Grimshaw, RHJ

Johnson, ER

UCL Discovery

The reduced modified Ostrovsky equation: integrability andbreakingE. R. JohnsonDepartment of Mathematics,University College London, U.K.R. H. J. GrimshawDepartment of Mathematical Sciences,Loughborough University, U.K.(Dated: May 2, 2013)AbstractThe reduced modified Ostrovsky equation is a reduction of the modified Korteweg-de Vriesequation, in which the usual linear dispersive term with a third-order derivative is replaced by alinear non-local integral term, representing the effect of background rotation. Here we study thecase when the cubic nonlinear term has the same polarity as the rotation term. This equation isintegrable provided certain slope constraints are satisfied. We demonstrate, through theoreticalanalysis and numerical simulations, that when this constraint is not satisfied at the initial time,then wave breaking inevitably occurs.1I. INTRODUCTIONIt is well-known that the extended Korteweg-de Vries, or Gardner, equation can be usedto model internal solitary waves in the atmosphere and ocean, see for instance the reviewsby Grimshaw [1, 2] and Helfrich and Melville [3],ut + µuux + νu2ux + λuxxx = 0 . (1)Here u(x, t) is the amplitude of an appropriate linear long wave mode, with linear long wavespeed c0, and (1) is expressed in a frame moving with that speed. The coefficients µ, ν, λ arefound from certain internal expressions involving the modal function and the backgrounddensity stratification. When the coefficient µ = 0, (1) becomes the modified KdV equation.This case can be realised in practice, for instance, when the background stratification isthat for a two-layer fluid, with equal layer depths. However, when the effects of backgroundrotation through the Coriolis parameter f need to be taken into account, an extra term isneeded, and (1) is replaced by(ut + µuux + νu2ux + λuxxx)x = γu , (2)where γ = f 2/2c0 6= 0. When (1) is just the KdV equation, that is the coefficient ν = 0,then (2) becomes the Ostrovsky equation, see Ostrovsky [4], Grimshaw [5], or the review byGrimshaw et al. [6].Our concern here is with the reduced modified Ostrovsky equation which is obtained bysetting λ = 0 and µ = 0 in (2),ut + νu2ux = γu . (3)Importantly, we note that when γ = 0, equation (3) reduces to a modified Hopf equation. Itis then easily demonstrated that all localized solutions, or all periodic solutions, will break.That is the solution will develop an infinite slope in finite time. The issue then is how thisbreaking is affected when γ 6= 0. This is to be contrasted with the regularisation by themKdV equation when breaking is replaced by the emergence of modulated periodic waves,see Kamchatnov et al. [7]. This present study is motivated by the recent article by Grimshawet al. [8] who examined the reduced Ostrovsky equation, that is (2) with λ, ν = 0. Theyshowed that then the equation is either integrable and solutions exist for all time, or wavebreaking occurs, depending on a certain criterion on the curvature of the initial condition.2Since (3) can be regarded as the reduced Ostrovsky equation with cubic nonlinearity ratherthan quadratic nonlinearity, we expect that a similar type of analysis can be applied here,and that is indeed the case, with the curvature criterion there being replaced here with slopecriteria.A canonical form of the reduced modified Ostrovsky equation is obtained by settingu(γ/2|ν|)1/2u˜ , t = t˜/γ , x = x˜, so that(ut ± u2ux2)x = u . (4)Here the ˜ has been removed. There are two equations, called respectively MROI, MROIIaccording to the sign ± in the coefficient in the cubic term, corresponding to the sign ofγν in (2). MROII is also known as the short pulse equation arising in nonlinear optics, seeSakovich and Sakovich [9, 10], Liu et al. [11], Pelinovsky and Sakovich [12] and the referencestherein. Note that equation (4) has the symmetry that if u is a solution, so also is −u. Also,the equation is invariant under the transformation u(x, t) = Du˜(x˜, t˜) where x˜ = x/D, t˜ = Dtfor any D > 0, and in particular, the slope ux = u˜x˜ is invariant. The equation has a zeromass constraint for periodic or localised solutions,∫Du dx = 0 , (5)where D is the periodic, or infinite, domain. There are also conservation laws for “momen-tum” and “energy”, similar to those for the Ostrovsky equation, see Grimshaw and Helfrich[13]. Further, multiplying (4) by ux yields{ ∂∂t± u22∂∂x}u2x + 2uux(±u2x − 1) = 0 ,and hence there is another conservation law,Ft ± (u2F2)x = 0 , F = |1∓ u2x|1/2 . (6)This conservation law is crucial for the issue of integrability and breaking.II. INTEGRABILITYTo establish integrability, we follow the same procedure used for the reduced Ostrovskyequation, see Grimshaw et al. [8]. Thus we transform to characteristic coordinates,x = θ(X,T ) = X ±∫ T0U2(X,T ′)2dT ′ , t = T , u(x, t) = U(X,T ) . (7)3Equivalently these are defined bydxdt= ±u22, where x = X at t = 0 . (8)It then follows thatUT = ut ± u2ux2, UX = φux ,φ = θX = 1±∫ T0U(X,T ′)UX(X,T ′) dT ′ , φT = ±UUX .(9)The Jacobian of the transformation isJ =∂(x, t)∂(X,T )= φ .Thus equation (4) becomesUXT = φU , and so UUXTT − UTUXT = ±U3UX . (10)This is more conveniently written as the systemφT = ±WU , WT = φU , W = UX . (11)Eliminating U yields the conservation law(φ2 ∓W 2)T = 0 , (12)showing that φ2 ∓W 2 is conserved along the characteristics X = constant. Importantly,this is just the conservation law (6) sinceφ2 ∓W 2 = φ2(1∓ u2x) = φ2F 2 . (13)Next consider the evolution from an initial profile u(x, 0) = u0(x) for some smooth u0(x).Since X(x, 0) = x, φ = 1, T = 0 when t = 0,φ2 ∓W 2 = 1∓W 20 , φF = F0 , T ≥ 0 , (14)where W0 = W (X, 0) = UX(X, 0) = u0x(x) , F0(X) = |1∓ u20x|1/2 .We next introduce the transformations associating a unique Z(X,T ) with each pair (φ,W ),φ = F0(X) coshZ , W = F0(X) sinhZ , for MROI , u20x < 1 , (15)φ = F0(X) sinhZ , W = F0(X) coshZ , for MROI , u20x > 1 , (16)φ = F0(X) cosZ , W = F0(X) sinZ , for MROII . (17)4These imply in turnZT = U , ZXT = F0(X) sinhZ , for MROI , u20x < 1 , (18)ZT = U , ZXT = F0(X) coshZ , for MROI , u20x > 1 , (19)ZT = U , ZXT = F0(X) sinZ , for MROII . (20)Thus Z satisfies the sinh-Gordon, cosh-Gordon or sine-Gordon equation respectively. Allare integrable equations, but only (18, 20) have soliton solutions. For further progress weexamine each of MROI, MROII separately.III. ANALYSIS AND NUMERICAL RESULTSA. The reduced modified Ostrovsky equation MROI1. |u0x| < 1 everywhereFirst we examine the case of a + sign in (4), which is the main case of interest here. Ifthe initial slope |u0x| < 1 for all x, then 0 < F0(X) ≤ 1 for all X. It follows from (14) thatthen W 20 < 1 for all X, and so φ > |W | ≥ 0 for all X,T . Breaking cannot occur and sincethe slope ux = UX/φ = W/φ, |ux| < 1 for all x, t. The equation remains integrable for all t.2. |u0x| > 1 somewhereNext suppose that there is a set of intervals X1 < x < X2 where |u0x| > 1 with equalityonly at the end points, so that F0(X1,2) = 0. Since F is conserved, see (14), F = 0 isconserved on characteristics, that is, F (X1,2, T ) = 0 for all T ≥ 0. When the initial value of1− u20x, takes both positive and negative values, then as long as the solution exists, that is0 < φ <∞, the arguments above show that the X,T domain is divided into regions where|W/φ| = |ux| > 1, namely the region between the characteristic boundaries X = X1,2 andthe remaining regions where |ux| < 1 with |ux| = 1 on X = X1,2. Moreover, the wave cannotbreak in the regions of the (X,T ) domain where |ux| < 1.In the region X1 < X < X2,F0(X) = [u20x − 1]1/2 = [W 20 − 1]1/2 > 0 , (21)5and it follows that |ux| > 1, throughout the region. In this case, we can use (16) to obtainthe cosh-Gordon equation (19). This can be written in terms of φ or F alone, as follows{ ψX[1 + ψ2]1/2}T = { ψT[1 + ψ2]1/2}X = F0(1 + ψ2)1/2 , ψ = φF0=1F. (22)Integrating (22) yieldsψX = (1 + ψ2)1/2{− F0XF0(1 + F 20 )1/2+∫ T0(1 + ψ2(X,T ′))1/2 dT ′} . (23)Hence, in any region where F0X ≤ 0, ψX > 0 and hence ψ cannot take its minimum valuein any such region. Instead a minimum can only be reached where F0X > 0, that is where|u0x| > 1 and is increasing.The system (11) subject to the initial conditions that φ = 1,W = u0x at T = 0 is solved tospectral accuracy following Esler et al. [14] and Grimshaw et al. [8]l. As noted by Esler et al.[14] solving in characteristics space is particularly useful for investigating wave breaking asthe wave breaks when an order unity quantity passes smoothly through zero. Esler et al.[14] show that the characteristic integrations can be carried smoothly past breaking wherethey agree closely with finite volume integrations which fit “equal area” shocks to the wavesafter breaking.The system (11) was integrated numerically with spectral accuracy by performing theintegration in Fourier space and then normalising in real space using the result that thetrapezium rule is spectrally accurate for periodic functions. Integrations up to T = 100 with2048 nodes showed F was conserved with an accuracy of 10−6. Figure 1 shows the waveprofile u(X,Tb) and the Jacobian φ(X,Tb) at the instant of breaking, T = Tb = 108.25 forthe initial profileu0x(x) = b cosx , (24)with b = 1.00005 so that max[u20x − 1]1/2 = 10−2 For this initial condition F0 has periodpi and thus so too does φ, while u0x has period 2pi and hence so too do W and U . Thewave first breaks in small intervals surrounding X = pi/2 and X = 3pi/2 where |u0x > 1.In particular, the wave first breaks just below the points of maximum |u0x| at X = pi/2and X = 3pi/2. Let the minimum of the Jacobian φ in (0, pi) at time T be φm(T ) andlie at Xm(T ). Figure 2a shows φmT ) for the same evolution as Figure 1, with φm(T ) firstvanishing at Tb = 108.25 and Figure 2b shows that Xm(T ) lies below, but close to X = pi/2as expected. Figure 3 shows the variation in the time to breaking, Tb with Fm, the value of60 0.5 1 1.5 2−1−0.8−0.6−0.4−0.200.20.40.60.81x/piuu at time t = 108.250 0.5 1 1.5 200.511.522.5J at time t = 108.25X/piφ(X)(a) (b)FIG. 1. (a) The wave profile u(x, tb) at the instant of breaking, T = t = tb = 108.25, for the initialprofile (24) with b = 1.00005 so that max[u20x− 1)1/2] = 10−2 computed with N = 1024 nodes. (b)The Jacobian φ(X, tb) at the instant of breaking. The thinner curve shows F0(X) = |1−u20x(X))|1/2.The initial profile slope satisfies |u0x| > 1 in small intervals surrounding X = pi/2 and X = 3pi/2.As expected from (14), the Jacobian exceeds F0(X) outside these intervals and so breaking firstoccurs inside these intervals..F0 at the point of initial maximum gradient for the initial profile (24) with varying b. Thebehaviour with decreasing Fm is consistent with the relationTb = AF−4/3m (25)for some constant A.B. The modified reduced Ostrovsky equation MROIIAlthough this case is not our main interest here, we record here some known results forcompleteness and comparison with the case MROI, see Liu et al. [11] and Pelinovsky andSakovich [12] for instance. Now the initial condition is such that F0(X) ≥ 1 is defined forall X as a bounded smooth function of X. Hence MROI is integrable provided that φ 6= 0,70 20 40 60 80 10000.050.10.150.20.250.30.350.4φm(T)T0 20 40 60 80 10000.020.040.060.080.10.120.140.160.180.2Tpi/2−Xm(T)(a) (b)FIG. 2. (a) φm(T ), the minimum of the Jacobian in (0, pi) at time T for the evolution of figure 1.φm(T ) first vanishes at tb = 108.25. (b) The distance Xm(T ), the position of the minimum of theJacobian, lies below pi/2.which is the case unless ux →∞, that is unless breaking occurs. Also W = UX is boundedfor all X,T , and so the case φ→∞ cannot occur. At T = 0, φ = 1, |Z| < pi/2, see (17). andso φ > 0 until breaking occurs, when φ = 0, Z = ±pi/2. Hence the issue of breaking/notbreaking in MROI (4) from an initial condition u(x, 0) = u0(x) can be restated in terms ofthe sine-Gordon (SG) equation (20), which can be written asZY T = sinZ , Y =∫ XF0(X′) dX ′ . (26)That is u0(x) generates a Z(X, 0) = Z0(X) initial condition for the SG equation (20) suchthat−pi/2 < Z0(X) < pi/2. Then if the evolving solution is such that−pi/2 < Z(Y, T ) < pi/2for all T ≥ 0, then there is no breaking. In particular, note that the kink solution of the SGequation violates this condition, and hence any initial condition for the SG equation whichgenerates a kink, corresponds to a breaking solution for the MROII equation. Equation (26)has an infinite set of conservation laws inherited from the SG equation, and especially we80 50 100 150 200 250 300 3500.080.10.120.140.160.180.20.220.240.260.281/FmTbF4/3mFIG. 3. The scaled time to breaking, F4/3m tb, as a function of 1/Fm, where Fm is the value of F0at the point of initial maximum gradient for the initial profile (24) with varying b.note(Z2Y2)T − (1− cosZ)Y = 0 ,(1− cosZ)T − (Z2T2)Y = 0 ,(−Z2Y Y +Z4Y4)T + (Z2Y cosZ)Y = 0 ,(Z2T cosZ)T + (−Z2TT +Z4T4)Y = 0 .(27)Using these and other conservation laws, it can be shown that, on the one hand, there isnow breaking and solutions exist for all time if the initial condition has sufficiently smallamplitude and slope, while on the other hand, an initial condition with a large amplitudeand slope will break, see Liu et al. [11] and Pelinovsky and Sakovich [12]. However, a moreprecise breaking condition analogous to that for MROI in section 3.1 is elusive.9IV. CONCLUSIONFor the MROI equation (4), we have shown that when the initial condition u(x, 0) = u0(x)is such that |u0x| > 1 somewhere, then wave breading inevitably occurs, and does so in alocation close to the left-hand end point of the region where |u0x| > 1. On the otherhand, if u0x| < 1 everywhere, then the equation is integrable, since it can be transformedto the sinh-Gordon equation (18). The precision of this result is analogous to that foundfor the reduced Ostrovsky equation, see Grimshaw et al. [8], where the slope condition isreplaced by a curvature condition. On the other hand, although the MROII equation (4)can be transformed to the integrable sine-Gordon equation (20), we have not found anysimilar precise wave-breaking conditions, although the available numerical and analyticalevidence suggests that initial conditions with sufficiently steep slopes lead to breaking, butsolutions exist for all time when the initial slopes are sufficiently small, see Liu et al. [11]and Pelinovsky and Sakovich [12].Finally, we recall that the MROI, MROII equations are obtained from extended Ostrovskyequation (2) by omitting the quadratic nonlinear term, and the third-order linear dispersiveterm. When wave-breaking occurs, then this latter term needs to be restored, with theoutcome that the potential discontinuity with an infinite slope is, in practice, resolved by theformation of an undular bore, see Kamchatnov et al. [7]. Hence, we infer that the breakingcondition, now expressed in terms of the original variables, |u0x| > |γ/2ν|1/2 implies theformation of an undular bore in the full equation (2) (with µ = 0). Thus, as expected strongrotation, (large γ) inhibits the formation of undular bores.[1] R. Grimshaw, in Environmental Stratified Flows, edited by R. Grimshaw (Kluwer, 2001) pp.1–27.[2] R. H. J. Grimshaw, in Nonlinear waves in fluids: Recent advances and modern applications.,CISM Courses and Lectures, Vol. 483, edited by R. H. J. Grimshaw (Springer, 2005) pp. 1–28.[3] K. R. Helfrich and W. K. Melville, Ann. Rev. Fluid Mech. 38, 395 (2006).[4] L. Ostrovsky, Oceanology 18(2), 119 (1978).[5] R. Grimshaw, Stud. Appl. Math. 73, 1 (1985).[6] R. H. J. Grimshaw, L. A. Ostrovsky, V. I. Shrira, and Y. A. Stepanyants, Surveys in Geo-10physics 19, 289 (1998).[7] A. M. Kamchatnov, Y. H. Kuo, T. C. Lin, T. L. Horng, S. C. Gou, R. Clift, G. A. El, andR. H. J. Grimshaw, Physical Review E 86 (2012).[8] R. H. J. Grimshaw, K. Helfrich, and E. R. Johnson, Studies In Applied Mathematics 129,414 (2012).[9] A. Sakovich and S. Sakovich, J. Phys. Soc. Japan 74, 239 (2004).[10] A. Sakovich and S. Sakovich, J. Phys.A, Mtah. Gen. 39, L361 (2006).[11] Y. Liu, D. Pelinovsky, and A. Sakovich, Dynamics of PDE 6, 291 (2009).[12] D. Pelinovsky and A. Sakovich, Comm. Partial Diff. Eq. 35, 613 (2010).[13] R. Grimshaw and K. R. Helfrich, IMA J. Appl. Math. 77, 326 (2012).[14] J. G. Esler, O. J. Rump, and E. R. Johnson, Phys. Fluids 21, 066601 (2009).11

The reduced modified Ostrovsky equation: integrability and breaking

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The reduced modified Ostrovsky equation: integrability and breaking

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