Chapter 5 Asymptotic Waves for Quasilinear Systems In this chapter, we shall assume the existence of appropriate asymptotic ex pansions, and derive asymptotic equations for hyperbolic PDEs These equa[.]
Chapter Asymptotic Waves for Quasilinear Systems In this chapter, we shall assume the existence of appropriate asymptotic expansions, and derive asymptotic equations for hyperbolic PDEs These equations display the qualitative effects of dissipation or dispersion balancing nonlinearity, and are easier to study analytically In this way, the study of complicated models reduces to that of models of asymptotic approximations expressed by a hierarchy of equations, which facilitate numerical calculations; this is often the only way that progress can be made to analyze complicated systems Essential ideas underlying these methods may be found in earlier publications; see for example Boillat [19], Taniuti, Asano and their coworkers ([7], [194]), Seymour and Varley [166], Germain [61], Roseau [155], Jeffrey and Kawahara [82], Fusco, Engelbrecht and their coworkers ([58], [59]), Cramer and Sen [43], Kluwick and Cox [94], and Cox and Kluwick [42] An account of some of the rigorous results which deal with convergence of such expansions may be found in [47] 5.1 Weakly Nonlinear Geometrical Optics Here we shall discuss the behavior of certain oscillatory solutions of quasilinear hyperbolic systems based on the theory of weakly nonlinear geometrical optics (WNGO), which is an asymptotic method and whose objective is to understand the laws governing the propagation and interaction of small amplitude high frequency waves in hyperbolic PDEs; this method describes asymptotic expansions for solutions satisfying initial data oscillating with high frequency and small amplitude The propagation of the small amplitude waves, considered over long time-intervals, is referred to as weakly nonlinear Most waves are of rather small amplitude; indeed, small amplitude high frequency short waves are frequently encountered Linearized theory satisfactorily describes such waves only for a finite time; after a sufficiently long time-interval, the cumulative nonlinear effects lead to a significant change in the wave field the basic principle involved in the methodology used for treating weakly nonlinear waves lies in a systematic use of the method of multiple scales Each 133 134 Asymptotic Waves for Quasilinear Systems dissipative mechanism present in the flow, such as rate dependence of the medium, or geometrical dissipation due to nonplanar wave fronts, or inhomogeneity of the medium, defines a local characteristic length (or time) scale, that arises in a natural way Indeed, WNGO is based on the assumption that the wave length of the wave is much smaller than any other characteristic length scale in the problem When this assumption is satisfied, i.e., the time (or length) scale defined by the dissipative mechanism is large compared with the time (or length) scale associated with the boundary data, the wave is referred to as a short wave, or a high frequency wave For instance, consider the motion excited by the boundary data u(0, t) = af (t/τ ) on x = 0, where a represents the size of the boundary data, while τ is the applied period or pulse length When we normalize the boundary data, by a suitable nondimensional quantity and t by the time scale τr , which is defined by the dissipative mechanism present in the flow, the geometrical acoustics limit then corresponds to the high frequency condition = τ /τr 1; in fact, there is a region in the neighborhood of the front where the nonlinear convection associated with the high frequency characteristics is important For both significant nonlinear distortion and dissipation, in this high frequency limit, we must have the nondimensional amplitude |a| = O() In terms of these normalized variables, the boundary condition becomes u(0, t) = g(t/), which describes oscillations of small amplitude and of frequency −1 For data of size , the method of WNGO yields approximations, which are valid on time intervals typically of size −1 Following the pioneering work of Landau [99], Lighthill [108], Keller [91] and Whitham [210], a vast amount of literature has emerged on the development, both formal and rigorous, in the theory of WNGO However, the development of these methods through systematic self-consistent perturbation schemes in one and several space dimensions is due to Chouqet-Bruhat [34], Varley and Cumberbatch ([204], [206]), Parker [140], Mortell and Varley [128], Seymour and Mortell [165], Hunter and Keller [75], Fusco [60], Mazda and Rosales [115], and Joly, Metivier and Rauch [88] For results based on numerical computations of model equations of weakly nonlinear ray theory, the reader is referred to Prasad [145] Using the systematic procedure, alluded to and applied by the above mentioned authors, we study here certain aspects of WNGO, which are largely based on our papers ([146], [171], [173], [190], and [192]) 5.1.1 High frequency processes Consider a quasilinear system of hyperbolic PDEs in a single space variable u,t + A(u)u,x + b(u) = 0, −∞ < x < ∞, t > (5.1.1) where u and b are n-component vectors and A is a n × n matrix Let uo be a known constant solution of (5.1.1), such that b(uo ) = We consider small amplitude variations in u from the equilibrium state u = uo , which are of the size , described earlier; and look for a small amplitude high frequency wave 5.1 Weakly Nonlinear Geometrical Optics 135 solution of (5.1.1), representing a single wave front, and valid for times of the order of O(1/) Then the WNGO ansatz for the situation described above is the formal expansion for the solution of (5.1.1) u = uo + u1 (x, t, θ) + 2 u2 (x, t, θ) + (5.1.2) where θ is a fast variable defined as θ = φ(x, t)/ with φ as the phase function to be determined The wave number k and wave frequency ω are defined by k = φ,x and ω = −φ,t By considering the Taylor expansion of A and b in the neighborhood of uo , and taking into account (5.1.2), equation (5.1.1) implies that O(o ) : (Ao − λI)u1,θ = O(1 ) : (Ao − λI)u2,θ = −(∇A)o u1 u1,θ − {u1,t + Ao u1,x + (∇b)o u1 }φ−1 ,x , (5.1.3) where the subscript o refers to the evaluation at u = uo , I is the n × n unit matrix, λ = −φ,t /φ,x , and ∇ is the gradient operator with respect to the components of u The equation (5.1.3)1 implies that for a particular choice of the eigenvalue λo (assuming that it is simple), u1 = π(x, t, θ)Ro , (5.1.4) where π is a scalar oscillatory function to be determined, and Ro is the right eigenvector of Ao corresponding to the eigenvalue λo The phase φ(x, t) is determined by φ,t + λo φ,x = (5.1.5) It may be noticed that if φ(x, 0) = x, then (5.1.5) implies that φ(x, t) = x−λo t Let Lo be the left eigenvector of Ao corresponding to its eigenvalue λo , satisfying the normalization condition Lo ·Ro = Then if (5.1.3)2 is multiplied by Lo on the left, we obtain along the characteristic curves associated with (5.1.5) the following transport equation for π ; π,τ + ν(π /2),θ = −µπ, (5.1.6) where ∂/∂τ = ∂/∂t + λo ∂/∂x, and µ = Lo · ∇b|o · Ro , ν = Lo · ∇A|o · Ro Ro (5.1.7) If the initial condition for π is specified, i.e., π|τ =0 = πo (xo , θo ) with xo = x|τ =o and θo = −1 φ|τ =0 , then the minimum time taken for the smooth solution to breakdown can be computed explicitly In fact, the initial conditions lead to a shock only when ν∂πo /∂θo < and −ν∂πo /∂θo > µ; in passing, we remark that the presence of source term b in (5.1.1) makes the solution exist for a longer time relative to what it would have been in the absence of a source term Further, if (5.1.1) has an associated conservative form, then (5.1.6), which is the proper conservation form, can be used to study the propagation of weak shocks (see [75]) 136 5.1.2 Asymptotic Waves for Quasilinear Systems Nonlinear geometrical acoustics solution in a relaxing gas The foregoing asymptotic analysis can be used to study the small amplitude high frequency wave solution to the one dimensional unsteady flow of a relaxing gas described by the system (4.6.1), i.e., ui,t + Aij uj,x + Bi = 0, i, j = 1, 2, 3, 4, where the symbols have the same meaning, defined earlier Here we are concerned with the motion consisting of only one component wave associated with the eigenvalue λ = u + a The medium ahead of the wave is taken to be uniform and at rest The left and right eigenvectors of A corresponding to this eigenvalue are L = (0, 1/(2a), 0, 1/(2ρa2)), R = (ρ, a, 0, ρa2 ), (5.1.8) and the phase function φ(x, t) is given by φ(x, t) = x−ao t, where the subscript o refers to the uniform state uo = (ρo , 0, σo , po ) ahead of the wave The transport equation for the wave amplitude π is given by (5.1.6) with coefficients µ and ν determined as µ = (α + (ao Ω)/2) and ν = (γ + 1)ao /2, (5.1.9) where α is the same as in (4.6.14)1 The characteristic field associated with π in (5.1.6) is defined by the equations dx/dt = ao , dθ/dt = ((γ + 1)ao π)/2 (5.1.10) In view of (5.1.10), equation (5.1.6) can be written as dπ/dt = −ao (α+(Ω/2))π along any characteristic curve belonging to this field, and yields on integration π = π o (xo , θo )(A/Ao )−1/2 exp{−αao t}, (5.1.11) along the rays x − ao t = xo (constant) We look for an asymptotic solution of the hyperbolic system (4.6.1) u = uo + u1 (x, t, θ) + O(2 ), (5.1.12) satisfying the small amplitude oscillatory initial condition u(x, 0) = g(x, x/) + O(2 ), (5.1.13) where g is smooth with a compact support; indeed, expansion (5.1.12) with u1 given by (5.1.4), where the wave amplitude π is given by (5.1.11), is uniformly valid to the leading order until shock waves have formed in the solution (see Majda [116]) Using (5.1.11) in (5.1.10)2 we obtain the family of characteristic curves parametrized by the fast variable θo as (γ + 1) θ− ao π o (xo , θo ) Z t A(xo + ao tˆ) A(xo ) −1/2 exp (−αao tˆ)dtˆ = θo (5.1.14) 5.2 Far Field Behavior 137 It may be noticed that equations (5.1.11) and (5.1.14) are similar to the equations (4.6.16) and (4.6.17), and therefore the discussion with regard to the shock formation and its subsequent propagation follows on parallel lines It may be recalled that for plane (m = 0) and radially symmetric (m = 1, 2) flow configurations, (A/Ao ) = (x/xo )m , and therefore the integral in (5.1.14) converges to a finite limit as t → ∞ Thus, an approximate solution (5.1.12), satisfying (5.1.13), is given by u = u0 + Ro π o (xo , θo )(A(xo + ao t)/A(xo ))−1/2 exp (−ao αt), where R is given by (5.1.8)2 ; the fast variable θo , given by (5.1.14), is chosen such that θo = x/ at t = 0, and the initial value of π is determined from (5.1.13) as π o (x, ξ) = g(x, x/)/ao This completes the solution of (4.6.1) and (5.1.13); any multivalued overlap in this solution has to be resolved by introducing shocks into the solution 5.2 Far Field Behavior When the characteristic time τ associated with the boundary data is large compared with the time scale τr defined by the dissipative mechanism present in the medium (i.e., δ = τr /τ 0} = Ω+ , implying thereby (∂j Φ)dS ∂ j χΩ + = (5.3.10) |∇x Φ| Similarly, we define Ω− {x : Φ(x) < 0} Let us now consider a weak solution u of a scalar conservation law (5.3.1) in n independent variables in a region Ω 143 5.3 Energy Dissipated across Shocks enclosing a single discontinuity of the solution u We write u = f χΩ+ + gχΩ− for some smooth maps f, g Then n X n X ∂j Fj (u) = j=0 ∂j Fj (f )χΩ+ + j=0 + n X n X ∂j Fj (g)χΩ− j=0 Fj (f )∂j χΩ+ + n X Fj (g)∂j χΩ− (5.3.11) j=0 j=0 The left-hand side as well as the first two terms on the right-hand side vanish as functions in L1loc , so that on evaluating the distribution equation (5.3.11) against a test function φ gives in view of (5.3.10), the formula * n + Z X (Fj (f ) − Fj (g))nj dS (Fj (f ) − Fj (g))nj dS, φ := φ=0 j=0 = Z φ φ=0 n X [Fj (u)]nj dS = (5.3.12) j=0 Equation (5.3.12) is the classical Rankine-Hugoniot condition that we have demonstrated in the spirit of distribution theory Note that if linear source terms G(x)u are present in the conservation law, they cancel out from either side of equation (5.3.11) since (G(x)f + Σnj=1 ∂j Fj (f ))χΩ+ and (G(x)g + Σnj=1 ∂j Fj (g))χΩ− both vanish, and similarly the formula is unaffected by integro-differential terms [154] However, the analysis shows how nonhomogeneous terms, which are distributions singular along Ψ = 0, are to be handled Theorem 5.3.2 Let β(x) be a smooth function defined on the manifold Φ = and consider a weak solution u for the system n X ∂j Fj (u) = β(x)Φ∗ (δ0 ), (5.3.13) j=0 with a jump discontinuity along Φ = Then the jump in u satisfies the following modified Rankine-Hugoniot condition Z φ φ=0 n X j=0 [Fj (u)]nj dS = hβ(x)Φ∗ (δ0 ), φi, (5.3.14) where φ is an arbitrary test function A detailed proof of theorem (5.3.2) is available in [191] As an application of theorem (5.3.2), recall that in classical potential theory, the electric field vector E and potential V , satisfy the Poisson’s equation divE = ∇2x V = 4πρ (5.3.15) 144 Asymptotic Waves for Quasilinear Systems The density of the electrostatic medium ρ is often singular along manifolds and, in the case of layer potentials is given by ρ = cΦ∗ (δ0 ) Applying formula (5.3.14) to the Poisson’s equation, we recover as a special case of the Rankine-Hugoniot condition, the following classical theorem [85] giving the jump in the normal derivative of the layer potential ∂V = [E].ˆ n = 4πρ ∂ˆ n 5.3.3 (5.3.16) Application to nonlinear geometrical optics We now proceed to apply the formula (5.3.8) to the transport equations of nonlinear geometrical optics which, at leading order, govern the propagation of high frequency monochromatic waves propagating into a given background state Let us consider the small amplitude high frequency wave solutions of a quasilinear hyperbolic system of partial differential equations in N unknowns u and n space variables: n ∂u X ∂ + (Fj (x, t, u)) = 0, ∂t ∂xj j=1 (5.3.17) where the n flux functions Fj are assumed to depend smoothly on their arguments Let u = u0 (x, t) be a smooth solution of (5.3.17) which we call the background state and denote by A0j the derivative Du Fj (x, t, u) evaluated at u0 and λ0q (k, x, t, u0 ) the qth eigenvalue of Σnj=1 kj A0j ; the left and right eigenvectors being respectively L0q (k, x, t, u0 ) and R0q (k, x, t, u0 ) Without any loss of generality (with densities and fluxes redefined, if necessary), we may assume u0 ≡ We assume that only one eigen-mode, the qth mode, in the initial data is excited Motivated by the linear theory we seek an asymptotic series solution in which only the qth eigen mode is present at the leading order namely, u (φ/, x, t) = u0 + π(φ/, x, t)R0q + 2 u2 (φ/, x, t) + · · · (5.3.18) The phase function φ satisfies the qth branch of the eikonal equation ∂φ = −λ0q (∇x φ, x, t), ∂t (5.3.19) with the wave number vector k = ∇x φ The validity of (5.3.18) over time scales of order O(−1 ) requires the use of multiple time scales and we denote by ξ the fast variable ξ = φ/ The equation governing the evolution of the principal amplitude π(ξ, t, x) is given by (see [153]): dπ ∂ + πχ + dt ∂ξ bπ aπ + = 0, (5.3.20) 5.3 Energy Dissipated across Shocks 145 d dt denotes the ray derivative, namely, the directional derivative along Pn ∂(L0q A0j R0q ) the characteristics of (5.3.19), χ is the smooth function 12 j=1 that ∂xj is related to the geometry of the wave front, and a and b are known functions of x and t On multiplying (5.3.20) by π, we get n π X ∂(L0q A0j R0q ) ∂(aπ /2 + bπ /3) d π2 + + + E = (5.3.21) dt 2 j=1 ∂xj ∂ξ where Formula (5.3.8) with F0 = π, F1 = aπ + 21 bπ , F˜0 = 12 π and F˜1 = 12 aπ + 31 bπ gives the following distribution density ∗ b Φ (δ0 ) b 22 b [π ] − [π ][π] = [π]3 Φ∗ (δ0 ) (5.3.22) E=− [π] 12 Remarks 5.3.2: (i) The motivation for multiplying (5.3.20) by π to get (5.3.21) lies in the observation that, away from the loci of discontinuities, the expression n π2 X ∂ d π2 ∂ aπ bπ + , (L0q A0j R0q ) + + dt 2 j=1 ∂xj ∂ξ is an exact divergence, namely, X n d π2 ∂ π 0 ∂ aπ bπ + , L A R + + dt ∂xj q j q ∂ξ j=1 implying the blow-up of intensity π as the ray tube collapses (ii) In [153], the distribution E given by equation (5.3.22) is interpreted as the energy dissipated across the shock We note here that (5.3.22) may be recast in zero divergence form and implies the blow-up of the intensity as the ray tube collapses (iii) The distributional term E in (5.3.21) may be regarded as a singular forcing term which is yet another motivation for the study carried out in [191] In certain systems exhibiting anomalous thermodynamic behavior, namely, when the so called fundamental derivative is small [94], the nonlinear distortions in the solution of the Cauchy problem for (5.3.17) are noticeable over much longer time scale of O(−2 ) This calls for a different scaling in the fast variable, namely ξ = φ/2 In place of (5.3.20) we get the following equation governing the evolution of π (see [94]): ∂π dπ γ + π + Σ1 π + χπ = 0, (5.3.23) dt ∂ξ 146 Asymptotic Waves for Quasilinear Systems where, γ and Σ1 are known constants Equation (5.3.23) can be written as d dt π2 n ∂ γπ Σ1 π π X ∂(L0q A0j R0q ) + + + + E = j=1 ∂xj ∂ξ Applying Formula (5.3.8), we get expression for E: γ[π ][π]2 Σ1 [π]3 E= Φ∗ (δ0 ) + 24 12 (5.3.24) If there are several shocks located at discrete locations {Φi = 0}i , the formula (5.3.8) must be modified to X [F1 ]i ˜ ˜ (5.3.25) [F ]i − [F1 ]i Φ∗i (δ0 ) − [F0 ]i i Equations (5.3.22) and (5.3.24) get modified to E= and E= X b [π]3i Φ∗i (δ0 ), 12 i X γ[π ]i [π]2 i i 24 Σ1 [π]3i + 12 (5.3.26) Φ∗i (δ0 ), (5.3.27) respectively Note that equation (5.3.27) differs from (5.3.26) by a correction term with coefficient γ, thereby generalizing the result of [153] Here the notation [π]i denotes the jump across the ith shock with locus Φi (x, t) = 5.4 Evolution Equation Describing Mixed Nonlinearity It has been found that in certain systems with singular thermodynamic behavior, where the so-called fundamental derivative is small, the effect of nonlinearity is perceptible over time scales longer by an order of magnitude, necessitating the use of fast variables of a higher order of magnitude, namely of order O(−2 ) This has been studied in an abstract setting by Kluwick and Cox [94], and on applying it to the usual equations of gas-dynamics, the authors have found that the transport equations governing the asymptotic behavior contain, in addition to the usual quadratic nonlinearity, cubic correction terms Srinivasan and Sharma [192] have generalized the work of [94] to the case of multiphase expansions The analysis in [192], which parallels the one in [77], is complicated due to the fact that the amplitudes at the leading order and the next order appear together in the transport equations, and to separate 5.4 Evolution Equation Describing Mixed Nonlinearity 147 them a two stage averaging process has to be employed rather than the single stage process detailed in [77] Although this complexity is absent in [77], the requisite analytic apparatus is developed in [77] In as much as the results obtained in [94] are important and interesting, we have rederived the transport equation (equation (5.4.23) below) for the leading amplitude in a spirit closer to ([115], [77]), expressing the result in terms of the Glimm interaction coefficients Γkij Besides, we have employed a method similar to the one used by Cramer and Sen [43] in contrast to the approach in [94], where the perturbative parameter is introduced in the differential equations In particular, we rederive the coefficients of the nonlinear terms of the (cubic) Burgers equation (5.4.23) in terms of the coefficients Γkij In the case of a symmetric and isotropic systems, the mean curvature of the wave front, which appears as the coefficient of the linear term in the transport equation, is related to the areal derivative along the bicharacteristics It is known from physical considerations and proved rigorously in the context of linear geometrical optics that the amplitude near a caustic becomes unbounded inversely as the area of cross section of the ray tube; we provide a short proof in Section 5.4.1 imitating the classical proof of Liouville type theorems on integral invariants It turns out that the square of the amplitude is a multiplier in the sense of Jacobi−Poincar´e [144], implying thereby the blow-up of intensities in a neighborhood of caustics [153] However, the analysis involves the manipulation of conservation laws through multiplication by polynomials in the unknown which is known to change the Rankine-Hugoniot conditions across discontinuities It is well known that these manipulations are invalid across discontinuities; in order to restore the validity, one has to modify the equations so obtained by adding singular terms which are distributions supported along discontinuity loci These singular terms have been interpreted as the energy dissipated across shocks in [153] To simplify the exposition somewhat, we have assumed throughout that the hyperbolic system of equations are in conservation form, which has the effect of making the coefficients Γkij symmetric in i and j In the final section we present an application to the system governing the propagation of acoustic waves through a nonuniform medium stratified by gravitational source terms 5.4.1 Derivation of the transport equations We derive the transport equation (5.4.23) governing the propagation of the high frequency monochromatic waves, that includes both quadratic and cubic nonlinearities inherent in the system of conservation laws In the first stage we introduce the fast variable corresponding to the frequency of the wave and set up the stage for the perturbative analysis In the second stage we obtain an -approximate equation (equation (5.4.16) below), at the second level of perturbation, balancing the error at the third level resulting in the transport equation (equation (5.4.23) below), satisfied by the primary amplitude π of the wave solution.We have in the last paragraph of this section, briefly compared our derivation of (5.4.23) with that in [94] 148 Asymptotic Waves for Quasilinear Systems We consider the hyperbolic system of conservation laws with a source term, n ∂u X ∂Fi (x, t, u) + + G (x, t, u) = ∂t ∂xi i=1 (5.4.1) The small amplitude high frequency solution u propagates in a background state u0 , a spatially dependent known solution of (5.4.1), i.e., n X ∂Fi (x, t, u0 ) j=1 ∂xj + G (x, t, u0 ) = The independent variable x varies over an open set in IR n , and densities Fi as well as the unknown u are IRm valued The hyperbolicity of system Pn (5.4.1) means that for each nonzero n-vector (ϑ1 , ϑ2 , , ϑn ), the matrix j=1 ϑj A0j , where A0j = Du Fj (u0 ), is diagonalizable with real eigenvalues We assume a highly oscillatory initial Cauchy data with one excited eigen-mode, say, the q th , φ (5.4.2) u|t=0 = u0 + π x, R0q , where the initial amplitude π o is a smooth function which is bounded with bounded first derivative; R0j and L0j are, respectively, the right and left eigenvectors corresponding to the eigenvalue λ0j (ϑ1 , , ϑn ), ≤ j ≤ n, and the zero superscript denotes their values at u = u0 Denoting by Bk [v1 , v2 ] and Ck [v1 , v2 , v3 ], respectively, the second and third derivatives D Fk [v1 , v2 ] and D3 Fk [v1 , v2 , v3 ], with a zero superscript indicating evaluation at u = u0 , we expand the fluxes in a Taylor series Fj = F0j + A0j (x, t)(u − u0 ) + B0j (x, t)[u − u0 , u − u0 ] + Cj (x, t)[u − u0 , u − u0 , u − u0 ] + O(|u − u0 | ) Likewise the source term G (x, t, u) may be developed as (5.4.3) G (x, t, u) = G (x, t, u0 ) + Du G (x, t, u0 ) (u − u0 ) + O(|u − u0 | ) (5.4.4) Introducing the fast variable of order O(−2 ), ξ = θ(x, t)/2 , where the phase function θ(x, t) is the solution, corresponding to λ0q , of the eikonal equation n X ∂θ ∂θ (5.4.5) A0j + I = 0, Det ∂x ∂t j j=1 the PDE (5.4.1) may be recast in the form " # n X ∂A0j 1 ∂θ ∂ 0 ∂v Aj Aj v+ Bj [v, v]+ Cj [v, v, v] +E0 v = + v+ ∂xj ∂xj ∂xj ∂ξ j=0 (5.4.6) 5.4 Evolution Equation Describing Mixed Nonlinearity 149 In Eqs (5.4.1), (5.4.2), and (5.4.6), the expressions such as ∂Fj /∂xj and ∂A0j /∂xj refer to the xj partial of the composite function Fj (x, t, u(x, t)) (respectively, A0j (x, t, u0 (x))) In (5.4.6) we have retained the terms of order at most O(3 ), using the notations v = u − u0 , F0 (v) ≡ v, x0 = t, and E0 = Du G (x, t, u0 ) We assume that the multiplicities of the eigenvalues remain constant, which implies that the eigenvalues depend smoothly on (ϑ1 , , ϑn ), thereby ensuring that the various branches are algebraic functions of ϑ1 , , ϑn Since the excited eigen-mode is the qth, we take the qth branch namely, ∂θ = −λ0q (x, t, ∇x θ) ∂t (5.4.7) The characteristics of (5.4.7), are the linear bicharacteristics of (5.4.1), which will be referred to as rays We seek a solution to (5.4.6) as a perturbation series v = u−u0 = u1 +2 u2 +3 u3 + = π(x, t, ξ)R0q +2 u2 +3 u3 +· · · (5.4.8) Substituting (5.4.8) into (5.4.6), multiplying through by , we get at levels O(k ) for k = 1, 2, the following equations: n X A0j j=0 n X j=0 ∂θ ∂π R ∂xj ∂ξ q = 0, ∂θ ∂u2 ∂xj ∂ξ = − A0j n X ∂θ ∂u3 Aj ∂x ∂ξ j j=0 (5.4.9) n X ∂θ 0 B [R , R ]ππ,ξ , (5.4.10) ∂xj j q q j=0 n X ∂θ ∂ Bk [u1 , u2 ] + Ck [u1 , u1 , u1 ] = − ∂xk ∂ξ k=0 − n X k=0 A0k ∂u1 − E u1 , ∂xk (5.4.11) where (5.4.9) holds by the choice of θ(x, t) 5.4.2 The -approximate equation and transport equation The solvability condition for (5.4.10) obtained by pre-multiplying with L0q is the vanishing of Σ0 := n X ∂θ 0 0 L B [R , R ] = ∂xj q j q q j=0 (5.4.12) In certain applications to media with mixed nonlinearity, condition (5.4.12) generally holds only approximately with an error of O() This is seen from physical considerations of the parameters involved, and seems to have been 150 Asymptotic Waves for Quasilinear Systems the main motivation in [94] for implicitly assuming an -dependence for the matrices A0j allowing a development of Bk and Ck as a series in Cramer and Sen [43], in a different context, cope with this problem by manipulating the perturbation expansion by introducing -corrections to the coefficients of 2 and 3 as follows When (5.4.8) is substituted in (5.4.6) the result is an asymptotic expansion Z1 + 2 Z2 + 3 Z3 + · · · = 0, (5.4.13) where Z1 = holds since the O() term in (5.4.8) is proportional to R0q ; however Z2 = 0, which is equation (5.4.10), only holds with an error of order O() Rewriting (5.4.13) as Z1 + 2 (Z2 − Z30 ) + 3 (Z3 + Z30 ) + = 0, (5.4.14) where Z30 = Z2 / and Z3 = Z30 In other words (5.4.10) is replaced by an approximate equation (5.4.16) given below which we solve exactly, thereby compensating the error at the O(3 ) level The smallness of the physical parameters involved makes this procedure licit Let us write − n n X X ∂θ 0 µj R0j , Bk [Rq , Rq ] = ∂xk j=1 (5.4.15) k=0 so that µq = O() and µj L0j R0j = − n X ∂θ L Bk [R0q , R0q ] ∂xk j k=0 We now solve exactly the -approximate equation n X X ∂θ ∂u2 Aj = −ππ,ξ µj R0j , ∂x ∂ξ j j=0 (5.4.16) j6=q namely, n X X ∂θ ∂u2 ππ,ξ =− L0 B0 [R0 , R0 ]R0 , ∂ξ ∂xk j k q q j (λ0j − λ0q )L0j R0j (5.4.17) j6=q k=0 which implies u2 = − n XX ∂θ 0 0 π2 Lj Bk [Rq , Rq ]Rj ∂xk 2(λ0j − λ0q )L0j R0j (5.4.18) j6=q k=0 These expressions are unique up to an additive multiple of R0q With the notations ∆0j = L0j R0j , and Γkij = n X ∂θ 0 0 L B [R , R ], ∂xl k l i j l=0 5.4 Evolution Equation Describing Mixed Nonlinearity 151 the formula for u2 can be stated as u2 = − X j6=q X Γjqq ππ,ξ R0j ∂u2 =− ∂ξ ∆0j (λ0j − λ0q ) Γjqq R0j π , 2∆0j (λ0j − λ0q ) (5.4.19) j6=q In terms of Γkij , we get Σ0 = Γqqq We note here that the general system considered in [94] is not in conservation form and hence the multilinear maps B0k , C0k would be unsymmetric in general Multiplying (5.4.11) by L0q and using (5.4.8) we get the compatibility condition n n X X ∂θ 0 0 ∂θ ∂ 0 (Lq Bk [u1 , u2 ]) + C [R , R , R ]π π,ξ Lq ∂xk ∂ξ ∂xk k q q q k=0 k=1 + n X L0q A0k k=0 ∂(πR0q ) + L0q E0 R0q π = (5.4.20) ∂xk On Substituting in (5.4.20) the values of ∂u2 /∂ξ and u2 from equation (5.4.19) we get − n X 3Γjqq Γqqj π π,ξ X ∂θ 0 0 Lq + C [R , R , R ]π π,ξ 2∆0j (λ0j − λ0q ) ∂xk k q q q j6=q k=0 dπ + (χ + h)π = 0, + dt (5.4.21) where h = L0q E0 R0q and d/dt denotes the ray derivative along the characteristics of (5.4.7), and n X ∂(R0q ) χ= L0q A0k (5.4.22) ∂xk k=0 Note that we have obtained the transport equation (32) of [94] except for the quadratic terms of the Burgers equation We must now incorporate the corrections indicated in equation (5.4.14) Since Z30 is the R0q component of Σnj=1 (∂θ/∂xj )B0j [R0q , R0q ], namely q Γqq ππ,ξ /∆0q , we must add Z30 = (1/)Γqqq ππ,ξ /∆0q = O(1) to the right-hand side of (5.4.21); thus we get dπ γ + (χ + h)π + π π,ξ + Σ1 ππ,ξ = 0, (5.4.23) dt which is precisely the equation obtained in [94] taking into account the contribution due to the source term in (5.4.1), where Σ1 = Σ0 /(∆0q ) = Γqqq /(∆0q ) and n X X 3Γjqq Γqqj ∂θ 0 0 + L0q C [R , R , R ] (5.4.24) γ=− ∆0j (λ0j − λ0q ) ∂xk k q q q j6=q k=0 152 Asymptotic Waves for Quasilinear Systems 5.4.3 Comparison with an alternative approach We compare our derivation of (5.4.23) with the derivation in [94], where for simplicity we have assumed that the system in [94] arises out of a system of conservation laws Let us denote by A(λ) the matrix Σnj=1 (∂θ/∂xj )A0j − λI and aτ (λ) its rows Let b be the row vector that represents the functional Σnj=0 (∂θ/∂xj )L0q B0j [R0q , ·]/∆0q Condition (5.4.12) can be stated as vanishing of b · R0q The authors in [94] proceed to observe that this holds if b is in the row space of the matrix A(λq ), i.e., b = Στ βτ aτ (λ) The coefficients of the cubic Burgers equation satisfied by π, as derived in [94], involve βτ , and hence there is a need to compute them explicitly To this, notice that aτ R0p = (λ0p − λ)R0p,τ , where R0p,τ denotes the τ entry in the column vector R0p , and similarly L0p,τ denotes the τ component of the left eigenvector L0p Multiplying by βτ and summing over τ we get ¯ · R0 , b · R0p /(λ0p − λ) = Γqqp /∆0q (λ0p − λ) = β p p = 1, 2, , n, (5.4.25) ¯ denotes a row matrix, from which it follows that where β ¯= β X l Γqql L0l ∆0l ∆0q (λ0l − λ) , and hence βτ = X l Γqql L0l,τ ∆0l ∆0q (λ0l − λ) On using this in equation (32) of [94], we recover (5.4.23) It may be noticed that the summand with l = q in the above formula for βτ vanishes, since (5.4.10) is assumed in [94] at the level A0i |=0 5.4.4 Energy dissipated across shocks Throughout this section we assume that u0 is constant, the matrices A0j are symmetric, and ∂A0j /∂xj = With the normalization L0q R0q = we get on using the relation ∂λq /∂ϑj = L0q A0j R0q obtained by differentiating Lq (−λq I + Σnj=1 ϑj A0j )R0q = with respect to ϑj , n X j=0 L0q ∂ (A0 R0 π) ∂xj j q = = n n X 1X ∂ ∂π π (L0q A0j R0q ) + (L0q A0j R0q ) , j=0 ∂xj ∂xj j=0 n n X ∂π X ∂ ∂λq π + (L0q A0j R0q ) j=1 ∂xj ∂ϑj ∂x j j=0 ϑ=∇x θ Numerous physical systems are governed by isotropic conservation laws, where the eigenvalues λ0q have the form λ0q (x, t, ϑ) = |ϑ|c0q (x, t) In particular, this is ... (5 .2. 5) γ{1 + (γ − 1)c}p1,θ φ,t + ρo a2o γu1,θ φ,x − (γ − 1)ca2o ρ1,θ φ,t = (5 .2. 6) c cp2 cpo ? ?2 ? ?2 po ? ?21 where Q2 = − p ρ1 + − − τ ρ2o ρo τ ρo τ ρ2o τ Eliminating Q2 between (5 .2. 5)1... Equations (5.3 .22 ) and (5.3 .24 ) get modified to E= and E= X b [π]3i Φ∗i (δ0 ), 12 i X γ[π ]i [π ]2 i i 24 Σ1 [π]3i + 12 (5.3 .26 ) Φ∗i (δ0 ), (5.3 .27 ) respectively Note that equation (5.3 .27 ) differs... ρ2o τ Eliminating Q2 between (5 .2. 5)1 and (5 .2. 5 )2 , and using (5 .2. 5)3 in the resulting equation, we get Equation (5 .2. 6), together with (5 .2. 4)1 and (5 .2. 4 )2 , constitutes a system of three equations