aa r X i v : . [ m a t h . A P ] A p r Interaction of 3 solitons for the GKdV-4equation
Georgy A. Omel’yanov ∗ Abstract
We describe an approach to construct multi-soliton asymptotic so-lutions for non-integrable equations. The general idea is realized inthe case of three waves and for the KdV-type equation with nonlin-earity u . A brief review of asymptotic methods as well as results ofnumerical simulation are included. Key words : generalized Korteweg-de Vries equation, soliton, interaction,weak asymptotics method, weak solution, non-integrability : 35Q53, 35D30
It is well known that an arbitrary number of solitary waves collide for inte-grable non-linear equations in an enormous manner: they pass through eachother almost as linear waves. The aim of this paper is the consideration: canwe believe that such type of interaction is conserved (in a sense) for someessentially non-integrable equations?Needless to recall that the integrability implies both the possibility tofind exact solutions of complicated structures and that the equation hassome special properties. Conversely, the non-integrability implies that we ∗ University of Sonora, Rosales y Encinas s/n, 83000, Hermosillo, Sonora, Mexico,[email protected]
1o not have, at least nowadays, neither explicit solutions, nor special usefulproperties of the problem.We consider the general techniques and averaging method by a specificbut typical example of the Generalized Korteweg-de Vries-4 equation withsmall dispersion, that is: ∂ u∂ t + ∂ u m ∂ x + ε ∂ u∂ x = 0 , m = 4 , x ∈ R , t > , (1)where ε << m = 2 and m = 3, that is theKdV and MKdV equations) and essentially non-integrable ( m ≥
4) equa-tions. The last means that there is not any method at present to constructexact solutions of the Cauchy problem with more or less general initial data.More in detail, there is known that (1) for m ≥ m = 5 is conditionally stable (unstable for solitons) [1, 2]. As for m = 4, this case is stable and there are known some exact particular solutionsincluding the solitary wave: u = Aω (cid:18) β x − V tε (cid:19) , ω ( η ) = c cosh − / ( η ) , A = 1 γ β / , (2)where β > c is such that Z ∞−∞ ω ( η ) dη = 1 , (3)and V = a γ β . (4)Here and in what follows we use the notation a k = Z ∞−∞ ω k ( η ) dη, k ≥ , a ′ = Z ∞−∞ (cid:16) dωdη (cid:17) dη (5)and the identities γ = (cid:16) a a a ′ (cid:17) / , a a = 75 a . (6)In view of the general wave propagation theory, there appear questionsboth about the stability of the solitary wave solution (2) with respect tosmall perturbations of the equation, and about the character of the solitarywave collision. 2 .2 Prehistory: single-phase asymptotic solutions The non-integrability implies the use of asymptotic approaches. We considerwaves of arbitrary amplitude (of the value O (1)), treating the dispersion ε as a small parameter. Thus, there appear ”fast” x/ε , t/ε and ”slow” x , t variables. Of course, it is possible to rescale η = x/ε , τ = t/ε and pass to”fast” η , τ and ”slow” εη , ετ variables. However we prefer the first version.The modern asymptotic technique, which is based on ideas by Poincare,van der Pol, Krylov-Bogoliubov and others, has been created firstly by G.E.Kuzmak (ODE, 1959, [3]) and G.B. Whitham, (PDE, 1965, [4, 5], see also[6]) for rapidly oscillating asymptotic solutions of non-linear equations. Thefamous Whitham method deals with a Lagrangian formulation and allows tofind slowly varying amplitude and wave number of non-uniform wave trains.This approach determined the development of the non-linear perturbationtheory in 1970s. At the same time, the passage from the original equationto the Lagrangian seemed to be artificial. For this reason J.C. Luke (1966,[7]) created a version of the Whitham method, which allows to constructasymptotic solutions with arbitrary precision appealing directly to the origi-nal equation. More in detail, for the equation L ( u, εu t , εu x , . . . ) = 0 we writethe ansatz u = Y ( τ, t, x ) + εY ( τ, t, x ) + . . . , (7)where τ = S ( x, t ) /ε , Y k ( τ + T, t, x ) = Y k ( τ, t, x ), T = const, and S ( x, t ), Y k ( τ, t, x ) are arbitrary functions from C ∞ . Since ε∂ t Y k (cid:0) S ( x, t ) /ε, t, x (cid:1) = { S t ∂ τ Y k ( τ, t, x ) + ε∂ t Y k ( τ, t, x ) }| τ = S/ε , we obtain the chain of ordinary (with respect to τ ) equations, the first ofthem is non-linear, L (cid:0) Y ( τ, t, x ) , S t ∂ τ Y ( τ, t, x ) , S x ∂ τ Y ( τ, t, x ) , . . . (cid:1) = 0 , (8)and others are non-homogenous linearization: L ′ (cid:0) Y ( τ, t, x ) , S t ∂ τ ,S x ∂ τ , . . . (cid:1) Y k ( τ, t, x )= F k (cid:0) Y ( τ, t, x ) , . . . , Y k − ( τ, t, x ) (cid:1) , k ≥ . (9)It is assumed that (8) has a T -periodic solution. Then there appear theorthogonality conditions Z T F k (cid:0) Y ( τ, t, x ) , . . . , Y k − ( τ, t, x ) (cid:1) Z i dτ = 0 , i = 1 , . . . , l, (10)3hich guarantee the solvability of (9) in the space of T -periodic smoothbounded functions. Here { Z i , i = 1 , . . . , l } is the kernel of the operatoradjoint to L ′ . Moreover, (10) allow to define the phase S ( x, t ) and all the”constants” of integration of the equations (8), (9).It seemed that the same procedure can be used to construct a perturbedsoliton-type solution (with trivial alterations). However, it is not true, anda mechanical repetition of the Whitham construction leads to some ”para-doxes” and senseless solutions (see, for example, [8], pp. 303 - 306). Thesituation has been improved by V. Maslov and G. Omel’yanov (1981, [9], seealso [10]). A little bit later a similar construction has been developed by I.Molotkov and S. Vakulenko (see e.g. [11]). To illustrate the modification [9]let us consider the perturbed GKdV-4 equation (1), ∂ u∂ t + ∂ u ∂ x + ε ∂ u∂ x = R, (11)where R = R ( x, t, u, εu x , ε u xx , . . . ) is ”small” in our scaling and R | u =0 = 0.To find a self-similar soliton-type asymptotics we restrict the soliton partof the solution on the zero-level set of the phase S ( x, t ) = x − ϕ ( t ) + O (( x − ϕ ( t )) ). This allows to avoid the appearance of some nonuniqueness effects(see [10], pp. 24 - 26). Next we take into account that a smooth small ”tail”can appear after the soliton. Therefore, instead of (7) we write the ansatz inthe form: u = Y ( τ, t ) + εY ( τ, t, x ) + . . . , (12)where τ = (cid:0) x − ϕ ( t ) (cid:1) /ε , Y k are smooth bounded function such that Y ( τ, t, x )tends to 0 as τ → ±∞ , Y k ( τ, t, x ) → τ → + ∞ , and Y k ( τ, t, x ) → Y − k ( x, t )as τ → −∞ for k ≥
1, and ϕ belongs to C ∞ .Similar to (8), substituting (12) into (11) we obtain the nonlinear modelequation − ϕ t dY dτ + dY dτ + d Y dτ = 0 (13)and define the shape of the leading term in (12), Y = A ( t ) ω (cid:0) βτ + ϕ ( t ) (cid:1) , β / ( t ) = γA ( t ) , (14)as well as the similar to (4) relation dϕdt = a A ( t ) . (15)4ere ϕ ( t ) is a ”constant” of integration. Next to find the deficient relationbetween ϕ and A we consider the first correction Y freezed on the solitonfront x = ϕ ( t ). Denoting ˇ Y ( τ, t ) = Y ( τ, t, x ) | x = ϕ ( t ) , we pass to the equation: ddτ n − ϕ t ˇ Y + 4 Y ˇ Y + d ˇ Y dτ o = R (cid:0) ϕ, t, Y , Y τ , Y ττ , . . . (cid:1) − Y t . (16)Respectively, to guarantee the existence of the desired correction ˇ Y , we ob-tain the following conditions: ddt Z ∞−∞ Y dτ = 2 Z ∞−∞ Y R (cid:0) ϕ, t, Y , Y τ , Y ττ , . . . (cid:1) dτ, (17) ϕ t ˇ Y | τ →−∞ = Z ∞−∞ n R (cid:0) ϕ, t, Y , Y τ , Y ττ , . . . (cid:1) − Y t o dτ. (18)Calculating the integrals in (17), we complete (15) by the equation a ddt A β = 2 Aβ R , (19)where R = Z ∞−∞ ω ( η ) R (cid:0) ϕ, t, Aω ( η ) , Aβω ( η ) η , Aβ ω ( η ) ηη , . . . (cid:1) dη. (20)This allows us to determine the phase and amplitude dynamics. The equa-tions (15), (19) have been called ”Hugoniot-type conditions” [9] since theydo not depend on ε , whereas the solitary wave (14) Y (cid:0) ( x − ϕ ( t )) /ε, t (cid:1) dis-appears (in D ′ sense) as ε →
0. Let us recall that the Rankine-Huginiotconditions remain the same both for parabolic regularization of shock waves,and for the limiting non-smooth solutions.Furthermore, returning to the asymptotic construction, we note that (18)implies the equality a A ˇ Y − = 1 β Z ∞−∞ R (cid:0) ϕ, t, Aω ( η ) , Aβω ( η ) η , Aβ ω ( η ) ηη , . . . (cid:1) dη − ddt Aβ , (21)where ˇ Y − = ˇ Y | τ →−∞ . Now we integrate the equation (16) and find thestructure of the first freezed correctionˇ Y ( τ, t ) = ˇ Y − ( t ) χ ( τ, t ) + Z ( τ, t ) + c ( t ) Y ′ τ ( τ, t ) , (22)5here χ and Z are some fixed functions such that Z → τ → ±∞ ,χ → τ → + ∞ , χ → τ → −∞ , and c is an arbitrary ”constant” of integration.The next step of the construction is the extension of ˇ Y ( τ, t ) to Y ( τ, t, x )in the following manner: Y ( τ, t, x ) = u − ( t, x ) χ ( τ, t ) + Z ( τ, t ) + c ( t ) Y ′ τ ( τ, t ) , (23)where u − is a smooth function such that ∂ u − ∂ t = u − R ′ u ( x, t, , . . . ) , x < ϕ ( t ) , t > , (24) u − | x = ϕ ( t ) = ˇ Y − , t > . (25)Continuing the procedure we can easily construct the one-phase self-similarasymptotic solution with arbitrary precision.Let us note finally that self-similarity implies the special choice of the ini-tial data. In particular, the initial function Y ( τ, , x ) should be of the specialform (23) with arbitrary c (0) and arbitrary u − ( x,
0) under the condition u − ( x, | x = ϕ (0) = ˇ Y − (0) . (26)If it is violated and, for example, u | t =0 = A (0) ω (cid:0) β ( x − ϕ (0)) /ε (cid:1) , then theperturbed soliton generates a rapidly oscillating tail of the amplitude o (1)(the so called ”radiation”) instead of the smooth tail εu − ( x, t ) (see [12] forthe perturbed KdV equation). However, εu − ( x, t ) describes sufficiently wellthe tendency of the radiation amplitude behavior (see e.g. [13]). Concerning the solitary wave collision, this problem is much more compli-cated. Indeed, to describe the interaction of two waves of the form (2), weshould look for the asymptotics as a two-phase function, u = W (cid:16) x − ϕ ε , x − ϕ ε , t (cid:17) + o (1) , (27)6here W ( τ , τ , t ) has properties similar to the two-soliton solution of theKdV equation. However, to construct W we obtain a non-linear PDE, whichis, in fact, equivalent to the original GKdV-4 equation (1). So, the existenceof such asymptotics remains unknown. The same is true for any essentiallynon-integrable equation. Respectively, there is not any possibility to con-struct a classical asymptotic solution (that is, with the remainder in the C -sense).At the same time it is easy to note that the solitary wave solutions (solitonor kink type) tend to distributions as ε →
0. This allows to treat the equationin the weak sense and, respectively, look for singularities instead of regularfunctions. Obviously, non-integrability implies that we cannot find neitherclassical nor weak exact solutions. However, we can construct an asymptoticweak solution considering the smallness of the remainder in the weak sense.Originally, such idea had been suggested by V. Danilov and V. Shelkovichfor shock wave type solutions (1997, [14]), and after that it has been devel-oped and adapted for many other problems (V. Danilov, G. Omel’yanov, V.Shelkovich, D. Mitrovic and others, [15] - [27] and references therein). Wecalled this approach the ”weak asymptotics method”.For the special case of soliton-type solutions we note now that they havethe value O ( ε ) in the weak sense. Thus, the remainder for the leading termof the asymptotic solution should be O ( ε ) in the weak sense. However, theGKdV equations (1) degenerate to a first-order PDE in D ′ for this preci-sion. The same fact has been noted by Danilov, Omel’yanov, and Radkevich(1997, [28]) by the consideration of a free boundary problem. There has beensuggested also a way about how to overcome this obstacle. Applying theseideas to the equation (1), we pass to the following definition of the weakasymptotic solution [17]: Definition 1.
A sequence u ( t, x, ε ) , belonging to C ∞ (0 , T ; C ∞ ( R x )) for ε =const > and belonging to C (0 , T ; D ′ ( R x )) uniformly in ε ≥ , is called aweak asymptotic mod O D ′ ( ε ) solution of (1) if the relations ddt Z ∞−∞ uψdx − Z ∞−∞ u ∂ ψ∂ x dx = O ( ε ) , (28) ddt Z ∞−∞ u ψdx − Z ∞−∞ u ∂ ψ∂ x dx + 3 Z ∞−∞ (cid:18) ε ∂ u∂ x (cid:19) ∂ ψ∂ x dx = O ( ε ) (29) hold uniformly in t for any test function ψ = ψ ( x ) ∈ D ( R ) . C ∞ -functions for ε = const > ε ≥
0. The estimates are understoodin the C (0 , T ) sense: g ( t, ε ) = O ( ε k ) ↔ max t ∈ [0 ,T ] | g ( t, ε ) | ≤ cε k . Definition 2.
A function v ( t, x, ε ) is said to be of the value O D ′ ( ε k ) if therelation Z ∞−∞ v ( t, x, ε ) ψ ( x ) dx = O ( ε k ) holds uniformly in t for any test function ψ ∈ D ( R x ) . The sense of the relation (28) is obvious: it is the adaptation of thestandard D ′ -definition to asymptotic mod O D ′ ( ε ) solution which belongs to C (0 , T ; D ′ ( R x )). Next we note again that (28) cannot be a unique satisfac-tory condition since here has been lost the difference between the GKdV-4equation and the limiting first order equation (with ε = 0). To involve thedispersion term into the consideration, we supplement (28) by the additionalcondition (29). It can be treated as a version of (28) but for special testfunctions u ψ ( x ), ψ ∈ D ( R x ), which vary rapidly together with the solution.It is important also that (29) duplicates the orthogonality condition whichappears for single-phase asymptotics. Indeed, the adaptation of the Defini-tion 1 to the perturbed equation (11) implies the transformation of (29) tothe following form: ddt Z ∞−∞ u ψdx − Z ∞−∞ u ∂ ψ∂ x dx + 3 Z ∞−∞ (cid:18) ε ∂ u∂ x (cid:19) ∂ ψ∂ x dx (30) − Z ∞−∞ uRψdx = O ( ε ) . Next for u of the form (12), (14) we calculate the weak expansion: Z ∞−∞ u k ( x, t ) ψ ( x ) dx = ε A k β Z ∞−∞ ω k ( η ) ψ ( ϕ + ε ηβ ) dx + O ( ε )= εa k A k β ψ ( ϕ ) + O ( ε ) , where k ≥
1. Thus u k = εa k A k β δ ( x − ϕ ) + O D ′ ( ε ) . (31)8n the same manner we obtain( εu x ) = εa ′ βA δ ( x − ϕ ) + O D ′ ( ε ) , (32) uR ( x, t, u, εu x , ε u xx , . . . ) = ε Aβ R δ ( x − ϕ ) + O D ′ ( ε ) , (33)where R has been defined in (20). Substitution of (31) - (33) into (30) impliesthe relation ε n − a A β dϕdt + a A β − a ′ βA o δ ′ ( x − ϕ )+ ε n a ddt A β − Aβ R o δ ( x − ϕ ) = O D ′ ( ε ) . (34)Since δ ( x − ϕ ) and δ ′ ( x − ϕ ) are linearly independent, their coefficients in(34) should be equal to zero. Taking into account the identities (4) we obtainagain the equations (15), (19) for the one-phase asymptotics (12).Let us revert to the two-wave interaction. Following [17] (see also [18]),we present the ansatz as the sum of two distorted solitons (2), that is: u = X i =1 G i ω (cid:18) β i x − ϕ i ε (cid:19) , (35)where G i = A i + S i ( τ ) , ϕ i = ϕ i ( t ) + εϕ i ( τ ) , τ = β (cid:0) ϕ ( t ) − ϕ ( t ) (cid:1) /ε, (36) A i are the original amplitudes and ϕ i = V i t + x i describe the trajectories ofthe non-interacting waves (2), β i = ( γA i ) / . We assume that A < A and x >> x , therefore, the trajectories x = ϕ and x = ϕ intersect at apoint ( x ∗ , t ∗ ). Next we define the ”fast time” τ to characterize the distancebetween the trajectories ϕ i and we assume that S i ( τ ), ϕ i ( τ ) are such that S i → τ → ±∞ , (37) ϕ i → τ → −∞ , ϕ i → ϕ ∞ i = const i as τ → + ∞ . (38)It is obvious that the existence of the weak asymptotics (35) with the prop-erties (37), (38) implies that the solitary waves (2) interact like the KdVsolitons at least in the leading term. 9o construct the asymptotics we should calculate again the weak expan-sions for u k and ( εu x ) . It is easy to check that u = ε X i =1 G i β i δ ( x − ϕ i ) + O D ′ ( ε ) . (39)At the same time Z ∞−∞ u ( x, t ) ψ ( x ) dx = ε X i =1 G i β i Z ∞−∞ ω ( η ) ψ ( ϕ i + ε ηβ i ) dx (40)+ 2 G G Z ∞−∞ ω (cid:18) β x − ϕ ε (cid:19) ω (cid:18) β x − ϕ ε (cid:19) ψ ( x ) dx. (41)We take into account that the integrand in (41) vanishes exponentially fastas | ϕ − ϕ | grows, thus, the main contribution gives the point x ∗ . We write ϕ i = x ∗ + V i ( t − t ∗ ) = x ∗ + ε V i β ( V − V ) τ and ϕ i = x ∗ + εχ i , (42)where χ i = V i τ / (cid:0) β ( V − V ) (cid:1) + ϕ i . Next we transform the integral in (41)to the following form: εβ Z ∞−∞ ω ( θ η − σ ) ω ( η ) ψ (cid:0) x ∗ + εχ + ε ηβ (cid:1) dη, (43)where θ = β /β , σ = β ( ϕ − ϕ ) /ε . It remains to apply the formula f ( τ ) δ ( x − ϕ i ) = f ( τ ) δ ( x − x ∗ ) − εχ i f ( τ ) δ ′ ( x − x ∗ ) + O D ′ ( ε ) , (44)which holds for each ϕ i of the form (42) with slowly increasing χ i and for f ( τ ) from the Schwartz space. Moreover, the second term in (44) is O D ′ ( ε ).Thus, under the assumptions (37) we can modify (39)-(41) to the final form: u = ε X i =1 A i β i δ ( x − ϕ i ) + ε X i =1 S i β i (cid:8) δ ( x − x ∗ ) − εχ i δ ′ ( x − x ∗ ) (cid:9) + O D ′ ( ε ) , (45) u = εa X i =1 A i β i δ ( x − ϕ i ) + εa n X i =1 β i (2 A i S i + S i )10 2 G G β λ , ( σ ) o δ ( x − x ∗ ) + O D ′ ( ε ) , (46)where the convolution λ , ( σ ) describes the product of two waves. In viewof further applications we present such type of convolutions in the generalversion: λ ( j ) m,k ( σ ln ) = 1 a m Z ∞−∞ η j ω m − k ( η ln ) ω k ( η ) dη, η ln def = θ ln η − σ ln , θ ln def = β l β n , (47)where 1 ≤ k < m , m ≥ j = 0 or j = 1, and we write λ m,k ( σ ln ) def = λ (0) m,k ( σ ln )simplifying the notation.To calculate the time-derivative of u with the accuracy O D ′ ( ε ) it isenough to use the expansion (45), the assumptions (37), (38), and to ap-ply the formula (44) again. Thus, ∂ u∂ t = X i =1 ˙ ψ β i dS i dτ δ ( x − x ∗ ) − ε X i =1 V i A i β i δ ′ ( x − ϕ i ) − ε ˙ ψ ddτ X i =1 n A i β i ϕ i + χ i K (1) i o δ ′ ( x − x ∗ ) + O D ′ ( ε ) , (48)where ˙ ψ = β ( V − V ).On the contrary, to find ∂ ( u ) /∂ t with the same accuracy we should addto the leading term (46) the next correction: − ε a ( X i =1 χ i β i (2 A i S i + S i )+ 2 G G β (cid:16) χ λ , ( σ ) + 1 β λ (1)2 , ( σ ) (cid:17)) δ ′ ( x − x ∗ ) + O D ′ ( ε ) . (49)Now we obtain ∂ u ∂ t = a ddτ n ψ β G G λ , ( σ ) + X i =1 ˙ ψ β i (cid:0) A i S i + S i (cid:1)o δ ( x − x ∗ ) − εa ddτ ( ˙ ψ X i =1 A i β i ϕ i + 2 ˙ ψ β G G (cid:0) χ λ , ( σ ) + 1 β λ (1)2 , ( σ ) (cid:1) (50)11 X i =1 ˙ ψ β i χ i (cid:0) A i S i + S i (cid:1)) δ ′ ( x − x ∗ ) − εa X i =1 V i A i β i δ ′ ( x − ϕ i ) + O D ′ ( ε ) . Calculating weak expansions for the other terms from the left-hand sides of(28), (29) and substituting them into (28), (29) we obtain linear combinationsof δ ′ ( x − ϕ i ), i = 1 , δ ( x − x ∗ ), and δ ′ ( x − x ∗ ). Therefore, we pass to thefollowing system of 8 equations for 8 unknowns: P i,j ( A i , β i , V i ) = 0 , j = 1 , , i = 1 , , (51) ddτ Q j ( S , S , σ ) = 0 , j = 1 , , (52) dϕ j dτ = R j ( S , S , σ ) , j = 1 , . (53)The first four algebraic equations (51) imply again the relations (4) between A i , β i , and V i . Furthermore, functional equations (52) allow to define S i withthe property (37), whereas an analysis of the ODE (53) justifies the existenceof the required phase corrections ϕ i with the property (38). By analogy with(15), (19) we call (51)-(53) the Hugoniot-type conditions again. Results ofFig. 1: Evolution of two solitary waves for ε = 0 . .4 The next problem: three wave interaction Further numerical investigation of the GKdV-4 equation showed that N soli-tary waves collide elastically (in the leading term) at least for N ≤
5, see[29, 30] and Figures 2–4. Thus, there appears the problem of describing theFig. 2: Evolution of the soliton triplet for ε = 0 . ∂ u∂ t + ∂∂ x n u + ε ∂ u∂ x o = 0 , (54) ∂ u ∂ t + ∂∂ x n u − ε (cid:16) ∂u∂ x (cid:17) + ε ∂ u ∂ x o = 0 . (55)Comparing the left-hand sides of (54), (55) with (28), (29) we conclude thatDefinition 1 calls a function to be a ”weak asymptotic solution” if it satisfiesthe conservation laws (54), (55) in the sense O D ′ ( ε ). Next note that pertur-bations of (1) imply corresponding perturbations of the conservation laws.For (11), instead of (54), (55), we have the relations ∂ u∂ t + ∂∂ x n u + ε ∂ u∂ x o = R, (56)13ig. 3: Evolution of 4 solitons for ε = 0 . ∂ u ∂ t + ∂∂ x n u − ε (cid:16) ∂u∂ x (cid:17) + ε ∂ u ∂ x o = 2 Ru. (57)Reverting to the single-phase asymptotic solution (12) one can easily es-tablish that the orthogonality condition (17), and therefore the equation (19),is the integral form of (57), calculated for (12) with the accuracy O D ′ ( ε ).At the same time (18), and thus the equality (21), is the integral form of(56) mod O D ′ ( ε ), calculated for (12), where Y has the form (23) and u − satisfies the equation (24). Note also that the second ”conservation law”Fig. 4: Evolution of 5 solitons for ε = 0 . Y . 14herefore, we see that to define the principal asymptotics term there hasbeen used only one conservation law for the single-phase solution, and twoconservation laws for the two-phase solution. So it is natural to assume thatto construct a three-phase asymptotics we should add to (54), (55) the thirdconservation law, namely ∂∂ t n(cid:16) ε ∂ u∂ x (cid:17) − u o − ∂∂ x n ε ∂ u∂ t ∂ u∂ x + (cid:16) u + ε ∂ u∂ x (cid:17) o = 0 . (58) Let us consider the equation (1) with the Cauchy data u | t =0 = X i =1 A i ω (cid:18) β i x − x i ε (cid:19) , (59)where A < A < A , x >> x >> x .The arguments considered in the previous subsection imply the following Definition 3.
A sequence u ( t, x, ε ) , belonging to C ∞ (0 , T ; C ∞ ( R x )) for ε =const > and belonging to C (0 , T ; D ′ ( R x )) uniformly in ε , is called the weakasymptotic mod O D ′ ( ε ) solution of the problem (1), (59) if the relations (28),(29), and ∂∂ t n Z ∞−∞ (cid:16) ε ∂ u∂ x (cid:17) ψdx − Z ∞−∞ u ψdx o (60)+ Z ∞−∞ n ε ∂ u∂ t ∂ u∂ x + u + (cid:16) ε ∂ u∂ x (cid:17) + 2 u ε ∂ u∂ x o ∂ψ∂ x dx = O ( ε ) hold uniformly in t for any test function ψ = ψ ( x ) ∈ D ( R ) . To construct the asymptotic solution we write the ansatz in the formsimilar to (35), namely u = X i =1 G i ω (cid:18) β i x − ϕ i ε (cid:19) , (61)where the same notation and hypothesis (36)-(38) are assumed with obviouscorrections: the ”fast time” is defined now using the distance between thefirst and third trajectories, τ = β (cid:0) ϕ ( t ) − ϕ ( t ) (cid:1) /ε, (62)15nd we suppose the intersection of all trajectories x = ϕ i ( t ), i = 1 , ,
3, atthe same point ( x ∗ , t ∗ ).The technic of our approach has been explained in Subsection 1.3. So weclarify here some new detail only using as an example u m . All others explicitformulas for the asymptotic expansions are presented in the Appendix. Lemma 1.
Let the assumptions (37), (38) for u of the form (61) be satisfied.Then the following asymptotic expansions hold: u m = εa m n X i =1 K ( m ) i δ ( x − ϕ i ) + R m δ ( x − x ∗ ) o + O D ′ ( ε ) , (63) ∂ u m ∂ t = a m ˙ ψ d R m dτ δ ( x − x ∗ ) − εa m X i =1 V i K ( m ) i δ ′ ( x − ϕ i ) − εa m ˙ ψ ddτ n X i =1 K ( m ) i ϕ i + R (1) m o δ ′ ( x − x ∗ ) + O D ′ ( ε ) , (64) where m ≥ , ˙ ψ = β ( V − V ) , K ( m ) i = G mi β i , K ( m ) i = A mi β i , K ( m ) i = K ( m ) i − K ( m ) i , (65) R m = X i =1 K ( m ) i + X l,n R m,ln + R m, , (66) R (1) m = X i =1 χ i K ( m ) i + X l,n ( χ n R m,ln + C m,ln ) + χ R m, + C m, , (67) R m,ln = β l m − X k =1 C km K ( m − k ) l K ( k ) n λ m,k ( σ ln ) , (68) R m, = β β m − X j =2 j − X k =1 C jm C kj K ( m − j )1 K ( j − k )2 K ( k )3 λ (0) , ( j,k ) m, , (69) C m,ln = m − X k =1 C km θ ln K ( m − k ) l K ( k ) n λ (1) m,k ( σ ln ) , (70) C m, = β θ m − X j =2 j − X k =1 C jm C kj K ( m − j )1 K ( j − k )2 K ( k )3 λ (1) , ( j,k ) m, , (71)16 km are the binomial coefficients, the notation (47) has been used, and X l,n f ln def = f + f + f , σ ln = β l ϕ l − ϕ n ε , χ i = V i ˙ ψ τ + ϕ i . (72) Furthermore, λ ( i ) , ( j,k ) m, = 1 a m Z ∞−∞ η i ω m − j ( η ) ω j − k ( η ) ω k ( η ) dη, i = 0 or i = 1 . (73)To prove the lemma let us separate all the terms of u m into three groups:one-phase, two-phase and three-phase functions: u m = X i =1 Y mi + X l,n m − X k =1 C km Y m − kl Y kn + m − X j =1 j − X k =1 C jm C kj Y m − j Y j − k Y k , (74)where Y i = G i ω (cid:0) β i ( x − ϕ i ) /ε (cid:1) . Now considering u m in the weak sense wechange the variable: x = ϕ i + εη/β i , x = ϕ n + εη/β n , and x = ϕ + εη/β re-spectively for the integrals of the groups. Next, preparing the same transfor-mations as in Subsection 1.3 and applying (44) again we pass to the formula(63).In the same manner one can prove the following proposition: Lemma 2.
Let the assumptions (37), (38) be satisfied for u of the form (61).Then the following asymptotic expansions hold: ( εu x ) = εa ′ n X i =1 β i K (2) i δ ( x − ϕ i ) + R (1) , δ ( x − x ∗ ) o + O D ′ ( ε ) , (75) ∂∂ t ( εu x ) = a ′ ˙ ψ d R (1) , dτ δ ( x − x ∗ ) − εa ′ X i =1 β i V i K (2) i δ ′ ( x − ϕ i ) − εa ′ ˙ ψ ddτ n X i =1 β i K (2) i ϕ i + R (1)(1) , o δ ′ ( x − x ∗ ) + O D ′ ( ε ) , (76)( ε u xx ) = εa ′′ n X i =1 β i K (2) i δ ( x − ϕ i ) + R (2) , δ ( x − x ∗ ) o + O D ′ ( ε ) , (77) ε u u xx = εa n − X i =1 β i K (5) i δ ( x − ϕ i ) + L δ ( x − x ∗ ) o + O D ′ ( ε ) , (78)17 u x u t = − εa ′ X i =1 β i V i K (2) i δ ( x − ϕ i ) − εa ′ P δ ( x − x ∗ ) + O D ′ ( ε ) , (79) where a ′ , ˙ ψ , and K (2) i are defined in (5), (65), P = ˙ ψ ( S + S G ) + M , S , S G , M , and other notation are deciphered in Attachment, Subsections 6.1and 6.2. Now we substitute the expansions (63), (64), and (75)-(79) into (28), (29),(60) and obtain the similar (51)-(53) system. Namely, the algebraic systemfor each i = 1 , , − V i K (1) i + a K (4) i = 0 , (80) − a V i K (2) i + 85 a K (5) i − a ′ β i K (2) i = 0 , (81) − a ′ β i V i K (2) i + 25 a V i K (5) i + 2 a ′ β i V i K (2) i − a K (8) i + 8 a β i K (5) i − a ′′ β i K (2) i = 0 , (82)the system of functional equations: X i =1 K (1) i = 0 , (83) R = 0 , (84) a ′ R (1) , − a R = 0 , (85)and the system of ordinary differential equations: − ˙ ψ ddτ n X i =1 K (1) i ϕ i + χ i K (1) i o + a R = 0 , (86) − a ˙ ψ ddτ n X i =1 K (2) i ϕ i + R (1)2 o + 85 a R − a ′ R (1) , = 0 , (87)˙ ψ ddτ n − a ′ (cid:16) X i =1 β i K (2) i ϕ i + R (1)(1) , (cid:17) + 25 a (cid:16) X i =1 K (5) i ϕ i + R (1)5 (cid:17)o + 2 a ′ P − a R − a L − a ′′ R (2) , = 0 . (88)Let us overcome the first obstacle: for each i the system (80)-(82) of threeequation contains only two free parameters A i , V i .18 emma 3. Let ω ( η ) , A i = A ( β i ) , and V i = V ( β i ) be of the form (2)-(6).Then the equalities (80)-(82) are satisfied uniformly in β i > .Proof. Obviously, equations (80), (81) coincide with (51) and imply againthe formulas (2)-(6). Substituting them into (82), we transform it to thefollowing form: a a − a − a ′′ γ + 8 a γ = 0 . (89)Next we note that ω ( η ) satisfies the model equation γ d ωdη = a ω − ω . (90)Multiplying (90) for ω ′′ and integrating, we obtain the identity4 a = γ a ′′ + a a ′ . (91)On the other hand, integrating the squares of the left-hand and right-handparts of (90), we pass to another identity: a = γ a ′′ − a a + 2 a a . (92)This and (6) verify the equality (89).Since the system of six equations (83)-(88) contains six free functions, weobtain the first formal result Theorem 1.
Let the system (83)-(88) have a solution which satisfies theassumptions of the form (37), (38). Then the solitary waves (61) collidepreserving mod O D ′ ( ε ) the KdV-type scenario of interaction. Moreover, similar to the Rankine-Hugoniot condition, which is simply theconservation law for the shock-wave solution, the Hugoniot-type conditions(80)-(88) imply the verification of some conservation laws:
Theorem 2.
Let the assumptions of Theorem 1 be satisfied. Then the ansatz(61) is a mod O D ′ ( ε ) asymptotic solution of the equation (1) if and onlyif (61) satisfies the conservation laws ddt Z ∞−∞ udx = 0 , ddt Z ∞−∞ u dx = 0 , ddt Z ∞−∞ n(cid:16) ε ∂ u∂ x (cid:17) − u o dx = 0 , (93)19 nd the energy relations ddt Z ∞−∞ xudx − Z ∞−∞ u dx = 0 ,ddt Z ∞−∞ xu dx − Z ∞−∞ u dx + 3 Z ∞−∞ (cid:18) ε ∂ u∂ x (cid:19) = 0 , (94) ddt (cid:26)Z ∞−∞ x (cid:16) ε ∂ u∂ x (cid:17) dx − Z ∞−∞ xu dx (cid:27) + 2 ε Z ∞−∞ ∂ u∂ t ∂ u∂ x dx + Z ∞−∞ (cid:18) u + ε ∂ u∂ x (cid:19) dx = 0 . To prove this conclusion it is enough to rewrite the equations (80)-(88)in the integral form.
Let us pay the attention to the equations (83)-(85). Normalization κ i = γβ / S i /β i (95)implies K ( m ) i = β m/ − γ m θ m/ − i Λ mi , where Λ i = 1 + θ / i κ i . (96)We denote R m = γ m R m /β m/ − and obtain: R m = X i =1 θ m/ − i (Λ mi −
1) + X l,n R m,ln + R m, (97) R m,ln = m − X k =1 C km θ m − k ) / l θ k/ − n Λ m − kl Λ kn λ m,k ( σ ln ) , (98) R m, = m − X j =2 j − X k =1 C jm C kj θ m − j ) / θ j − k ) / Λ m − j Λ j − k Λ k λ (0) , ( j,k ) m, . (99)20his and similar formulas for R (1) , = β / R (1) , /γ (see Attachment) allowus to transform (83)-(85) to the following form: X i =1 κ i = 0 , (100) X i =1 θ / i (Λ i −
1) + 2 X l,n θ / l θ / ln Λ l Λ n λ , ( σ ln ) = 0 , (101) X i =1 θ / i (Λ i −
1) + 2 X l,n θ / l θ / n Λ l Λ n λ (0) I ( σ ln ) − R = 0 , (102)where the equalities (6), the notation (97)-(99), and (140) have been used.Next let us simplify the equations (86)-(88). We note firstly that in viewof (83) and the identity β l ( χ l − χ n ) = σ ln (103)it is possible to eliminate χ i from the left-hand side of (86), since X i =1 K (1) i ϕ i + χ i K (1) i = X i =1 K (1) i ϕ i + σ β K (1)11 − σ β K (1)31 . In the same manner, applying (84) and (85), we simplify the equations (87),(88). Thus, we transform (86)-(88) to the following form:˙ ψ ddτ n X i =1 K (1) i ϕ i + σ β K (1)11 − σ β K (1)31 o = f, (104)˙ ψ ddτ n X i =1 K (2) i ϕ i + X l,n C ,ln + σ β K (2)11 − σ β (cid:0) K (2)31 + R , + R , (cid:1)o = F, (105)˙ ψ ddτ n X i =1 (cid:16) β i K (2) i − γ K (5) i (cid:17) ϕ i + K o − S = F , (106)where f = a R , F = a ′ a R (1) , , F = 2 M − a a ′ R − a a ′ L − a ′′ a ′ R (2) , , (107)21nd the function K is described in Attachment (see formula (142)).The second step is the elimination of ϕ i from the model system. To doit we divide σ ln into the growing ( σ ln ) and the bounded (˜ σ ln ) parts: σ ln = σ ln + ˜ σ ln , σ ln def = β l ˙ ψ ( V l − V n ) τ (108)and rewrite the identity (103):˜ σ ln = β l ( ϕ l − ϕ n ) . (109)Thus ϕ = ˜ σ β + ϕ , ϕ = − ˜ σ β + ϕ . (110)Substituting (110) into (104) we obtain˙ ψ dϕ dτ = − ˙ ψ r ddτ n ˜ σ β K (1)10 + σ β K (1)11 − ˜ σ β K (1)30 − σ β K (1)31 o + fr . (111)Here and in what follows we use the notation r j = X i =1 K ( j ) i for j = 1 and j = 2 . (112)Next we use the equalities (110), (111), and˜ σ = ˜ σ + θ ˜ σ , σ = σ + θ σ , (113)and rewrite (105), (106) as equations for new unknowns ˜ σ , ˜ σ . Afternormalization (95) we pass to the following model equations:˙ ψ ddτ n p ˜ σ β + p σ β − p ˜ σ β − p σ β + X l,n C ,ln o = F − r r f, (114)˙ ψ ( ddτ n e ˜ σ β + e σ β o + 2Ψ ddτ n K (1)11 σ β o − ddτ n e ˜ σ β + e σ β o − ddτ n K (1)31 σ β o + r d K C dτ − r S G ) = F , (115)where F = r F + (2Ψ + P i =1 q (2) i ) f and the coefficients p ( k ) i , e ki , q ( m ) ik , Ψ arepresented in Attachment (see formulas (144) - (148)).22 .2 Asymptotic analysis To simplify the further analysis let us assume that θ / = µ, θ / = µ α , where α ∈ [0 ,
1) and µ is sufficiently small . (116)We look for the asymptotic solution of the system (100)-(102) in the form: κ = 12 µ α ( y − µ − α x ) , κ = − µ α ( y + µ − α x ) , κ = µ x , (117)where x and y are free functions. Then (100) is satisfied, whereas (101)and (102) imply the system:2 x − µ α y n µλ , ( σ ) − µ α (1 − λ , ( σ ) + 14 y ) − µ α ) / λ , ( σ ) o = − λ , ( σ ) − µ α λ , ( σ ) − µ α λ , ( σ ) + O λ ( µ ) , (118) n µ λ , ( σ ) + 372 µ x o x − µ α y λ , ( σ )= − λ , ( σ ) − µ α λ , ( σ ) + O λ ( µ + µ (3+7 α ) / ) . (119)Here and in what follows we denote f ( σ, µ ) = O λ ( µ k ) if max σ | f ( σ, µ ) | ≤ cµ k (120)and f ( σ, · ) belongs to the Schwartz space.It is easy to see that the compatibility of the equations (118) and (119)requires the condition: 10 λ , ( σ ) = 7 λ , ( σ ) + O λ ( µ α ). Lemma 4.
Let ω ( η ) be of the form (2). Then λ , ( σ ln ) = 7 λ , ( σ ln ) + 3 θ ln λ (0) I ( σ ln ) (121) for all indices l, n . To prove the lemma it is enough to use again the equation (68) and theidentities (5), (6), (67).Now we set x = − λ , ( σ ) + µ α x (122)and transform (118), (119) to the final form:2 x − r y = − λ , ( σ ) − µλ , ( σ ) + O λ ( µ − α ) , (123)23 x − µλ , ( σ ) y = − µλ , ( σ ) − µ − α λ (0) I ( σ ) + O λ ( µ ) , (124)where r = 1 + µλ , ( σ ) − µ α (cid:0) − λ , ( σ ) + 14 y (cid:1) + O λ ( µ α ) / ) ,r = 1 + 194 µ λ , ( σ ) + O λ ( µ α ) . Solving this system we obtain the asymptotic representation: x = − µ (cid:0) λ , ( σ ) − λ , ( σ ) λ , ( σ ) (cid:1) + O λ ( µ α ) , (125) y = 2 λ , ( σ ) (cid:16) µ α (cid:0) − λ , ( σ ) (cid:1)(cid:17) + O λ ( µ − α ) . (126)Combining (117), (122), (125), and (126) we conclude: Lemma 5.
Let there exist functions ϕ i , i = 1 , , , with the properties (38)and let the condition (116) be realized. Then the system (100) - (102) hasthe unique solution κ = µ α λ , ( σ ) (cid:8) µ α (cid:0) − λ , ( σ ) (cid:1)(cid:9) + O λ ( µ ) , (127) κ = − µ α λ , ( σ ) (cid:8) µ α (cid:0) − λ , ( σ ) (cid:1)(cid:9) + O λ ( µ ) , (128) κ = − µ λ , ( σ ) + O λ ( µ α ) , (129) such that S i = β i κ i /γβ / satisfy the assumptions (37). To complete the analysis we should prove the solvability of the system(114), (115). Taking into account (95), (116), and (127) - (128), we obtain:˜ E d ˜ σ dτ − θ ˜ E d ˜ σ dτ = ˜ F , (130)˜ E d ˜ σ dτ − θ ˜ E d ˜ σ dτ = ˜ F , (131)where the coefficients ˜ E ij and right-hand sides ˜ F i are demonstrated in At-tachment, Subsection 6.4.It is easy to calculate thatdet( ˜ E ij ) = θ µ ∆ , (132)24here ∆ = 73 + O ( µ (1+3 α ) / ) + O λ ( µ (3+ α ) / + µ ) . Thus, we transform the system (130), (131) to the standard form d ˜ σ dτ = ˜ M ( τ, σ , σ , µ ) / ∆ , d ˜ σ dτ = ˜ M ( τ, σ , σ , µ ) / ∆ , (133)where ˜ M = − µz ′ ( σ ) + O λ ( µ (3+ α ) / ) , z ( σ ) def = σλ , ( σ ) , (134)˜ M = − µ − α λ , ( σ ) − z ′ ( σ ) + O λ ( µ (1 − α ) / + µ α ) , (135)and the equalities (6), (91), (92), as well as the functional relation (121) and a λ , ( σ ln ) = a a λ , ( σ ln ) − γ a λ , ( σ ln ) + θ ln γ a a ′ λ (0) I ( σ ln ) (136)have been taken into account.According to the notation (108) and the first assumption of the form (38)we add to (133) the ”initial” condition:˜ σ (cid:12)(cid:12) τ →−∞ → , ˜ σ (cid:12)(cid:12) τ →−∞ → . (137)Since ˜ M ij vanish with an exponential rate as τ → ±∞ , it is easy to provethe solvability of the problem (133), (137). Next we note that λ , ( σ ln ) = λ , ( ˙ σ ln τ + ˜ σ ln ). Since ˙ σ = O ( µ − α ) / ) we find from (133), (135) that˜ σ ( τ ) = O ( µ (3+ α ) / ) for sufficiently large τ , however it tends to the limitingvalue sufficiently slowly, with an exponent O ( µ (3+ α ) / ). Conversely, takinginto account that ˙ σ = O ( µ ), we obtain that ˜ σ ( τ ) = O (1) for sufficientlylarge τ and tends to the limit with an exponent O (1).The last step of the construction is the return to the phase corrections ϕ i . In view of (110), (111) it is obvious that the last assumption of the form(38) is justified. This implies our main proposition Theorem 3.
Under the assumption (116) the asymptotic solution (61) de-scribes mod O D ′ ( ε ) the KdV-type scenario of the solitary waves interac-tion. Conclusion
We looked for an approach to describe solitary wave collisions avoiding theuse of explicit multi-soliton formulas. Surprisingly, we came back to the an-cient Whitham’s idea to construct asymptotics with the help of conservationlaws and a reasonable ansatz, but in the framework of the weak asymp-totics method. In our case three conservation laws for three waves have beenutilized. It is clear now how to generalize the approach: for N waves N con-servation laws should be used. On contrary, the existence of N conservationlaws does not imply the existence of N -soliton type solution since some veryastonishing additional conditions appear to guarantee both the solvability ofmodel equations (like (121)) and the regularity of the solutions (like (136)).Furthermore, some questions remain open, the first of them: how to choosethe collection of conservation laws to describe N -soliton interaction and isit possible to change conservation laws to reasonable energy relations? Atthe same time we can formulate the main result of the paper: there is not asharp frontier between integrable and nonintegrable equations: similar sce-narios of the soliton interaction are realized, but with small corrections inthe nonintegrable case. The research was supported by SEP-CONACYT under grant 178690 (Mex-ico).
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( εu x ) and ( ε u xx ) R ( k ) , = X i =1 β ki K (2) i + X l,n R ( k )2 ,ln , R ( k )2 ,ln = 2 β kl β kn K (1) l K (1) n λ (0) I,k ( σ ln ) , (138) R (1)(2) , = X i =1 β i χ i K (2) i + X l,n ( χ n R (1)2 ,ln + C (1)2 ,ln ) , (139) C (1)2 ,ln = 2 β l K (1) l K (1) n λ (1) I, ( σ ln ) , λ ( i ) I, ( σ ln ) = 1 a ′ Z ∞−∞ η i ω ′ ( η ln ) ω ′ ( η ) dη, (140) λ I, ( σ ln ) = 1 a ′′ Z ∞−∞ ω ′′ ( η ln ) ω ′′ ( η ) dη, a ′′ = Z ∞−∞ (cid:16) ω ′′ ( η ) (cid:17) dη, (141)and ω ′ ( η ) = dω ( η ) /dη , ω ′′ ( η ) = d ω ( η ) /dη .29 .2 Formulas for ε u x u t and ε u u xx S = X i =1 β i K (2) i dϕ i dτ + X l,n β l β n K (1) l K (1) n (cid:16) dϕ l dτ + dϕ n dτ (cid:17) λ (0) I, ( σ ln ) , M = X i =1 β i V i K (2) i + X l,n M ln , M ln = β l β n K (1) l K (1) n (cid:0) V l + V n (cid:1) λ (0) I, ( σ ln ) , S G = X l,n (cid:16) G l dG n dτ − G n dG l dτ (cid:17) λ I ( σ ln ) , L = − X i =1 β i K (5) i + X l,n L ln + Q , L ln = X j =1 C j β l K (5 − j ) l K ( j ) n λ ,j (1) ( σ ln ) + X j =0 C j β l β n K (4 − j ) l K ( j +1) n λ ,j (2) ( σ ln ) , Q = X j =1 C j n β β β K (4 − j )1 K ( j )2 K (1)3 λ ,j (1) + β β K (4 − j )1 K (1)2 K ( j )3 λ ,j (2) + β β K (1)1 K (4 − j )2 K ( j )3 λ ,j (3) o + X j =2 j − X k =1 C j C kj β β K (4 − j )1 K ( j − k )2 K ( k )3 3 X m =1 β m K (1) m λ ,jkm ,λ I ( σ ln ) = 1 a ′ Z ∞−∞ ω ( η ln ) ω ′ ( η ) dη, a = Z ∞−∞ ω ( η ) (cid:16) ω ′ ( η ) (cid:17) dη,λ ,j (1) ( σ ln ) = 1 a Z ∞−∞ ω − j ( η ln ) ω j ( η ) ω ′′ ( η ln ) dη,λ ,j (2) ( σ ln ) = 1 a Z ∞−∞ ω − j ( η ln ) ω j ( η ) ω ′′ ( η ) dη,λ ,j (1) = 1 a Z ∞−∞ ω − j ( η ) ω j ( η ) ω ′′ ( η ) dη,λ ,j (2) ) = 1 a Z ∞−∞ ω − j ( η ) ω ′′ ( η ) ω j ( η ) dη,λ ,j (3) = 1 a Z ∞−∞ ω ′′ ( η ) ω − j ( η ) ω j ( η ) dη,λ ,jkm = 1 a Z ∞−∞ ω − j ( η ) ω j − k ( η ) ω k ( η ) ω ′′ ( η m ) dη. .3 Normalization R (1) , = X i =1 θ / i (Λ i −
1) + 2 X l,n θ / l θ / n Λ l Λ n λ (0) I, ( σ ln ) , K = σ β q (1)11 − σ β (cid:0) q (1)31 + Q (cid:1) + K C , q ( m ) ik = β i K (2) ik + ( − m γ K (5) ik , (142) K C = X l,n (cid:16) C (1)2 ,ln − γ C ,ln (cid:17) − γ C , , (143) Q = X j =1 (cid:16) R (1)2 ,j − γ R ,j (cid:17) − γ R , , (144) p ik = K (2) ik − r r K (1) ik + R ( k ) i , R (1)3 = X j =1 R ,j , R ( k ) i = 0 if i = 3 , k = 1 , (145) e i = − r q (2) i + K (1) i X j =1 q (2) j − r ( β i K (2) i + ζ + ζ i ) + 2 K (1) i Ψ , (146) e i = r ( q (1) i + Q i ) + K (1) i X j =1 q (2) j , Ψ = X i =1 β i K (2) i + 2 X ln ζ ln , (147) ζ ln = β l β n K (1) l K (1) n λ (0) I ( σ ln ) , ζ = ζ , ζ = ζ , Q = 0 . (148) ˜ E = − r + r θ / − r µ α ) / z ′ ( σ ) − r µ α ) / λ ′ ( σ )+ 2 r µ α (cid:16)(cid:0) σλ ( σ + θ σ ) (cid:1) ′ σ (cid:12)(cid:12) σ = σ − Λ z ′ ( σ ) (cid:17) + O λ ( µ α ) , ˜ E = r − r θ / − r µ α (cid:0) λ ( σ ) z ′ ( σ ) − Λ z ′ ( σ ) (cid:1) + r µ (7+3 α ) / z ′ ( σ ) + O λ ( µ ) , r = X i =1 θ / i , r = X i =1 θ / i ˜ E = 73 + 73 µ α ) / z ′ ( σ ) − µ λ ( σ ) + O λ ( µ α ) , ˜ E = − r + 73 µ (3+ α ) / + 4 µ (cid:0) λ ( σ ) − z ′ ( σ ) (cid:1) + O λ ( µ α ) , ˜ F = − µ n λ ( σ ) (cid:0) µ (1+ α ) / (cid:1) + µ (3+ α ) / (cid:0) λ ( σ )31 λ ( σ ) (cid:1) + O λ ( µ ) o , ˜ F = 2 µ n λ ( σ ) − µ α (cid:0) λ ( σ ) − λ ( σ ) λ ( σ ) (cid:1) + 283 µ (3+ α ) / (cid:0) λ ( σ ) − λ ( σ ) (cid:1) + O λ ( µ ) o ..