Explosive instability due to 4-wave mixing

Explosive instability due to 4-wave mixing Benjamin R. Safdi and Harvey Segur Department of Applied Mathematics, Univ ersity of Colorado, Boulder, CO 80309-0526 (April 12, 2007) Abstract: It is known that an explosive inst ability can occur when nonlinear waves propagate in certain media that admit 3-wave mixing. The purpose of this paper is to show that explosive instabili ties can occur even in media that adm it no 3-wave mixing. Instead, the instability is cause d by 4-wave mixing: four resona ntly interacting wavetra ins gain energy from a background, and all blow up in a finite time. Unlike singularities associated with self-focussing, these singulari ties can occur with no sp atial structure – the waves blow up everywhere in space, simultaneously. PACS numbers: 05.45.-a, 42.65.-k, 47.35.-i, 52.35.-g Among mathematical models that descri be nonlinear wave propagation without dissipation, certain “universal” models stand out – each of th ese models appears when one takes a specific limit, and each arises in m any physical situations. In all cases one first linearizes the governing equations about so m e trivial state, and obtains a (linearized) dispersion relation, ω ( k ), which relates the frequency ( ω ) of a signal to its wavenumber ( k ). A 3-wave resonance is possible if the dispersion relation admits pairs { ω m , k m } such that k 1 ± k 2 ± k 3 = 0, ω 1 ± ω 2 ± ω 3 = 0. (1) For a given problem, (1) may or m ay not be po ssible. For example, in nonlinear optics, (1) occurs only in so-called χ 2 materials [1]; for surface water waves, (1) is impossible for pure gravity waves, but it occurs if both gravity and su rface tension are included in the model [2]. When (1) occurs, then { A 1 ( x , t ), A 2 ( x , t ), A 3 ( x , t )}, the slowly-varying complex amplitudes of three wave modes, evolve accord ing to the “three-wave equations”: three coupled e quations of the form (with l,m,n = 1,2,3 cyclically) ∂ t A m + c m ⋅∇ A m = i δ m A n * A l * . ( 2 ) Here c m is the group velocity and δ m is a real-valued interaction coefficient, each corresponding to { ω m , k m } [3]. In the simplest model of 3-wave m ixing, one ignores the spatial dependence of the interacting modes, so that (2) reduces to three coupled, complex, ordinary differential equations (ODEs), dA 1 dt = i δ 1 A 2 * A 3 * , dA 2 dt = i δ 2 A 3 * A 1 * , dA 3 dt = i δ 3 A 1 * A 2 * . (3) If any two δ m in (3) differ in sign, then one can show that all solutions of (3) are bounded for all time. But this is not the on ly possibility: si tuations in which { δ 1 , δ 2 , δ 3 } all have the same sign occur in plasmas [ 4,5], in density-stratified shea r flows [6,7] and for vorticity waves [8]. If all δ m have the same sign, then solutions of (3) can blow up in finite time (like ( t - t 0 ) -1 ), including solutions that start with arbi trarily sm all initial data. This is the explosive instability . All three waves grow together, so all three waves draw energy from a background source and blow up in unis on. Thus, the rela tive signs of the δ m in (2) and (3) signal whether su ch an energy source is av ailable in the physical problem that (2) and (3) approximate. The main point of this pa per is to show that explosive ins tabilities can occur even in situations where a 3-wave resonan ce is im possible. In that case th e simplest nonlin ear interaction among wave modes is a 4-wave resonance, in which four pairs { ω m , k m } satisfy k 1 ± k 2 ± k 3 ± k 4 = 0, ω 1 ± ω 2 ± ω 3 ± ω 4 = 0. (4) A common special case, in which one wave mode interacts with two other m odes at nearly the same frequency and wave number, ( k + δ k ) + ( k - δ k ) - k = k , ( ω + δω ) + ( ω - δω ) – ω = ω , leads to the nonlinear Schr ödinger (NLS) equation for th e slowly varying, com plex amplitude of one wave mode [3] i ∂ t A + { α 1 ∂ x 2 + α 2 ∂ y 2 + α 3 ∂ z 2 } A + γ | A | 2 A = 0 . (5) Here { α m } are real-valued cons tants obtained from ω ( k ), and γ is a real-valued interaction coefficient (provided the origin al problem has no dissipation). In optics, (4) occurs in χ 3 materials [1]. With one additional term , (5) becom es the Gross-Pitaevski equation, a commonly used model for Bose-Ein stein condensates [9,10]. More complicated interactions, in whic h wave modes interact nonlinearly with themselves and also with other wave modes, lead to coupled NLS equations [11]. More complicated still are syst ems with self-int eractions, cross-interactions, and 4-wave- mixing (with m,p,q,r = 1,2,3,4 cyclically): i ( ∂ t A m + c m ⋅∇ A m ) + α m , l , n l , n ∑ ∂ x l ∂ x n A m + A m γ m , n n = 1 4 ∑ | A n | 2 + δ m A p * A q * A r * = 0 . (6) The system in (6) has four such equations. In each equation, c m is the group velocity and { α m,l,n } are real-valued constant s, all obtained from ω ( k ); { γ m n } are coefficients of NLS- type interaction terms; and { δ m } are real-valued coefficients of the 4-wave mixing term s. The general form of this system of eq uations was first recognized in [12]. In this paper, we show that an explosiv e instability of the ki nd usually associated with 3-wave interactions can also occur b ecause of 4-wave mixing. As with (3 ), a simpler model with 4-wave m ixing than (6) is obtained by ignoring any spatial dependence of the interacting modes, so th at (6) reduces to four coupled ODEs (with m,p,q,r = 1,2,3,4 cyclically): i dA m dt + A m γ m , n n = 1 4 ∑ | A n | 2 + δ m A p * A q * A r * = 0 . (7) Note that with no spatial dependence, th e self-focussing kind of singularity usually associated with NLS-type systems [13] cannot occur. Without the 4-wave mixing term s in (7), there is no blow-up: one shows directly from (7) that for each m , if δ m = 0 then | A m | 2 is constant. Hence we now assume that all δ m ≠ 0. Then (7) admits three independent cons tants of the motion in the f orm of Manley- Rowe [14] relations: J m = | A m | 2 δ m − | A 4 | 2 δ 4 , m = 1,2,3. (8) It follows from (8) that if any two δ m differ in sign, then every | A m | 2 is bounded for all time. This result parallels the corresponding result for (3): all solutions of (3) are bounded if any two δ m in (3) differ in sign. How ever, requiring that a ll the δ m have the same sign in (7) is necessary but no t sufficien t for an explosive instability: it is also necessary that the δ m be large enough relative to γ m , n ∑ , as we show next. If all δ m in (7) have the same sign, then change variables { A m ( t ) = | δ m | ⋅ B m ( t ), Γ m , n = γ m , n | δ n |, δ = sign ( δ 1 ) δ 1 δ 2 δ 3 δ 4 } to obtain an equivalent system of ODEs ( m,p,q,r = 1,2,3,4 cyclically): i dB m dt + B m Γ m , n n = 1 4 ∑ | B n | 2 + δ ⋅ B p * B q * B r * = 0 . (9) This system of ODEs is Hamiltonian , with conjugate variables { B m , B m * , m = 1,2,3,4} and Hamiltonian H = H 1 + H 2 , where H 1 =− i 2 Γ m , n m , n = 1 4 ∑ B m B m * B n B n * , H 2 =− i δ ( B 1 B 2 B 3 B 4 + B 1 * B 2 * B 3 * B 4 * ) . (10) Direct computation shows that H is a constan t of the moti on. In addition, in these variables, (8) becom es J m = | B m | 2 - | B 4 | 2 , m = 1,2,3. (8a) And one can verify directly that the usual Poisson bracket of any two of ( H , J 1 , J 2 , J 3 ) vanishes, so these constants are said to be in involution. Then it follow s that the system of four complex OD Es in (9) is completely in teg rable in the sense of Liouville [15]. Next, we show that solutions of (9) blow up in finite time if | Γ m , n m , n = 1 4 ∑ | ≤ 4| δ | . ( 1 1 a ) In terms of the variables in (7), its solu tions blow up in finite tim e if all four δ m have the same sign and | γ m , n m , n = 1 4 ∑ | δ n || ≤ 4 δ 1 δ 2 δ 3 δ 4 . (11b) In either notation, these are th e criteria for an explosive inst ability due to 4-wave mixing. They comprise the m ain result in this pa per. Assuming (11) hol ds, a four-parameter family of exact, singular solutions of (9) is B m ( t ) = c ⋅ e i θ m ( t 0 − t ) 1 2 + i φ m , m = 1,2,3,4 (12) where { c , t 0 , θ m , φ m } are real-valued constants, and Θ= θ m = arccos{ − 1 4 δ m = 1 4 ∑ Γ m , n m , n = 1 4 ∑ } , ( 1 3 a ) c 2 = 1 2 δ ⋅ si n Θ , φ m =− c 2 { Γ m , n n = 1 4 ∑ + δ cos Θ } . (13b,c) [These results hold for t 0 > t ; for t > t 0 , one changes the sign of ( t 0 - t ) in (12), and the sign of c 2 in (13b,c).] The four free constants in (12) are t 0 , and any three of the four θ m . Then the last θ m must be chosen to satisfy (13a). Subs titution of (12) into (9) shows that th is form of solution is possible if and only if (11) holds. One can also verify by substituting (12) into (8a) and (10) that { H , J 1 , J 2 , J 3 } all vanish for any solution in this f amily. Next we show that when (11) holds, all solutions of (9) blow up in finite time. To do so, we may consider the solution in (9) to be the first term in a Lauren t series, near ( t = t 0 ), and seek solutions of (9) in the form (for m = 1,2,3,4) B m ( t ) = c ⋅ e i θ m ( t 0 − t ) 1 2 + i φ m [1 + α m ( t − t 0 ) + β m ( t − t 0 ) 2 + ...] . (14) In (14), { c , t 0 , θ m , φ m } are defined as above, and { α m , β m ,…} are complex numb ers. Substituting (14) into (9) and requiring that the complex coefficient of each power of ( t - t 0 ) vanish shows that most of the coefficients in this expan sion ar e fixed, with four exceptions: the real parts of three α m can be chosen arbitrarily, as can the imaginary part of one β m . Thus, the family of solutions in (14 ) contains eight free, real constan ts. [For example, one can choose the 8 free constants to be { t 0 , θ 1 , θ 2 , θ 3 , Re( α 1 ), Re( α 2 ), Re( α 3 ), Im( β 4 )}.] Therefore the fam ily of solutions in (14) is the general solution of (9), so all solutions of (9) with nonzero in itial data blow up in finite tim e, pr ovided only that (11) holds. Because the solutions in (12) all occur with {0 = H = J 1 = J 2 = J 3 }, it follows that the four new constants in (14) must determine { H , J 1 , J 2 , J 3 }. One can show that for m = 1,2,3, J m = 2 c 2 [Re( α 4 ) − Re( α m ) ] . ( 1 5 ) Then Im( β 4 ) determines the value of H , but we have found no simple way to write this relation. It is known that the self-focussing (or “wave collapse”) singularity of an NLS- type equation occurs only for a range of H [13]. The singularity in (14) occurs for any (real) values of { H , J 1 , J 2 , J 3 }, provided only that (11) holds, so the two kinds of singularities differ in this respect. They al so differ because spatial structure plays an essential role in a self-focu ssing singularity, but it plays no role whatsoev er here. It remains to show that the solutions of (9) must be non-singul ar if (11) is not satisfied, so that (11) is both necessary and sufficient for an explosive instability. Suppose that | B 4 ( t )| → ∞ as t → t 0 . Then it follows from (8a) that all four | B m ( t )| must grow at the same rate. Hence as t → t 0 , the dominant terms in (10 ) are H 1 =− i 2 Γ m , n m , n = 1 4 ∑ | B 4 | 4 + O (| B 4 | 2 ) , H 2 =− 2 i δ | B 4 | 4 cos( ϕ ) + O (| B 4 | 3 ) , where ϕ ( t ) is some (unknown) phase. These dominant terms m ust balance as t → t 0 , so necessarily | 1 2 Γ m , n | ⋅ m , n = 1 4 ∑ | B 4 | 4 = | 2 δ | B 4 | 4 cos( ϕ )| ≤ 2| δ | ⋅ | B 4 | 4 (16) in this limit. Dividing by | B 4 | 4 shows that no explosive singularity can occur for | Γ m , n m , n = 1 4 ∑ | > 4| δ |. This completes th e proof. Explosive instabilities due to 3-wave mixing have been known for thirty years [4-8]. To our knowledge, no explosive inst ability caused by 4-wave mixing has ever been observed in a physical system. The an alysis above indicate s that it should be possible. As with 3-wave mixing, an explos ive instability in a 4-wave sy stem requires a background source of energy, so that all four wa ve modes can grow in intensity together. And as with 3-wave mixing, the indication th at such a background source is available is that all four δ m in (6) or (7) have the same si gn. One difference between the two processes is that for 4-wave mixing, this agreem ent in signs of the δ m in (7) by itself does not guarantee an explosive instability – the inter action coefficients must also satisfy (11). This work was funded in part by an NSF-MCTP grant, DMS-0602284. __________________ [1] R.W. Boyd, Nonlinear Optics (Academ ic Press, 2003) 2 nd ed . 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