c )+ 36 c C (2 − c + 9 c + 10 c ) − C (2 + c )( − c + 13 c ) (cid:1)(cid:17) . Appendix B. Collision Condition
Up to redefining m and n , (3.11) simplifies to Ω ( µ + n ) = Ω − ( µ + m ) = 0 . (8.1)With k = µ + n and p = m − n , (8.1) becomes Ω ( k ) = Ω − ( k + p ) = 0 . (8.2)We refer to (8.2) as the collision condition. We prove that, for each p ∈ Z \ { , ± } , there existsa unique k ( p ; α ) that satisfies the collision condition. These solutions k ( p ; α ) are distinct from eachother (for each α > ) and result in an infinite number of distinct collision points on the imaginaryaxis, according to (3.11). First, we establish important monotonicity properties of Ω σ ( k ) , defined in(3.9). Lemma 1.
The function ω α ( k ) = sgn( k ) p αk tanh( αk ) is strictly increasing for k ∈ R . If | k | > ,then ω ′ α ( k ) < α | c | , where c = tanh( α ) /α . Proof.
A direct calculation shows ω ′ α ( k ) = 12 r α tanh( αk ) k + α s αk sinh( αk ) sech / ( αk ) ! , (8.3)from which ω ′ α ( k ) > . This proves the first claim. Since tanh( αk ) / ( αk ) ≤ , αk/ sinh( αk ) ≤ , and sech( αk ) ≤ , (8.3) gives ω ′ α ( k ) ≤ r α tanh( αk ) k + α sech( αk ) ! . (8.4)Since α > , sinh( α ) /α > > sech( α ) , so that sech( α ) < | c | . Because sech( z ) is even and strictlydecreasing for z > , we have sech( αk ) < | c | , for | k | > . (8.5)Similarly, since tanh( z ) /z is even and strictly decreasing for z > , r α tanh( αk ) k < α r tanh( α ) α = α | c | , for | k | > . (8.6)Together with (8.4), inequalities (8.5) and (8.6) imply ω ′ α ( k ) < α | c | for | k | > . (cid:3) Lemma 2. If c > , Ω − ( k ) is strictly decreasing for k ∈ R , and Ω ( k ) is strictly decreasing for | k | > . If c < , Ω ( k ) is strictly increasing for k ∈ R , and Ω − ( k ) is strictly increasing for | k | > . Proof.
Suppose c > . By definition, Ω ′ σ ( k ) = − αc + σω ′ α ( k ) . If σ = − , we use ω ′ α ( k ) > fromLemma 1 to conclude Ω ′− ( k ) < . If σ = 1 and | k | > , we use ω ′ α ( k ) < α | c | from Lemma 1 toconclude Ω ′ ( k ) = − αc + ω ′ α ( k ) < , since c > . An analogous proof holds when c < . (cid:3) IGH-FREQUENCY INSTABILITIES OF A BOUSSINESQ-WHITHAM SYSTEM: A PERTURBATIVE APPROACH 21
In what follows, we consider c > , which corresponds to right-traveling Stokes waves. Similarstatements hold when c < if one rewrites the collision condition (8.2) as Ω − ( k ) = Ω ( k + p ) = 0 ,where k and p are redefined appropriately. Lemma 3.
For each p ∈ R and α > , there exists a unique k ( p ; α ) ∈ R such that Ω ( k ( p ; α )) =Ω − ( k ( p ; α ) + p ) . If p ∈ Z and c > , we have · · · < k (1; α ) < k (0; α ) < k ( − α ) < · · · . Moreover, | k ( p ; α ) | > | p | for p ∈ Z \ { , ± } and c > . Proof.
Fix p ∈ R and α > . Define F ( k, p ) = Ω ( k ) − Ω − ( k + p ) . Then, F ( k, p ) ∼ k r α | k | + O p | k | ! as | k | → ∞ . (8.7)Since F has opposite signs as k → ±∞ , there exists at least one root, denoted k ( p ; α ) . Since ∂ k F ( k, p ) = ω ′ ( k ) + ω ′ ( k + p ) > by Lemma 1, k ( p ; α ) is the only root of F in R , proving the firstclaim of the theorem.To prove the second claim, differentiate F ( k ( p ; α ) , p ) with respect to p . Using the definition of Ω σ , k ′ ( p ) = Ω ′− ( k ( p ; α ) + p ) ω ′ ( k ( p ; α )) + ω ′ ( k ( p ; α ) + p ) , (8.8)which is well-defined since ∂ k F ( k, p ) > . If c > , then Lemma 2 implies k ′ ( p ) < . If p is restrictedto Z , we have · · · < k (1; α ) < k (0; α ) < k ( − α ) < · · · , as desired.To prove the third claim, first consider p > . Suppose k ( p ; α ) ≥ − p . Since ω α ( k ) is odd andstrictly increasing by Lemma 1, ω α ( k ( p ; α )) ≥ − ω α ( p ) , (8.9a) ω α ( k ( p ; α ) + p ) ≥ ω α (0) = 0 . (8.9b)Using the definition of Ω σ , F ( k ( p ; α ) , p ) = 0 can be rewritten as ω α ( k ( p ; α )) + ω α ( k ( p ; α ) + p ) = − αc p. (8.10)Together with (8.10), inequalities (8.9a) and (8.9b) imply − ω α ( p ) ≤ − αc p ⇒ ω α ( p ) p ≥ αc = ω α (1)1 , (8.11)a contradiction since ω α ( z ) /z is strictly decreasing for z > . Therefore, k ( p ; α ) < − p for p > .Since Ω σ ( k ) is odd, k ( p ; α ) = − k ( − p ; α ) . Therefore, when p < − , k ( p ; α ) > − p . Combining the twocases yields | k ( p ; α ) | > | p | whenever p ∈ Z \ { , ± } and c > , as desired. (cid:3) Lemma 3 has several consequences:1. When c > , k ( p ; α ) < for p > , and k ( p ; α ) > for p < .2. When c > , k ( p ; α ) → ±∞ as p → ∓∞ . In fact, the sequence { k ( p ; α ) } must grow at leastlinearly as | p | → ∞ . Formal arguments suggest quadratic growth in this limit.3. The products k ( p ; α )( k ( p ; α ) + p ) > and ω α ( k ( p ; α )) ω α ( k ( p ; α ) + p ) > when c > .The latter of these products is related to the Krein signature condition proposed in [21]. Ineffect, Lemma 3 provides a different proof that collided eigenvalues (3.11) have opposite Kreinsignatures, consistent with [13].The above results lead to the following theorem. Theorem 4.
Let c > . If p ∈ { , ± } , then the collision condition (8.2) is not satisfied. If p ∈ Z \ { , ± } , then k ( p ; α ) solves the collision condition. Moreover, · · · < λ i, < λ i, < < λ i, − <λ i, − < · · · , where λ i,p is the imaginary part of the collision point corresponding to k ( p ; α ) . Proof.
When p = 0 or ± , we have k ( p ; α ) = 0 or ∓ , respectively, by inspection. It follows that Ω ( k ( p ; α )) = 0 in all three cases, and so (8.2) is not satisfied. This proves the first claim.To prove the second claim, consider the sequence { Ω ( k ( p ; α )) } , p ∈ Z \ { , ± } . From Lemma 3, { k ( p ; α ) } is a strictly decreasing sequence, and each element of this sequence satisfies | k ( p ; α ) | > | p | > . Thus, Lemma 2 holds, and the sequence { Ω ( k ( p ; α )) } is strictly increasing. Since Ω ( ±
1) = 0 , we
IGH-FREQUENCY INSTABILITIES OF A BOUSSINESQ-WHITHAM SYSTEM: A PERTURBATIVE APPROACH 22 have Ω ( k ( p ; α )) = 0 . This proves that k ( p ; α ) satisfies the collision condition (8.2) for the relevantvalues of p .The proof of the third claim is immediate since { Ω ( k ( p ; α )) } is strictly increasing. (cid:3) Let µ = k ( p ; α ) − [ k ( p ; α )] for p ∈ Z \ { , ± } , where [ · ] denotes the nearest integer function. Then, µ is the unique Floquet exponent in [ − / , / for which λ (1)0 ,µ ,n and λ ( − ,µ ,m satisfy (3.11) with n = [ k ( p ; α )] and m = n + p . References [1] Akers, B.; Nicholls, D. The spectrum of finite depth water waves.
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