On the resonant reflection of weak, nonlinear sound waves off an entropy wave
OON THE RESONANT REFLECTION OF WEAK, NONLINEARSOUND WAVES OFF AN ENTROPY WAVE
JOHN K. HUNTER AND EVAN B. SMOTHERS
Abstract.
We derive a degenerate quasilinear Schr¨odinger equation that de-scribes the resonant reflection of very weak, nonlinear sound waves off a weaksawtooth entropy wave. Introduction
Away from a vacuum, the Eulerian form of the one-dimensional, nonisentropicgas dynamics equations is a 3 × (cid:15) (cid:28) u ( x, t ), v ( x, t ) of the left and right moving sound waves,in separate reference frames moving with their linearized sound speeds, satisfy asystem of the form u t + (cid:18) u (cid:19) x = 12 π (cid:90) π − π K ( x − y ) v y ( y, t ) dy,v t + (cid:18) v (cid:19) x = 12 π (cid:90) π − π K ( y − x ) u y ( y, t ) dy. (1)Here, the kernel K ( x ) is a rescaled entropy-wave profile, which is not affected bythe sound waves to leading order, and all functions are 2 π -periodic in x with zeromean. The convolution terms in (1) describe the resonant reflection of the left andright moving sound waves off the entropy wave, while the Burgers terms describethe nonlinear deformation of the sound waves. The validity of these asymptotic Date : October 14, 2018.JKH was supported by the NSF under grant number DMS-1616988. a r X i v : . [ m a t h . A P ] O c t JOHN K. HUNTER AND EVAN B. SMOTHERS equations for weak solutions is proved in [17], and the global existence of entropysolutions of (1) is proved in [16].In general, the system (1) is spatially nonlocal and dispersive. One of the simplestcases, and one with particularly interesting dynamics, is when the entropy wave isa sawtooth wave K ( x ) = (cid:40) x + π for − π < x < x − π for 0 < x < π , K ( x + 2 π ) = K ( x ) , which leads to the local, nondispersive system [12] u t + (cid:18) u (cid:19) x = v, v t + (cid:18) v (cid:19) x = − u. (2)In this case, the linearized sound waves oscillate with frequency 1 independent oftheir wavenumber and have zero group velocity. We refer to (2) as the Majda-Rosales-Schonbeck (MRS) equation.In this paper, we consider the resonant reflection of sound waves off an entropywave when the sound waves are much weaker than the entropy wave. Specifically,we suppose that the entropy wave is a sawtooth wave with amplitude of the order (cid:15) (cid:28) (cid:15) / (cid:28) (cid:15) . This particularchoice of scaling leads to a dominant balance between weak nonlinearity and weakdispersion.There are then three timescales: on times of the order 1, the sound waves propa-gate at the linearized sound speed; on times of the order 1 /(cid:15) , the sound wave profilesoscillate linearly with constant frequency as a result of their reflection off the en-tropy wave; and on times of the order 1 /(cid:15) , the spatial profiles of these oscillationsare modulated by the nonlinear self-interaction of the sound waves.It is convenient to carry out the asymptotic expansion in Lagrangian coodinates,when the entropy wave is independent of time. As summarized in Section 3, thedeformation x (cid:55)→ χ ( x, t ) of the compressible fluid satisfies the quasilinear waveequation (12) with wave speed (13).In suitably nondimensionalized variables, our asymptotic solution for the La-grangian velocity ν = χ t of the fluid is given by(3) ν ( x, t ; (cid:15) ) = (cid:15) / (cid:8) a ( t − x, (cid:15) t ) − ia ( t + x, (cid:15) t ) (cid:9) e − i(cid:15)t + c.c. + O ( (cid:15) )as (cid:15) → + , where the complex-valued amplitude function a ( z, τ ) is 2 π -periodic in z with zero mean and satisfies the degenerate quasilinear Schr¨odinger equation i (cid:18) a τ + 12 ∂ − z a − π a z (cid:19) = ( γ + 1) | a | a z ) z . (4)Equation (3) describes right and left moving sound waves with linearized speed 1and complex amplitude functions a and − ia , respectively. The sound waves oscillateslowly with frequency (cid:15) and evolve over longer times of the order 1 /(cid:15) according to(4). As described in Section 2, the dispersionless version of (4), given in (6), alsoarises directly from the MRS equation (2).When | a | is bounded away from zero, one expects that the qualitative behaviorof (4) is similar to a Schr¨odinger equation. However, energy estimates for thederivatives of a fail when a →
0. Numerical solutions of (4) shown in Section 5.2suggest that a may become zero at some point, even if it is initially nonzero, leading EFLECTION OF SOUND WAVES OFF AN ENTROPY WAVE 3 to the generation of smaller-scale waves and a possible loss of smoothness in thesolution.From a more general perspective, the much richer dynamical possibilities forspatially periodic solutions of the nonisentropic gas dynamics equations in compar-ison with the isentropic equations is the simplest and most physically importantexample of the different behavior of large-variation solutions of n × n nonlinearhyperbolic conservation laws for n ≥ n = 2.Glimm [6] proved the existence of global weak entropy solutions of n × n systemsof strictly hyperbolic conservation laws in one space dimension for u : R × R + → R n , u t + f ( u ) x = 0 f : R n → R n , with genuinely nonlinear or linearly degenerate wave fields and initial data withsufficiently small total variation. These Glimm solutions were later shown to beunique by Bianchini and Bressan [1]. In this case, the different wave fields canundergo only a limited amount of significant interactions, and they eventually sep-arate into nondecaying linearly degenerate waves and decaying genuinely nonlinear N -waves [13, 14].For 2 × L ∞ (but have infinite total variation on R ) decayto zero as t → ∞ . Thus, as a consequence of shock-formation and the resultingdecay induced by shocks, the long-time dynamics of solutions is trivial in both ofthese casesFor n ≥
3, the behavior of solutions of n × n hyperbolic systems with small L ∞ -norm and large total variation may be very different, since significant waveinteractions can persist for all t >
0. In particular, resonant three-wave interactionsmay weaken shocks, or delay their formation, or conceivably prevent their formationentirely [21]. This may lead to solutions that do not decay to zero as t → ∞ andhave nontrivial long-time nonlinear dynamics. The formal asymptotic solutions forresonant gas dynamics support these suggestions [15, 18], as do the ones derivedhere. However, it should be noted that these asymptotic solutions only apply forlong times of some order 1 /(cid:15) n and need not remain valid as t → ∞ , leading todifficult KAM-type issues [22].An outline of this paper is as follows. In Section 2, we derive the asymptoticequation (6) for the MRS equation. In Section 3, we summarize the Lagrangiangas dynamics equations, and in Section 4, we derive the asymptotic equation (4).Finally, in Section 5, we present some numerical solutions.2. The MRS Equation
As a model problem, we first consider the MRS equation (2), which provides anapproximate description of the resonant reflection of sound waves off a sawtoothentropy wave. We will show that this system has an asymptotic solution of theform u ( x, t ; (cid:15) ) = (cid:15)a ( x, (cid:15) t ) e − it + c.c. + O ( (cid:15) ) ,v ( x, t ; (cid:15) ) = − i(cid:15)a ( x, (cid:15) t ) e − it + c.c. + O ( (cid:15) ) , (5) JOHN K. HUNTER AND EVAN B. SMOTHERS as (cid:15) →
0, valid on timescales t = O ( (cid:15) − ), where the complex-valued amplitude a ( x, τ ) satisfies a degenerate, quasilinear Schr¨odinger equation [8](6) ia τ + (cid:18) | a | a x (cid:19) x = 0 . Equation (5) describes a solution with an arbitrary spatial profile and amplitudeof the order (cid:15) that oscillates in time with frequency 1. Nonlinear effects lead to aslow deformation of the profile over long times of the order 1 /(cid:15) , according to (6).Using the method of multiple scales, we look for asymptotic solutions of (2) ofthe form u ( x, t ; (cid:15) ) = (cid:15)u ( x, t, (cid:15) t ) + (cid:15) u ( x, t, (cid:15) t ) + (cid:15) u ( x, t, (cid:15) t ) + O ( (cid:15) ) ,v ( x, t ; (cid:15) ) = (cid:15)v ( x, t, (cid:15) t ) + (cid:15) v ( x, t, (cid:15) t ) + (cid:15) v ( x, t, (cid:15) t ) + O ( (cid:15) ) . Substituting these expansions into (2), expanding derivatives, and equating coeffi-cients of (cid:15) , (cid:15) , and (cid:15) , we find that the functions u j ( x, t, τ ), v j ( x, t, τ ) with j = 1 , , u t = v , v t = − u (7) u t + (cid:18) u (cid:19) x = v , v t + (cid:18) v (cid:19) x = − u , (8) u t + u τ + ( u u ) x = v , v t + v τ + ( v v ) x = − u . (9)The solution of (7) for ( u , v ) is(10) u ( x, t, τ ) = a ( x, τ ) e − it + c . c ., v ( x, t, τ ) = − ia ( x, τ ) e − it + c . c ., where a ( x, τ ) is an arbitrary complex-valued function. The solution of (8) for( u , v ) is then u ( x, t, τ ) = B ( x, τ ) e − it − M ( x, τ ) + c . c .,v ( x, t, τ ) = C ( x, τ ) e − it + M ( x, τ ) + c . c .,B = − (cid:18) i (cid:19) (cid:0) a (cid:1) x , C = − (cid:18) − i (cid:19) (cid:0) a (cid:1) x ,M = 12 (cid:0) | a | (cid:1) x . (11)where we omit a homogeneous solution that does not enter into the final result.The solvability condition for the removal of secular terms from solutions ( u , v )of the system u t − v = F e − it , v t + u = Ge − it is F + iG = 0. We use (10)–(11) in (9), compute the coefficients of e − it , and imposethis solvability condition, which gives equation (6) for a ( x, τ ).We remark that one can derive similar asymptotic solutions for more generalbalance laws for u ( x, t ) ∈ R n of the form u t + d (cid:88) j =1 f j ( u ) x j = g ( u ) , where ∇ u f j (0) = 0, so that the fluxes f j : R n → R n are at least quadraticallynonlinear, and ∇ u g (0) has purely imaginary eigenvalues, so that the linearized EFLECTION OF SOUND WAVES OFF AN ENTROPY WAVE 5 equation has oscillatory solutions. See [20] for more details in the one-dimensionalcase d = 1.The MRS system (2) can be compared with the Burgers-Hilbert equation u t + (cid:18) u (cid:19) x = H [ u ] , where H is the Hilbert transform with symbol − i sgn k . This equation arises asa description of waves on a vorticity discontinuity and models unidirectional, realwave motions whose linearized frequency is independent of their wavenumber [2, 9].Since H = − I , the functions ( u, v ) with v = H [ u ] satisfy u t + (cid:18) u (cid:19) x = v, v t + | ∂ x | (cid:18) u (cid:19) = − u, where | ∂ x | = H∂ x has symbol | k | . This system has the same linear terms as theMRS equation, but has a nonlocal nonlinear term. Instead of (6), the complexamplitude a ( x, τ ) of weakly nonlinear solutions of the Burgers-Hilbert equationsatisfies a nonlocal, degenerate, cubically quasilinear Schr¨odinger-type equation ia τ + P (cid:2) | a | a x − ia | ∂ x | (cid:0) | a | (cid:1)(cid:3) x = 0 , where P denotes the projection onto positive wavenumber components, and P a = a [2].The degenerate quasilinear Schr¨odinger (DQS) equation (6) is of interest inits own right as a degenerate dispersive analog of the degenerate diffusive porousmedium equation u t = ( u u x ) x [23]. We will study the DQS equation in more detailelsewhere [10], but, by analogy with wetting fronts for the porous medium equa-tion, it is natural to ask about the propagation of finite-speed dispersive fronts intothe zero solution for the DQS equation. In Section 5.1, we show a numerical com-parison of front propagation for the MRS and DQS equations. Front propagationin a different degenerate quasilinear Schr¨odinger equation and related degeneratequasilinear KdV equations is analyzed in [4, 5].3. Gas Dynamics
We consider the one-dimensional flow of an inviscid, ideal gas with equation ofstate p = Kρ γ e s/C V , where p , ρ , s are the pressure, spatial density, and entropy, respectively, and K , C V , γ are positive constants.The Lagrangian form of the compressible Euler equations for the deformation x (cid:55)→ χ ( x, t ) of a Lagrangian particle x can be written as the quasilinear waveequation (see e.g., [3])(12) χ tt + 1 γ c ( x ) (cid:20) χ γx (cid:21) x = 0 , where we use a reference configuration in which p = p is constant when χ ( x, t ) = x ,and the sound speed c ( x ) is given in terms of the Lagrangian density ρ ( x ) by c ( x ) = (cid:18) γp ρ ( x ) (cid:19) / . JOHN K. HUNTER AND EVAN B. SMOTHERS
We use nondimensionalized ( x, t )-variables in which the mean sound speed isequal to 1 and the perturbations in the sound speed have wavelength π with respectto the Lagrangian variable x . Introducing a small parameter (cid:15) (cid:28)
1, we supposethat(13) c ( x ; (cid:15) ) = 1 + (cid:15)S (2 x ) , where the 2 π -periodic sawtooth function S ( x ) is given by(14) S ( x ) = (cid:40) x + π ) if − π ≤ x < x − π ) if 0 < x ≤ π , S ( x + 2 π ) = S ( x ) . The corresponding entropy perturbation is(15) s ( x ) = s + γC V log [1 + (cid:15)S (2 x )] = s + (cid:15)γC V S (2 x ) + O ( (cid:15) ) . One could include O ( (cid:15) ) perturbations in the expansion (13) of the wave speed, butfor simplicity we assume they are zero. The effect of such perturbations is discussedfurther at the end of Section 4.5.We consider weakly nonlinear asymptotic solutions of (12) which are 2 π -periodicin x . In that case, the phases of the left and right moving sound waves and theentropy wave satisfy ( t + x ) − ( t − x ) = 2 x , so the sound waves reflect resonantlyinto each other off the entropy wave. In the next section, we derive the asymptoticsolution of (12) for the Lagrangian velocity ν = χ t that is given in (3)–(4).4. Derivation of the asymptotic equation
Writing χ ( x, t ) = x + w ( x, t ) in (12), we get that the displacement w satisfies(16) w tt + 1 γ c ( x ) (cid:20) w x ) γ (cid:21) x = 0 , where c is given by (13). We look for asymptotic solutions of (16) of the form w ( x, t ; (cid:15) ) = (cid:15) / W ( x, t, T, τ ; (cid:15) ) , T = (cid:15)t, τ = (cid:15) t,W = W + (cid:15) / W + (cid:15)W + (cid:15) / W + (cid:15) W + O ( (cid:15) / ) , (17)where W ( x, t, T, τ ; (cid:15) ) is a 2 π -periodic function of x with zero mean.Using (13) and (17) in (16), expanding time derivatives, Taylor expanding theresult, and equating coefficients of powers of (cid:15) / , we obtain the perturbation equa-tions (cid:32)L W = 0 , (18) (cid:32)L W = 0 , (19) (cid:32)L W = S (2 x ) W xx − W tT , (20) (cid:32)L W = S (2 x ) W xx − W tT − ( γ + 1) W xx W x , (21) (cid:32)L W = S (2 x ) W xx − W tT − ( γ + 1)( W x W x ) x − W T T − W tτ , (22)where (cid:32)L = ∂ t − ∂ x . EFLECTION OF SOUND WAVES OFF AN ENTROPY WAVE 7
The W , W equations. We make a change of independent variables ( x, t ) (cid:55)→ ( ξ, η ), where ξ = t − x, η = t + x, (cid:32)L = 4 ∂ ξ ∂ η . Then the solutions of (18)–(19) are W = U ( ξ, T, τ ) + V ( η, T, τ ) ,W = U ( ξ, T, τ ) + V ( η, T, τ ) , (23)where U , U or V , V are arbitrary 2 π -periodic, zero-mean functions of ξ or η ,respectively.For j = 0 , , , . . . , we use the notation u j ( ξ, T, τ ) = U jξ ( ξ, T, τ ) , v j ( η, T, τ ) = V jη ( η, T, τ ) , so that(24) U jt = u j , U jx = − u j , V jt = v j , V jx = v j . The Lagrangian velocity ν = w t is given to leading order by(25) ν = (cid:15) / ( u + v ) + O ( (cid:15) ) . The W equation. Since η − ξ = 2 x , equation (20) becomes4 W ξη = S ( η − ξ )[ u ξ + v η ] − u T + v T ] . (26)In order for (26) to have a solution for W that is 2 π -periodic in ξ , η , the right-handside must have zero mean with respect to ξ , η . We use (cid:104) f (cid:105) ξ to denote the averageof a 2 π -periodic function f ( ξ, η ) with respect to η and (cid:104) f (cid:105) η to denote the averagewith respect to ξ , (cid:104) f (cid:105) ξ = 12 π (cid:90) π − π f ( ξ, η ) dη, (cid:104) f (cid:105) η = 12 π (cid:90) π − π f ( ξ, η ) dξ. Using the fact that S (cid:48) ( x ) = 2 − πδ ( x ), together with the zero-mean periodicityof S , u , v , we compute that (cid:104) S ( η − ξ ) u ξ ( ξ ) (cid:105) η = − u ( η ) , (cid:104) S ( η − ξ ) v η ( η ) (cid:105) η = 0 , (cid:104) S ( η − ξ ) u ξ ( ξ ) (cid:105) ξ = 0 , (cid:104) S ( η − ξ ) v η ( η ) (cid:105) ξ = 2 v ( ξ ) , (27)where we suppress the dependence on T and τ .Taking the averages of (26) with respect to ξ and η , we therefore obtain that u T = v , v T = − u . (28)The solution of (28) is u ( ξ, T, τ ) = a ( ξ, τ ) e − iT + c.c. ,v ( η, T, τ ) = − ia ( η, τ ) e − iT + c.c. , (29)where a ( z, τ ) is an arbitrary complex-valued function that is 2 π -periodic in z withzero mean. The use of (29) in (25) gives (3).Using (27)–(28), we can rewrite equation (26) for W as(30) 4 W ξη = S ( η − ξ ) u ξ − (cid:104) S ( η − ξ ) u ξ (cid:105) η + S ( η − ξ ) v η − (cid:104) S ( η − ξ ) v η (cid:105) ξ . JOHN K. HUNTER AND EVAN B. SMOTHERS
We Fourier expand the functions in this equation as u ξ ( ξ ) = (cid:88) k ∈ Z (cid:98) u k e ikξ , v η ( η ) = (cid:88) l ∈ Z (cid:98) v l e ilη ,W ( ξ, η ) = (cid:88) m,n ∈ Z (cid:99) W mn e i ( mξ + nη ) , S ( η − ξ ) = (cid:88) p ∈ Z (cid:98) S p e ip ( η − ξ ) . (31)Since (cid:98) S = 0, it follows that S ( η − ξ ) u ξ − (cid:104) S ( η − ξ ) u ξ (cid:105) η = (cid:88) m,n ∈ Z ∗ (cid:98) S n (cid:98) u m + n e i ( mξ + nη ) ,S ( η − ξ ) v η − (cid:104) S ( η − ξ ) v η (cid:105) ξ = (cid:88) m,n ∈ Z ∗ (cid:98) S − m (cid:98) v m + n e i ( mξ + nη ) , where Z ∗ = Z \ { } . Hence, the Fourier coefficients of a particular solution of thenonhomogeneous equation (30) are given by (cid:99) W mn = − (cid:98) S n (cid:98) u m + n + (cid:98) S − m (cid:98) v m + n mn for m, n (cid:54) = 0, and the general solution of (30) is W ( ξ, η ) = U ( ξ ) + V ( η ) − (cid:88) m,n ∈ Z ∗ (cid:98) S n (cid:98) u m + n + (cid:98) S − m (cid:98) v m + n mn e i ( mξ + nη ) , (32)where U , V are arbitrary functions of integration.4.3. The W equation. Next, we examine equation (22) for W , which can bewritten as 4 W ξη = − ( γ + 1)[ u ξ + v η ][ v − u ]+ S ( η − ξ )[ u ξ + v η ] − u T + v T ] . (33)As with W , we impose the solvability conditions that follow by averaging theequation with respect to ξ and η . Using (27) and the fact that (cid:104) F G η (cid:105) ξ = 0, (cid:104) F ξ G (cid:105) η = 0 for any functions F ( ξ ), G ( η ), we find that u and v satisfy u T − v = γ + 12 u ξ u , v T + u = − γ + 12 v η v . Using (29) in these equations and computing the solution, we get that u = γ + 16 ( − i ) aa x e − iT − γ + 14 | a | z + c.c. v = γ + 16 (1 + 2 i ) aa x e − iT − γ + 14 | a | z + c.c. , (34)where we omit a solution of the homogeneous equation, which does not enter intothe final result. We can then solve (33) for W , but we do not need an explicitexpression since W does not appear in the equation for W . EFLECTION OF SOUND WAVES OFF AN ENTROPY WAVE 9
The W equation. We derive an equation for a ( z, τ ) by imposing the previoussolvability conditions on equation (22) for W , which has the form4 W ξη = ff = − ( γ + 1)( W x W x ) x + S (2 x ) W xx − W tT − W T T − W tτ . The first solvability condition is(35) (cid:104) f (cid:105) ξ = 0 , (cid:104) f (cid:105) η = 0 , which ensures that there is a solution for W that is 2 π -periodic in ( ξ, η ). Wecompute the averages of each of the terms in f separately.Using (23)–(24), we have (cid:104) W T T (cid:105) ξ = U T T , (cid:104) V T T (cid:105) η = V T T , (cid:104) W tτ (cid:105) ξ = u τ , (cid:104) W tτ (cid:105) η = v τ . In addition, ( W x W x ) x = ( ∂ η − ∂ ξ )[( v − u )( v − u )] , so that (cid:104) ( W x W x ) x (cid:105) ξ = − ( u u ) ξ , (cid:104) ( W x W x ) x (cid:105) η = ( v v ) η . From (32) and (24) we also obtain that (cid:104) W tT (cid:105) ξ = u T , (cid:104) W tT (cid:105) η = v T . To compute the averages of S (2 x ) W xx , we use the previous Fourier expansions,which gives S (2 x ) W xx = S ( η − ξ )[ u ξ + v η ]+ (cid:88) p ∈ Z (cid:98) S p e ip ( η − ξ ) (cid:88) m,n ∈ Z ∗ ( n − m ) mn [ (cid:98) S n (cid:98) u m + n + (cid:98) S − m (cid:98) v m + n ] e i ( mξ + nη ) (36)In the η average, we retain only the η = 0 Fourier coefficients, which corresponds totaking p + n = 0, and in the ξ average, we retain only the ξ = 0 Fourier coefficients,which corresponds to taking m − p = 0. Applying this observation to (36), rewritingthe result as sum over k = m + n , and using (27) for the first term, we get that (cid:104) S ( η − ξ )[ ∂ η − ∂ ξ ] W (cid:105) ξ = 2 v ( ξ )+ (cid:88) k ∈ Z (cid:88) n ∈ Z \{ ,k } (2 n − k ) k − n ) n (cid:98) S − n [ (cid:98) S n (cid:98) u k + (cid:98) S n − k (cid:98) v k ] e ikξ (37) (cid:104) S ( η − ξ )[ ∂ η − ∂ ξ ] W (cid:105) η = − u ( η )+ (cid:88) k ∈ Z (cid:88) n ∈ Z \{ ,k } (2 n − k ) k − n ) n (cid:98) S k − n [ (cid:98) S n (cid:98) u k + (cid:98) S n − k (cid:98) v k ] e ikη . (38)For the sawtooth wave (14), we have (cid:98) S k = 12 π (cid:90) π − π S ( x ) e − ikx dx = 2 ik . Using this value in (37), computing the resulting sums, and using (31), we find that (cid:104) S ( η − ξ )[ ∂ η − ∂ ξ ] W (cid:105) ξ = 2 v ( ξ ) − (cid:88) k ∈ Z ∗ (cid:26) π (cid:98) u k + (cid:18) π k (cid:19) (cid:98) v k (cid:27) e ikξ = 2 v ( ξ ) − π u ξ ( ξ ) − π v ξ ( ξ ) − ∂ − ξ v ( ξ ) , where ∂ − ξ is the Fourier multiplier operator on periodic, zero-mean functions withsymbol 1 /ik . A similar computation in (38) gives (cid:104) S ( η − ξ )[ ∂ η − ∂ ξ ] W (cid:105) η = − u ( η ) − π v η ( η ) − π u η ( η ) − ∂ − η u ( η ) . Combining these results, we find that the solvability condition (35) is u T − v = γ + 12 ( u u ) ξ − π u ξ − π v ξ − ∂ − ξ v − ∂ − ξ u T T − u τ ,v T + u = − γ + 12 ( v v ) η − π v η − π u η − ∂ − η u − ∂ − η v T T − v τ . (39)The first equation is an equation in ξ and the second is an equation in η . Wereplace ξ by z in the first equation and η by z in the second equation to get asystem for functions of z .We now impose the second solvability condition, which comes from the require-ment that (39) has solutions for ( u , v ) that are 2 π -periodic in T , to avoid secularterms in T . If u T − v = F e − iT + c.c. + n.r.t. , v T + u = Ge − iT + c.c. + n.r.t. , where n.r.t. stands for non-resonant terms, then this solvability condition is F + iG = 0. We use (29) and (34) in the right-hand side of (39), with ξ or η replacedby z , compute the coefficients of e − iT , and impose the solvability condition. Aftersome algebra, we find that a ( z, τ ) satisfies the degenerate Schr¨odinger equation (4).4.5. Comparison with the MRS and linearized equations.
Equation (4) con-sists of the asymptotic equation (6) for the MRS equation (1) with a lower-orderdispersive term.When the MRS equation is derived from gas dynamics, with u and v the Eulerianvelocity perturbations in the right and left moving sound waves, there are additionalfactors of ( γ + 1) / | a | a x ) x in (6) of − (cid:18) γ + 12 (cid:19) = − ( γ + 1) , which agrees with the one in (4) found by expanding the gas dynamics equationsdirectly.The linearization of the Lagrangian wave equation (16) with sound speed (13) is(40) w tt − [1 + (cid:15)S (2 x )] w xx = 0 . We suppose that S ( x ) is a general real-valued, 2 π -periodic function with zero meanand Fourier expansion S ( x ) = (cid:88) k ∈ Z ∗ S k e ikx , EFLECTION OF SOUND WAVES OFF AN ENTROPY WAVE 11 and look for spatially periodic, resonantly reflected solutions of (40) of the form w = (cid:2) Ae ikx + Be − ikx (cid:3) e − iω ( k ; (cid:15) ) t + O ( (cid:15) ) , with wavenumber k ∈ Z ∗ .A perturbation expansion similar to the one above for the nonlinear equation,whose details we omit, shows that (up to an overall sign) ω ( k ; (cid:15) ) = k + (cid:15)ω ( k ) + (cid:15) ω ( k ) + O ( (cid:15) ) ,ω = ± k | S k | ,ω = − ω k − k (cid:88) n ∈ Z \{ ,k } (2 n − k ) n ( n − k ) (cid:12)(cid:12) S n ± e iσ S n − k (cid:12)(cid:12) , where S k = | S k | e iσ and A = ± e iσ B .For a sawtooth wave with S k = 2 i/k and e iσ = i sgn k , we find that ω = ± , ω = − k − π k. The first order perturbation ω is independent of the wavenumber k , correspondingto the nondispersive, constant frequency oscillations in the resonant reflection ofsound waves off a sawtooth entropy wave. (In contrast, other entropy profiles leadto dispersive oscillations.) The second order correction ω is the dispersion relationof a τ + 12 ∂ − z a − π a z = 0 , which corresponds to the linear terms in (4).We remark that the inclusion of higher order corrections in the sound speedperturbation S , say S ( x, (cid:15) ) = S ( x ) + (cid:15)R ( x ) + O ( (cid:15) ) , gives ω = ± k (cid:12)(cid:12) S k + (cid:15)R k + O ( (cid:15) ) (cid:12)(cid:12) = ± k | S k | (1 + (cid:15)q k ) + O ( (cid:15) ) ,q k = 12 (cid:18) R k S k + R ∗ k S ∗ k (cid:19) , which would lead to additional dispersive terms in (4) if q k (cid:54) = 0. Since the sawtoothwave S ( x ) is odd and S k is imaginary, no such terms appear when R ( x ) is evenand R k is real. In particular, this is the case for the sound speed perturbationscorresponding to a pure sawtooth entropy wave of the form s ( x ) = s + (cid:15)S (2 x ) , instead of (15), so we would get an asymptotic equation of the same form as (4) inthat case also. 5. Numerical solutions
In this section, we show two sets of numerical solutions, one for front propagationin the MRS and DQS equations, the other for periodic solutions of the asymptoticgas dynamics equation.
Figure 1.
Left: Contour plot of ( u + v ) / for the solution ofthe MRS equation (2) with initial data (41), where (cid:15) = 0 . ≤ t ≤ · π . Right: Contour plot of the amplitude | a | , rescaledto the MRS solution, for the solution of the DQS equation (6) onthe corresponding slow time interval 0 ≤ τ ≤ . Front propagation for the MRS equation.
We consider the MRS equation(2) with the compactly supported initial data u ( x,
0) = (cid:40) (cid:15) (cid:2) π / − ( x − π ) (cid:3) if | x − π | < π/ , π/ < | x − π | < π,v ( x,
0) = 0 , (41)where (cid:15) · · π = 0 .
15, or (cid:15) ≈ . u + v ) / for the solution of the MRS equation The right-handside of Figure 1 shows a contour plot of the corresponding solution of the DQSequation (6) rescaled to the MRS variables.The numerical solutions were computed by the method of lines, using an explicitforth order Runge-Kutta method in time and a forth order WENO flux in space forthe MRS equation [19] with 2 spatial grid points, and an implicit second ordermethod in time and a pseudo-spectral method in space for the DQS equation with2 Fourier modes. The pseudo-spectral method includes second-order spectralviscosity, which is required in order to get a convergent numerical solution for theDQS equation with this compactly supported initial data.The numerical solution of the MRS equation shows that the fronts at x = π/ , π/ t ≈ t ≈ t ≈ x − π (cid:55)→ − ( x − π ) , u (cid:55)→ − v, v (cid:55)→ − u, so complete reflectional symmetry is not to be expected. The mechanism of frontexpansion for the MRS equation is the propagation of shocks in u or v into the zerostate.The right-hand side of Figure 1 shows the corresponding numerical solution ofthe DQS equation. The solution agrees well with the MRS solution, although, EFLECTION OF SOUND WAVES OFF AN ENTROPY WAVE 13
Figure 2.
Left: Solution on 0 < x < π . Right: Detail nearthe front. The red line is the solution of the MRS equation (2)for u ( x, t ) with initial data (41) at t = 800 · π . The black lineis the corresponding asymptotic solution for u ( x, t ) obtained fromthe DQS equation (6) and rescaled to the MRS variables.as shown in Figure 2, the DQS solution under-estimates the speed of the fronts.This discrepancy may be the result of greater numerical dissipation in the spectralviscosity scheme than in the WENO scheme. The DQS solution is symmetric in x , since averaging over the ( u, v )-oscillations in the MRS solutions eliminates theirreflectional asymmetry.The structure of the DQS solution and the mechanism by which a DQS frontpropagates into the zero solution is less clear than for the MRS equation. In par-ticular, compactly supported traveling waves or self-similar solutions of the DQSequation (6) that are dispersive analogs of the self-similar Barenblatt solutions forthe porous medium equation are not weak solutions of (6) at the front [10, 20]and do not agree with the numerical simulations. These questions will be analyzedfurther in [10].5.2. Two-harmonic initial data for the asymptotic gas dynamics equation.
First, we normalize the asymptotic gas dynamics equation (4). The dimensionlessparameter (cid:15) in (13) measures the strength of the perturbation in the sound speedrelative to the mean sound speed ¯ c . A dimensionless parameter that measuresthe strength of the sound waves is M = U/ ¯ c where U is a typical size of velocityperturbations in the sound waves. In the asymptotic expansion, we assume that M = O ( (cid:15) / ).We make the change of variables a = √ M ( γ + 1) (cid:15) / ˜ a, ˜ z = z + 12 π τ, ˜ τ = M (cid:15) τ, in (4) and, after dropping the tildes, get(42) i (cid:0) a τ + µ∂ − z a (cid:1) = (cid:0) | a | a z (cid:1) z , where µ = (cid:15) / M measures the relative strength of the dispersion and nonlinearity.Up to an unimportant coefficient, the dispersionless limit of this equation with µ = 0is (6). Figure 3.
A contour plot of the solution of (42) with µ = 1 andinitial data (43). The solution is computed by a pseudo-spectralmethod using 2 Fourier modes. Left: | a | . Right: Re a . Figure 4.
Left: Graph of the solution of (42) with µ = 1 andinitial data (43) at τ = 0 .
5. Right: Detail of the cusp in | a | at τ = 0 . | a | in the numerical solution is | a | ≈ . | a | ; Red: Re a ; Green: Im a .)The solution of (42) with initial data a ( z,
0) = a e inz is a ( z, τ ) = a e inz − i Ω τ , Ω = − µn − n | a | , where the only effect of nonlinearity is a frequency shift. Thus, we need at leasttwo initial harmonics for the cubic nonlinearity to generate a resonant cascade tohigher harmonics and nontrivial dynamics.In Figure 3, we show contour plots of | a | and Re a for the solution of (42) with µ = 1 and initial data(43) a ( z,
0) = − e iz + 12 e i ( z +2 π ) . In Figure 4, we show a graph of the solution at time τ = 0 .
5. The solution iscomputed by the method of lines, using an implicit second order method in timeand a pseudo-spectral method in space. No spectral viscosity is required for theinitial data (43).
EFLECTION OF SOUND WAVES OFF AN ENTROPY WAVE 15
The effects of dispersion on this solution are small. Numerical solutions fordifferent values of µ , including µ = 0, are qualitatively similar to the one for µ = 1,and we do not show them here.The numerical solution for a becomes close to 0 at τ ≈ . z ≈ .
7, when | a | develops a cusp that propagates to the left; a graph of the solution near the cuspat τ = 0 . | a | is boundedwell away from zero.We remark that, although one cannot tell from the numerical solutions whether | a | beomes exactly zero, the dispersionless version of (42), with µ = 0, has travelingwaves in which a passes through zero [10]. After normalization, this traveling waveis given implicitly by a ( z, τ ) = 1 − e − ic Φ( z − cτ ) , Φ( ξ ) − c sin [ c Φ( ξ )] = 12 ξ. This solution has a cube-root dependence on ( z − cτ ) / near z = cτ , which isconsistent with the qualitative behavior of the numerical solution, and suggeststhat the solution for a may become H¨older continuous with exponent 1 / a vanishes at a point. References [1]
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Department of Mathematics, University of California at Davis
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