On asymptotic solutions of integrable wave equations

27 August 2001

Physics Letters A 287 (2001) 223–232 www.elsevier.com/locate/pla

On asymptotic solutions of integrable wave equations A.M. Kamchatnov a,b,∗ , R.A. Kraenkel b , B.A. Umarov b,c a Institute of Spectroscopy, Russian Academy of Sciences, Troitsk 142190, Moscow Region, Russia b Instituto de Física Teórica, Universidade Estadual Paulista — UNESP, Rua Pamplona 145, 01405-900 São Paulo, Brazil c Physical-Technical Institute, Uzbek Academy of Sciences, G. Mavlyanov str., 2-b, 700084 Tashkent-84, Uzbekistan

Received 12 July 2001; accepted 17 July 2001 Communicated by V.M. Agranovich

Abstract Asymptotic ‘soliton train’ solutions of integrable wave equations described by inverse scattering transform method with second-order scalar eigenvalue problem are considered. It is shown that if asymptotic solution can be presented as a modulated one-phase nonlinear periodic wavetrain, then the corresponding Baker–Akhiezer function transforms into quasiclassical eigenfunction of the linear spectral problem in weak dispersion limit for initially smooth pulses. In this quasiclassical limit the corresponding eigenvalues can be calculated with the use of the Bohr–Sommerfeld quantization rule. The asymptotic distributions of solitons parameters obtained in this way specify the solution of the Whitham equations.  2001 Elsevier Science B.V. All rights reserved.

1. Introduction and background material It is well known that for many evolutional equations an initially large and smooth pulse evolves eventually into a train of solitons. In many physical applications it is important to know the distribution functions of the parameters of solitons representing this final state of evolution. For the case of the KdV equation ut + 6uux + ε2 uxxx = 0

(1)

this problem was solved long ago by Karpman [1,2] (see also [3,4]). The solution was based on the fact that the parameters λn = −an which characterize the soliton solutions of Eq. (1), us (x, t) =

−2λn 2an = , √ √ cosh2 [ −λn (x + 4λn t)/ε] cosh2 [ an (x − 4an t)/ε]

coincide with eigenvalues of the spectral problem

 ε2 ψxx = − u(x, t) + λ ψ = − u(x, t) − a ψ, * Corresponding author.

E-mail address: [email protected] (A.M. Kamchatnov). 0375-9601/01/$ – see front matter  2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 1 ) 0 0 4 7 8 - 9

(2)

(3)

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and the spectrum of Eq. (3) does not change during the KdV evolution of the potential u(x, t) and can therefore be calculated from the initial distribution u0 (x) = u(x, 0). The isospectrality of the KdV evolution corresponds to the time dependence of ψ(x, t) governed by the equation ψt = ux ψ + (4λ − 2u)ψx ,

(4)

so that Eq. (1) becomes a compatibility condition (ψxx )t = (ψt )xx of two linear equations (3) and (4). In the limit of small ε one can apply to Eq. (3) the WKB method and find the eigenvalues λn = −an from the Bohr–Sommerfeld quantization rule     1  1  1 , n = 0, 1, 2, . . . , N, (5) u0 (x) + λn dx = u0 (x) − an dx = 2π n + ε ε 2 where the integral is taken over the cycle around two turning points corresponding to the eigenvalue λn = −an . The total number of solitons N corresponds to the minimum value of an . For initial pulses u0 (x) such that u0 (x) → 0 as |x| → ∞, the number of solitons can be estimated as 1 N πε

∞ 

u0 (x) dx.

(6)

−∞

The maximal amplitude a is equal approximately to the maximum value um of the potential u0 (x) so that amplitudes of solitons are distributed inside the interval
 0 < a < um = max u0 (x) . (7) Differentiation of Eq. (5) with respect to a gives the number of solitons with amplitudes in the interval (a, a + da),    1 dx dn = (8) √ da ≡ f (a) da. 4πε u0 (x) − a To find how solitons are distributed along the space coordinate and how this distribution evolves with time, we suppose that for asymptotically large values of time the wavetrain can be represented as a modulated periodic wave with slowly changing wavelength L, wavenumber k = 2π/L, amplitude a, and frequency ω = kV , where V is a phase velocity of the wave. Since in slowly modulated periodic wave the wavenumber and the frequency can be presented as derivatives of the phase θ (x, t) (see [3,4]), ω = −θt ,

k = θx ,

they are connected by the relation of conservation of ‘number of waves’, kt + ωx = 0.

(9)

In the asymptotic limit, when solitons are well separated from each other, the phase velocity is equal to velocity of solitons, V = 4a, and we arrive at the equation for the parameters k and a: kt + 4(ka)x = 0.

(10)

To obtain another equation, we represent the KdV equation (1) as a conservation law, 
ut + 3u2 + ε2 uxx x = 0,

(11)

and average the density and the flux over the wavelength. Again in the asymptotic limit with well separated solitons the full derivative term ε2 uxx can be omitted and the other terms can be estimated with the aid of Eq. (2) as ∞

u=k −∞

us (x) dx = 4ka

1/2

,

u 2 = k

∞

−∞

u2s (x) dx =

16 3/2 ka , 3

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225

whence Eq. (11) transforms into the Whitham equation 

1/2 + 4 ka 3/2 x = 0. ka t

(12)

From Eqs. (11) and (12) we find the equation for the amplitude,
1/2
 a + 4a a 1/2 x = 0, t

(13)

with the well-known solution x − 4at = F (a),

(14)

where F (a) is an arbitrary function to be found from the initial conditions. However, at asymptotically large values of time the solitons propagate independently from each other with velocities determined by their amplitudes and differences in their ‘initial positions’ can be neglected, i.e., x  4at. Thus, we obtain the asymptotic solution a(x, t) = x/4t,

0 < x < 4um t,

(15)

where we have used condition (7). The number of solitons in the interval (x, x + dx) is given by dx/L = (k/2π) dx = f (a) da = f (x/4t)(dx/4t), and hence the distribution function equals to k(x, t)/2π = (1/4t)f (x/4t),

(16)

where f (a) is defined by Eq. (8). The same expression (16) for k(x, t) with arbitrary function f can be obtained as a solution of Eq. (10) with a(x, t) given by Eq. (15), that is, Eq. (8) specifies solution (15), (16) of the Whitham equations (10), (12). The aim of this Letter is to generalize the above theory on a wider class of nonlinear completely integrable equations.

2. Connection of the Baker–Akhiezer function with the WKB function The starting point for the generalization we are looking for is an observation [5] that the WKB function can be obtained as a certain limit of the Baker–Akhiezer function which is a solution of the associated linear spectral problem with periodic potential. From more physical point of view it means that since, on the one hand, the Bohr– Sommerfeld quantization rule (5) corresponds to the quasiclassical form  x i  ψWKB (x, λ) ∼ exp (17) u0 (x) + λ dx ε of the solution of Eq. (3), and, on the other hand, the asymptotic form of modulated wavetrain corresponds to the Baker–Akhiezer solution ψBA of Eq. (3) with slowly modulated periodic potential u(x, t), then these two forms of ψ, corresponding to different stages of evolution of u(x, t), should be connected with each other. To clarify the connection between ψWKB (x, λ) and ψBA (x, λ), let us recall that for the KdV equation case the Baker–Akhiezer function can be expressed in terms of the ‘squared basis function’ g = ψ+ ψ− ,

(18)

where ψ± are two basis solutions of Eq. (3). As follows from Eqs. (3) and (4), g(x, t) satisfies the equations ε2 gxxx + 2ux g + 4(u + λ)gx = 0,

(19)

gt = (4λ − 2u)gx + 2ux g.

(20)

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Upon multiplication by g, Eq. (19) can be integrated once to give   1 1 ε2 ggxx − gx2 + (u + λ)g 2 = P (λ), 2 4

(21)

where the integration constant P (λ) can only depend on λ. From Eq. (20) we find the relation [6]     1 4λ − 2u = g t g x

(22)

which can be considered as a generating function of conservation laws arising as coefficients in series expansion of Eq. (22) in powers of 1/λ. The periodic solutions of the KdV equation (1) are distinguished by the condition that P (λ) in Eq. (21) be a polynomial in λ, and the most important for physical applications one-phase periodic solution, which we are interested in, corresponds to the third-degree polynomial P (λ), P (λ) = λ3 − s1 λ2 + s2 λ − s3 ,

(23)

and the first-degree polynomial g(λ), g = λ − µ(x, t).

(24)

On substituting Eqs. (23) and (24) into (21) and comparing coefficients of

λ2 ,

we find

1 µ = (s1 + u), 2 and hence the polynomial P (λ) can be expressed in terms of the potential u(x, t):    1 1 2 ε2
P (λ) = λ − s1 − u (λ + u) + 2(s1 + u)uxx − u2x . 2 2 16 Now with the use of the well-known expression (see, e.g., [4]) for the Baker–Akhiezer function 1  x √ P (λ) i √ dx , ψBA = g exp ± ε g we obtain



i ψBA ∼ exp ± ε

x 

(λ − s1 /2 − u/2)2 (λ + u) + (ε2 /16)(2(s1 + u)uxx − u2x ) dx . λ − s1 /2 − u/2

(25)

(26)

(27)

(28)

For a strictly periodic solution u(x, t) the polynomial P (λ) is constant and its zeros λi are the parameters defining the solution. For a slowly modulated wave the zeros λi , i = 1, 2, 3, become slow functions of x and t, and for this reason we have inserted P (λ) under the integration sign. This corresponds to the well-known method of derivation of Whitham modulation equation [4,8,9]. We suppose that expression (28) is valid approximately for all moments of time including the initial state. Then, since the initial potential is supposed to be a smooth function of x, we can neglect its derivatives with respect to x in Eq. (28) (or take the limit ε → 0 in the expression under the integration sign) to obtain immediately the WKB expression (17). We want to show in this Letter that similar approach can be used for a wide class of evolution equations (see also [5]) what permits one to investigate asymptotic solutions of evolution equations in the way similar to that for the KdV equation case. 1 Note that this expression, as well as some other facts from the finite-gap integration method for the KdV equation, were discovered as

early as 1919 by J. Drach in his remarkable but forgotten papers [7]. We are grateful to Yu.V. Brezhnev for information about these papers.

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3. Asymptotic solutions of integrable wave equations corresponding to spectral problems with ‘energy-dependent potentials’ Many physically interesting integrable equations arise as compatibility conditions of the second-order scalar scattering problem ε2 ψxx = Aψ

(29)

and the corresponding equation for t-evolution of ψ(x, t), 1 ψt = − Bx ψ + Bψx . 2 It is easy to check by direct calculation that the function  x √  P (λ) i dx ψ = g(x, t) exp ± ε g(x, t) satisfies Eq. (29) provided the following relation fulfils:   1 2 2 1 ggxx − gx − Ag 2 = P (λ). ε 2 4

(30)

(31)

(32)

These relations generalize Eqs. (27) and (21). The squared basis function g satisfies also the equation gt = Bgx − Bx g,

(33)

analogous to Eq. (20), and the equation     1 B = g t g x

(34)

yields an infinite sequence of conservation laws. We shall consider here the case of ‘energy-dependent potential’ A = ±λM −

M−1

λr ur (x)

(35)

r=0

and confine ourselves to several examples with M = 2. This problem was considered for the first time by Kaup [10] and Jaulent and Miodek [11], and its connection with Zakharov–Shabat problem was established in [12]. Further generalizations and mathematical properties of arising hierarchies were considered in [13–17]. Periodic and soliton solutions of arising equations were obtained in [5,18–20]. 3.1. Kaup–Boussinesq system The Kaup–Boussinesq equations [10] 1 (36) ht + (hu)x − ε2 uxxx = 0 4 describe shallow water dispersive waves and can be represented as a compatibility condition of two linear equations   1 2 2 ε ψxx = − λ − u − h ψ, (37) 2   1 1 ψt = ux ψ − λ + u ψx , (38) 4 2 ut + uux + hx = 0,

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where A in Eq. (37) has the form of Eq. (35) with M = 2. Introducing the squared basis function g = ψ+ ψ− , we write Eq. (32) in the form [20]     1 2 1 2 2 1 ggxx − gx + λ − u − h g 2 = P (λ). ε 2 4 2 Now the one-phase periodic solution corresponds to the fourth-degree polynomial P (λ) = 4 k 4−k , s = 1, λ > λ > λ > λ , 0 1 2 3 4 k=0 (−1) sk λ g = λ − µ = λ − s1 /2 + u/2,

(39)

4

i=1 (λ

− λi ) =

4

(40)

and phase velocity of the periodic wave is equal to V = (1/2)s1 = (1/2) i=1 λi . The soliton limit corresponds to λ3 = λ2 , and if we choose u → 0 as |x| → ∞, then we must take λ4 = −λ1 , so that the soliton solution depends on two constants λ1 and λ2 : us (θ ) = −

2(λ21 − λ22 )  ,
 2λ1 cosh2 λ21 − λ22 θ − (λ1 + λ2 )

θ = x − λ2 t,

(41)

and h(θ ) can be expressed in terms of us (θ ): 2 1 1
hs (θ ) = λ21 + λ22 − us (θ ) − λ2 . (42) 2 2 The Baker–Akhiezer solution of Eq. (37) corresponding to nonlinear periodic wave solution u(x, t), h(x, t) of Eqs. (36) can be written in form (31), and supposing again that this form holds also for the modulated wave, we obtain for smooth enough initial pulses u0 (x) and h0 (x) the WKB solution   2 x  1 i ψWKB ∼ exp ± (43) λ − u0 (x) − h0 (x) dx . ε 2 (This formula follows also directly from the scattering problem (37).) In quasiclassical limit ε → 0, the eigenvalues can be obtained from the Bohr–Sommerfeld quantization rule,  2     1 1 1 (n) λ2 − u0 (x) − h0 (x) dx = 2π n + , n = 0, 1, 2, . . . , N, (44) ε 2 2 and they correspond to the values of the parameter λ2 in the soliton solution (41), (42) for solitons in the train arising eventually from the initial pulse. The parameter λ1 for all these soliton solutions is determined, according to Eq. (42), by constant value of h∞ ≡ h(x)||x|→∞ , namely, by the relation h∞ = λ21 . If initial velocity u0 (x) is equal to zero, then Eq. (44) reduces actually to the KdV equation case and determines (n) 2 2 values of (λ(n) 2 ) for dark initial pulses h∞ > h0 (x) > 0. The lowest value of (λ2 ) equals approximately to the minimal value of h0 (x), and the highest value equals approximately to h∞ . Thus we arrive at two symmetrical sets of λ2 : −λmax < λ2 < −λmin < 0 and 0 < λmin < λ2 < λmax ,

(45)

corresponding to two symmetrical wavetrains propagating in opposite directions. In case of nonzero initial velocity u0 (x) this symmetry is destroyed and we obtain two sets of eigenvalues λ˜ min < λ2 < λ˜ max

(46) and λmin < λ2 < λmax , √ where again we have estimates |λ˜ min |  λmax  h∞ , and λ˜ max and λmin depend on details of behaviour of two potentials h0 (x) and u0 (x). For large enough velocities u0 (x) one of sets (46) may disappear and then all solitons move in one direction.

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The eigenvalues λ = λ2 are equal to velocities of solitons and determine their amplitudes: u(n) s (0) = −2(λ1 − (n) (n) and hs (0) = λ1 (2λ2 − λ1 ) (see Eqs. (41), (42)). Differentiation of Eq. (44) with respect to λ = λ2 gives the number of solitons with velocities in the interval (λ, λ + dλ),    1 λ − u0 (x)  dn = (47) dx dλ ≡ f (λ) dλ. 2πε (λ − u0 (x)/2)2 − h0 (x) λ(n) 2 )

As in the KdV equation case (see Eq. (15)), we obtain the asymptotic distribution of separately moving solitons, λ(x, t) = x/t

for λ˜ min t < x < λ˜ max t

and λmin t < x < λmax t,

and the number of solitons in the interval (x, x + dx) is given by
 k(x, t)/2π dx = (1/t)f (x/t) dx

(48)

(49)

with f (λ) defined by Eq. (47). Thus, we have obtained a simple description of asymptotic solution of the Kaup–Boussinesq equations for initially large pulse consisting of ‘many’ solitons. 3.2. Nonlinear Schrödinger equation Usually NLS equation is studied in framework of the Zakharov–Shabat matrix scattering problem [21]. As was mentioned above, this problem is connected with the second-order scalar problem (29), and the NLS equation 1 iεut + ε2 uxx ∓ |u|2 u = 0, (50) 2 where upper and lower signs correspond to defocusing and focusing nonlinearities, respectively, can be represented as a compatibility condition of linear equations (29) and (30) with     iε ux 2 ε2 uxx u2x iε ux 2 − 2 , A=− λ− (51) ± |u| − B=λ+ 2 u 2 u u 2 u (see, e.g., [22]). Quasiclassical limit ε  1 of Eq. (50) is achieved by means of Madelung’s substitution  x  i u(x, t) = ρ(x, t) exp v(x  , t) dx  , ε so that NLS equation transforms into a system of equations for ρ(x, t) and v(x, t): 
vt + vvx ± ρx + ε2 ρx2 /8ρ 2 − ρxx /4ρ x = 0. ρt + (ρv)x = 0,

(52)

(53)

Thus, the initial pulse is given by distributions of intensity ρ0 (x) and ‘chirp’ v0 (x), whereas u(x, t) is a fast oscillating function, if v0 (x) = 0. We are interested in asymptotic solutions of this problem for smooth enough distributions ρ0 (x) and v0 (x). Again we suppose that the final state can be represented as a modulated soliton train. Then the squared basis function g satisfies the condition       1 iε ux 2 uxx u2x 1 1 − 2 g 2 = P (λ), ∓ |u|2 + ε2 ε2 ggxx − gx2 + λ − (54) 2 4 2 u 2 u u

where the one-phase solution corresponds to the fourth-degree polynomial P (λ) = 4i=1 (λ − λi ) = 4 k 4−k , s = 1, and 0 k=0 (−1) sk λ g = λ − s1 /2 + iεux /2u.

(55)

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Substitution of Eqs. (54) and (55) into (31) yields the expression for the Baker–Akhiezer function in terms of the complex ‘potential’ u(x, t) whose evolution is governed by the NLS equation (50). As before, we suppose that Eq. (31) holds for slowly modulated periodic wave and in the limit of small t transforms into the WKB function corresponding to smooth initial distributions ρ0 (x) and v0 (x). From Eq. (52) we find in the main approximation iεux /u  −v0 (x) and, hence, the WKB function takes the form   2 x  1 i ψWKB ∼ exp ± (56) λ + v0 ∓ ρ0 dx . ε 2 Here we have to distinguish defocusing and focusing cases and consider them separately. 3.2.1. Defocusing NLS equation In this case the spectrum is real and in our quasiclassical limit it is determined by the Bohr–Sommerfeld quantization rule   2    1 1 1 (57) , n = 0, 1, 2, . . . , N, λ(n) + v0 − ρ0 dx = 2π n + ε 2 2 where the integration is taken over the cycle between two real turning points x± corresponding to the eigenvalue λ(n) . This equation coincides actually with Eq. (44) for the Kaup–Boussinesq system and leads to similar results. 3.2.2. Focusing NLS equation Now there is no real turning points in the integrand of Eq. (56) and the spectrum becomes complex. Nevertheless, it can be calculated again by means of the complex WKB method [23] for initial distributions ρ0 (x) and v0 (x) given by analytical functions which can be continued analytically into the complex x plane. Such analysis has been done in the recent paper by Miller [24] for a particular case ρ0 (x) = 1/ cosh2 (2x), v0 (x) = −2 sinh(2x)/ cosh2 (2x) studied earlier numerically by Bronski [25], and excellent agreement between the WKB theory and numerical results was found. The spectrum consists of a purely imaginary part and two branches with nonzero real parts, λ = α + iγ , α = 0. For this particular case with even function ρ0 (x) and odd function v0 (x) the complex branches are located symmetrically with respect to the imaginary axis α = 0, so that the whole spectrum has in the complex λ = α + iγ plane a ‘Y-shaped’ form. It is natural to suppose that for other choice of ρ0 (x) and v0 (x) the spectrum may be deformed but remains qualitatively the same. Then we can make the following conclusions (see also [26]). Since velocities of solitons, u(x, t) =

2γ exp[2iαx − 4i(α 2 − γ 2 )t] , cosh[2γ (x − 4αt)]

(58)

are determined by the real parts α of the eigenvalues, the purely imaginary part of the spectrum does not contribute into asymptotic soliton trains and leads rather to quite complicated behaviour in vicinity of x = 0 studied numerically in [27,28]. The nonzero real parts α of the eigenvalues λ = α + iγ determine velocities V = 4α of the solitons in the trains and in the asymptotic limit we obtain α(x, t) = x/4t.

(59)

The dependence of γ on α, γ = γ (α), in the two branches of the spectrum with α = 0 determines the amplitudes 2γ of the solitons: γ = γ (x/4t).

(60)

The number of solitons in the interval (x, x + dx) is given by (k/2π) dx, k being the wavenumber, and general form of this function can be found from the Whitham equations. To this end, we average the conservation laws of

A.M. Kamchatnov et al. / Physics Letters A 287 (2001) 223–232

the NLS equation,
2  
|u| t = i ux u∗ − uu∗x x ,     i
1
∗ ∗ 2 ∗ ∗ 4 ux u − uux = |ux | − uxx u + uuxx − |u| , 2 2 t x

231

(61)

over the wavelength L = 2π/k with the help of the soliton solution (58) and with account of conservation of ‘number of waves’ (9) arrive at the Whitham system kt + 4(kα)x = 0, (kγ )t + 4(kαγ )x = 0, (kαγ )t + 4(kα 2 γ )x = 0.

(62)

Its asymptotic solution is given by Eqs. (59), (60) and k(x, t) = (1/t)f (x/4t).

(63)

As in the previous cases, the function f (α) can be found by differentiation of the Bohr–Sommerfeld quantization rule. These formulas give asymptotic solution for the wavetrains arising from initially large and smooth pulses with chirp.

4. Conclusion We have shown that the quasiclassical quantization rules, which determine the spectrum of the linear eigenvalue problem corresponding to the integrable wave equations, can be obtained not only from investigation of the linear scattering problem, but also from the corresponding limit of the Baker–Akhiezer function. This approach has an advantage of connecting directly the asymptotic soliton trains with the WKB function and emphasizing such properties of the spectrum which permit one to distinguish initial conditions leading to asymptotic soliton trains from other conditions leading to a more complex behaviour. We have illustrated this approach by several simple examples. Our numerical simulations confirmed the theoretical predictions and more detailed expositions will be published elsewhere. We hope that the suggested approach applies not only to equations connected with the second-order linear problem (29), but also to integrable equations corresponding to higher-order problems.

Acknowledgements We thank Yu.A. Brezhnev and M.V. Pavlov for useful discussions. A.M.K. and B.A.U. are grateful to the staff of Instituto de Física Teórica, UNESP, where this work was done, for kind hospitality. The authors were partially supported by FAPESP (Brazil). A.M.K. thanks also RFBR (grant 01-01-00696), R.A.K. thanks CNPq (Brazil) and B.A.U. thanks CRDF (grant ZM2-2095) for partial support.

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