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Soliton perturbation theory for a higher order Hirota equation
Available online at www.sciencedirect.com
Mathematics and Computers in Simulation 80 (2009) 770–778
Soliton perturbation theory for a higher order Hirota equation S.M. Hoseini a , T.R. Marchant b,∗ b
a Mathematics Department, Vali-e-Asr University, Rafsanjan, Iran School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, 2522 NSW, Australia
Available online 2 September 2009
Abstract Solitary wave evolution for a higher order Hirota equation is examined. For the higher order Hirota equation resonance between the solitary waves and linear radiation causes radiation loss. Soliton perturbation theory is used to determine the details of the evolving wave and its tail. An analytical expression for the solitary wave tail is derived and compared to numerical solutions. An excellent comparison between numerical and theoretical solutions is obtained for both right- and left-moving waves. Also, a two-parameter family of higher order asymptotic embedded solitons is identified. © 2009 IMACS. Published by Elsevier B.V. All rights reserved. Keywords: Higher order Hirota equation; Embedded solitons; Soliton perturbation theory; Solitary wave tails
1. Introduction The Hirota equation is an integrable equation which has a number of physical applications, such as the propagation of optical pulses in nematic liquid crystal waveguides, see Rodriguez et al. [14], and for a certain parameter regime, femtosecond pulse propagation in optical fibres, see Yang [16]. The Hirota equation ψt + 6|ψ|2 ψx + ψxxx = 0,
(1)
is integrable, see Hirota [8] for a derivation of the N-soliton solution. The Hirota equation is closely related to both the Nonlinear Schrödinger (NLS) and modified Korteweg-de Vries (mKdV) equations, as it is a complex generalisation of the mKdV equation and it is part of the NLS hierarchy of integrable equations The Hirota equation (1) has a two-parameter soliton family, with arbitrary amplitude and velocity, which are embedded in the linear wave spectrum. Rodriguez et al. [14] explains why these embedded solitons do not emit radiation and proves their stability. The Hirota equation is also called the complex modified Korteweg-de Vries (cmKdV) equation although this name is shared with a non-integrable variant, in which the nonlinear term in (1) is replaced by (|ψ|2 ψ)x . The non-integrable cmKdV equation has physical applications such as Langmuir solitons in a plasma, see Dysthe et al. [2], and transverse waves in a elastic solid, see Erbay [3]. The analysis of how soliton solutions of integrable equations behave under integrable and non-integrable perturbations has a long and rich history. Grimshaw [4,5] considered slowly varying solitary waves solutions of the Korteweg-de Vries (KdV) and NLS equations, using a multiple scales expansion. He found expressions for the variation in amplitude, ∗
Corresponding author. E-mail addresses: [email protected] (S.M. Hoseini), tim [email protected] (T.R. Marchant).
0378-4754/$36.00 © 2009 IMACS. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.matcom.2009.08.012
S.M. Hoseini, T.R. Marchant / Mathematics and Computers in Simulation 80 (2009) 770–778
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correction to the wavespeed and the trailing tail. The connection with conservation laws was also explored. Grimshaw and Mitsudera [6] revisited the perturbed KdV problem and found results for a more general form of the perturbation. Kivshar and Malomed [10] comprehensively surveyed the use of perturbation theory for nearly integrable equations, using inverse scattering and other techniques. They considered multiple applications for the KdV, NLS, sine-Gordon and Landau-Lifshitz equations with both conservative and dissipative perturbation terms. Some other important examples of soliton perturbation theory in the literature include Herman [7], for applications to perturbed KdV, NLS and mKdV equations, or Yang and Kaup [17] for an application to a generalised NLS equation. A higher order Hirota equation of the form ψt + 6|ψ|2 ψx + ψxxx + H(ψ) = 0, H(ψ) = c1
|ψ|4 ψ
2 x + c2 (ψ|ψx | )x
1,
∗ +c ψ , + c3 ψ∗ (ψψxx )x + c4 ψ∗ ψx ψxx + c5 ψψx ψxx 6 5x
(2)
is considered in this paper, where starred quantities are complex conjugates and is a measure of the importance of the higher order terms. If the higher order coefficients are given by 2 2 2 2 , (3) (c1 , c2 , c3 , c4 , c5 , c6 ) = 1, , , , 0, 3 3 3 15 then (2) is a member of the NLS integrable hierarchy, see Kano [9]. Hence (2) represents a generalisation of the integrable hierarchy (3). Similar generalisations of the integrable hierarchy for the KdV, NLS and mKdV equations has been the focus of much theoretical interest, due to the concept of asymptotic integrability and the usefulness of asymptotic transformations in obtaining analytical results, see, for example, Kodama and Mikhailov [11] and Marchant and Smyth [12]. Yang [16] considered solitary wave evolution for a higher order Hirota equation of the form ψt + 6|ψ|2 ψx + ψxxx + i(c1 |ψ|2 ψ + c2 ψ(|ψ|2 )x ),
1.
(4)
Using soliton perturbation theory he found that there exists a one-parameter family, of arbitrary amplitude, of embedded solitons. An expression for the amplitude of the solitary wave tail and ordinary differential equations (odes), describing the evolution of the wave amplitude and frequency, were found. In this paper we examine solitary wave evolution for the higher order Hirota equation (2). In general, resonance between the linear radiation and the solitary wave causes radiation loss and the formation of a solitary wave tail. In Section 2 soliton perturbation theory is used to derive the details of the solitary wave tail, for (2). An analytical expression is found for the tail and an excellent comparison is found between the theoretical and numerical solutions. Moreover, the perturbation theory shows that the tail vanishes when c1 − 3c3 − 2c4 − 2c5 + 20c6 = 0,
(5)
is satisfied and confirms the existence of the two-parameter family of higher order asymptotic embedded solitons. 2. Soliton perturbation theory There are many applications of soliton perturbation theory in the literature. A popular approach is the direct method, which will be used here. This is based on the complete set of eigenfunctions for the linearised equation, expanded about the single soliton solution. Suppression of the secular growth then allows the slow variations in the soliton parameters to be determined. See Hereman [7] or Yang [15,16] for details. In this section the evolving solitary wave tail is determined, to first-order. In particular, an analytical expression is found for the amplitude of the solitary wave tail, which allows the special cases in which no tail occurs to be identified. These special cases correspond to a two-parameter family of higher order asymptotic embedded solitons. The Hirota soliton is ψ0 = κei(aθ+ϕ) sech κθ, where v =
κ2
− 3a2 , λ
ϕ = λt − ϕ0 , =
−2(a2
+ κ2 )a,
θ = x − vt − θ0 ,
(6)
where κ, a, θ0 and ϕ0 are free parameters. κ is the amplitude, v the velocity, a and λ are phase properties, while θ0 and φ0 describe the initial location of the wave and its initial phase.
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For the higher order Hirota equation (2) the perturbed solution can be written as t t v dt − θ0 , ϕ = λ dt − ϕ0 . ψ = eiϕ η(θ, t, T1 , T2 , . . .), θ = x − 0
(7)
0
By substituting (7) in the higher order Hirota equation (2) we obtain ηt + iλη − vηθ + ηθθθ + 6|η|2 ηθ = −H(η) − (ηT1 − iηϕ0T1 − ηθ θ0T1 ) + O(2 ).
(8)
The solution η(θ, t) can be expanded as a perturbation series η = η0 (θ) + η1 (θ, t, T1 , T2 , . . .) + . . . ,
(9)
which is substituted into (8). The method describes the slow variation of the soliton parameters κ, a, θ0 and ϕ0 and the development of the (small) tail behind the solitary wave, due to the perturbation terms in H. The main interest here is the development of the solitary wave tail and the fact that certain combinations of the terms in H lead to no tail being generated at all. At O() the equation η1t + iλη1 + (6|η0 |2 − v)η1θ + η1θθθ + r(θ)η1 + q(θ)η∗1 = ω1 ,
where r(θ) = 6η∗0 η0θ , q(θ) = 6η0 η0θ , η0 = ψ0 e−iϕ = κeiaθ sech κθ, ω1 = −H(η0 ) − η0T1 + iη0 ϕ0T1 + η0θ θ0T1 , is found. The linear equation (10) can be written in the matrix form (∂t + L)w1 = H where ω1 η1 G(θ) q(θ) , H= , , w1 = L= η∗1 ω1∗ q∗ (θ) G∗ (θ)
(10)
(11)
G(θ) = ∂θθθ + (6|η0 − v)∂θ + r(θ) + iλ. |2
The technique used to solve (10) is similar to that applied to the perturbed KdV and NLS equations. The solution is expanded as the set of eigenfunctions for the linearisation operator. These eigenfunctions are just the squared eigenstates of the Zakharov–Shabat system with the soliton potential, see Yang [15] and Zakharov and Shabat [18]. We define the inner product between functions f (θ) and g(θ) as ∞ f (θ), g(θ) = f (θ)T g(θ) dθ, (12) −∞
and define L† , the adjoint of L, as −G∗ (θ) q∗ (θ) † , L = q(θ) −G(θ)
(13)
Note that T † g − vf T g]|∞ f, Lg = [f T gθθ − fθT gθ + fθθ −∞ + L f, g.
The non-localized (continuous) eigenstates of L are −κ2 sech2 κθei(2a−ξ)θ , w2 = (κ tanh κθ + iξ − ia)2 e−iξθ −(κ tanh κθ − iξ + ia)2 e+iξθ w3 = , κ2 sech2 κθe−i(2a−ξ)θ
(14)
(15)
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and the localized (discrete) eigenstates of L are −eiaθ −η0 = κ sech κθ, w4 = η∗0 e−iaθ η0θ (−κ tanh κθ + ai)eiaθ w5 = = κ sech κθ, η∗0θ (−κ tanh κθ − ai)e−iaθ (−3a(1 − κθ tanh κθ) − iκ2 θ)eiaθ w6 = sech κθ, (−3a(1 − κθ tanh κθ) + iκ2 θ)e−iaθ (−2iaκ2 θ + (κ2 + 3a2 )(1 − κθ tanh κθ))eiaθ w7 = sech κθ, (2iaκ2 θ + (κ2 + 3a2 )(1 − κθ tanh κθ))e−iaθ
773
(16)
where Lw4 = Lw5 = 0, Lw6 =
+ 9a2 )w4 , Lw7 = −2κ(κ2 + 9a2 )w5 , Lw2 = iϑw2 , Lw3 = −iϑw3 ,
2iκ(κ2
(17)
and the parameter ϑ is defined as ϑ = (ξ + 2a)k1 ,
k1 = κ2 + (ξ − a)2 .
(18)
L†
The eigenstates of can be related to the corresponding eigenstate of L. If an eigenstate of L has the vector form † † T (a, b) then the corresponding eigenstate of L† is (−a∗ , b∗ )T . Note that w6 , w7 , w6 and w7 are called generalised eigenstates, as they satisfy the modified eigenvalue equations, in (17). The continuous and discrete eigenstates (both true and generalised) form a complete set in L2 , in which the solitary wave solution (20) can be expanded, see Yang and Kaup [17]. The only non-zero inner products of the (bounded) eigenstates and adjoint (bounded) eigenstates are †
†
w4 , w6 = 6a = −w6 , w4 , †
†
†
†
w5 , w6 = −2i(κ2 + 3a2 ) = −w6 , w5 , w4 , w7 = −2(κ2 + 3a2 ) = −w7 , w4 , †
†
w5 , w7 = −2ia(κ2 + 3a2 ) = −w7 , w5 , †
(19)
w2 (θ; ξ), w2 (θ; ξ ) = 2πk12 δ(ξ − ξ ), †
w3 (θ; ξ), w3 (θ; ξ ) = −2πk12 δ(ξ − ξ ), where δ is the Dirac-delta function. The eigenfunctions (16) and the inner products (19) are used to find expansion coefficients of the first-order solitary wave solutions of (2). Note that in the case a = 0 (and (2) becomes a higher order mKdV equation) Yang [15] found the equivalent expressions to the eigenfunctions (16) and the adjoints. Let us start by expanding the first-order solitary wave solution of (11) ∞ w1 = [g1 (t; ξ)w2 + g2 (t; ξ)w3 ] dξ + h1 (t)w4 + h2 (t)w5 + h3 (t)w6 + h4 (t)w7 , (20) −∞
and also right hand site of (11), ∞ H= [Γ1 (ξ)w2 + Γ2 (ξ)w3 ] dξ + p1 w4 + p2 w5 + p3 w6 + p4 w7 , −∞
(21)
as a linear combination of all the eigenstates. Then it is not difficult to show that the coefficients pj , Γ1 (ξ) and Γ2 (ξ) can be determined by applying inner product of the adjoint eigenfunctions L† on (21) and then using orthogonality
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relations (19) †
†
†
†
†
†
†
†
†
†
†
†
†
†
†
†
†
†
H, w7 w5 , w6 − H, w6 w5 , w7
p1 =
†
†
w4 , w7 w5 , w6 − w4 , w6 w5 , w7 †
†
H, w7 w4 , w6 − H, w6 w4 , w7
p2 =
†
†
w5 , w7 w4 , w6 − w5 , w6 w4 , w7 †
†
†
w6 , w5 w7 , w4 − w6 , w4 w7 , w5 †
†
†
w7 , w5 w6 , w4 − w7 , w4 w6 , w5 H, w2 , 2πk12
,
†
†
Γ1 (ξ) =
(22)
,
†
H, w5 w6 , w4 − H, w4 w6 , w5
p4 =
,
†
H, w5 w7 , w4 − H, w4 w7 , w5
p3 =
,
Γ2 (ξ) = −
H, w3 . 2πk12
If the coefficients hi and pi are non-zero then secular growth occurs in η1 . From (20), the conditions to eliminate the secular terms are †
†
†
†
H, w4 = H, w5 = H, w6 = H, w7 = 0.
(23)
as these imply that the hi and pi are all zero. Alternatively, the Fredholm alternative for the linear equation (11) directly implies the solvability conditions (23). Thus, the first-order correction (20) can be written as an integral of the continuous eigenstates of L. We note that this fact is a general feature of the direct perturbation approach. Applying the secularity conditions (23) gives the solitary wave parameters as v= θ0 = ϕ0 =
λ = −2(κ2 + a2 )a, κ2 − 3a2 , 1 1 5c6 a4 + 10c6 + c5 + c2 − 2c3 − c4 κ2 a2 + (5c2 − c4 − 34c3 − c5 + 8c1 + 105c6 )κ4 t, 3 45 4 1 a −4c6 a4 − c5 κ2 a2 + (420c6 + 32c1 − 24c4 − 96c3 − 44c5 )κ4 t. 3 45 (24)
These represent first-order corrections to the velocity and the initial position and phase of the solitary wave. The first-order solution has the form ∞ iΓ1 (ξ) w1 = (g1 w2 + g1∗ w3 ) dξ, where g1 = − (25) (1 − e−iϑt ). ϑ −∞ The quantity Γ1 (ξ) (i.e. Γ2 (ξ) = Γ1 (ξ)), found using the residue theorem, is 360κ6 Γ1 (ξ) = i(45κ3 (A1 + A2 ) − κ2 (15A3 + 9A4 )(ξ − a) + 5κA1 (ξ − a)2 − A4 (ξ − a)3 ) sech
πξ−a 2 κ
, (26)
where the various coefficients are A1 =
(−c1 + c2 + 10c3 + 4c4 − 120c6 )κ5 a,
A2 =
(−c2 + 2c3 + c4 + c5 − 20c6 )κ3 a3 + (−c2 − 6c3 − 3c4 + c5 + 100c6 )κ5 a,
A3 =
(3c2 − 6c3 − 3c4 + c5 + 60c6 )κ4 a2 + (3c2 + 2c3 + c4 + c5 − 60c6 )κ6 ,
A4 =
(c1 − 5c2 − 8c3 − 2c4 − 2c5 + 120c6 )κ6 ,
S.M. Hoseini, T.R. Marchant / Mathematics and Computers in Simulation 80 (2009) 770–778
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Rearranging (25) gives ∞ ∞ 1 − eiϑt 1 − e−iϑt 2 η1 (θ, t) = (κ tanh κθ − i(ξ − a))2 eiξθ dξ − κ sech2 κθe−i(ξ−2a)θ dξ, iΓ1 (ξ) iΓ1 (ξ) ϑ ϑ −∞ −∞ (27) as an integral expression for the first-order correction to the solitary wave profile. We also note that (1 − e±iϑt ) = ∓it, ϑ→0 ϑ lim
(28)
so the integrand of (27) is singular at ϑ = 0 (ξ = −2a) in the limit of large time, as t → ∞. 2.1. The solitary wave tail As radiation is shed from the solitary wave, it propagates in the negative x-direction, according to cg = −3k2 , the group velocity of the linearised version of the Hirota equation. Hence for right-moving waves, κ2 > 3a2 , the radiation forms a tail behind the evolving solitary wave. For left-moving waves, 3a2 > κ2 , and the radiation and the wave move in the same direction, with the radiation forming a front ahead of the solitary wave. For convenience we call the region, θ −1, the solitary wave tail for both right- and left-moving solitary waves. The asymptotic analysis considered here is similar to that of Herman [7] (see his Appendix B) and Pelinovsky and Yang [13] (see their Section 4). The Riemann–Lebesgue lemma implies that the integral (27) will decay to zero, as t → ∞, except near any singular points. Hence for large time, the leading-order behaviour of (27) can be found by considering contributions near ξ = −2a. We substitute the transformation ξ = −2a + z/t into (27) and obtain ∞ iz(γ(k2 +9a2 )+cp ) ∞ eizcp e (κ(±1) + i3a)2 −2iaθ η1 (θ, t)∼ie Γ1 (−2a) dz − dz , |θ| 1, t → ∞, 2 2 γ(κ + 9a ) z −∞ z −∞ (29) where the ± in the expression refers to the regions well in front (+) and behind (−) the solitary wave. Also we let θ/t = cp , an O(1) constant. Also, the integral ∞ izp e dz = iπsgn(p), (30) −∞ z is needed to evaluate (29). Well in front of the soliton, cp > 0, so both integrals within (29) have the same value. So the cancellation of the two integrals implies η∼0 well ahead of the solitary wave. This is physically consistent with the fact that the group velocity of the linear radiation is negative. Behind the wave we choose −(k2 + 9a2 ) < cp < 0. Then the integrals within (29) have the opposite sign and 1 iaπ(κ2 + a2 ) 10 κ − 3ia 3πa (c1 − 3c3 − 2c4 − 2c5 + 20c6 ) sech , κ + 3ia 2κ
η1 ∼he−2iaθ ,
where h =
(31) θ −1,
t → ∞,
is obtained as the steady-state tail of the solitary wave at first-order. The tail (31) is qualitatively similar to the expression found by Yang [16] for the tail of an evolving solitary wave governed by (4). For the higher order Hirota equation (4) it was found that a one-parameter family of embedded solitons occurred for fixed values of the wavenumber a. For (2) the tail (31) is zero if (5) is satisfied and a two-parameter family of asymptotic embedded solitons occurs. Note that this two-parameter family only exists up to first-order, as the second-order tail, at O(2 ), may be non-zero. The tail amplitude (31) is also zero if a = 0, hence a one-parameter family of embedded solitons (the real mKdV soliton family) also exists. This family are part of the one-parameter family of embedded solitons found by Yang [16]. The leading-order transient terms in the tail region will occur due to contributions from (27) at points of stationary phase. The relevant phase of (27) is ϑ1 = ϑ + ξcp = (ξ + 2a)(κ2 + (ξ − a)2 ) + ξcp ,
(32)
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and points of stationary phase, which occur when dϑ1 /dξ = 0, are 1 ± ξs = ± − (κ2 − 3a2 + cp ). 3 Using the method of stationary phase gives η1 ∼he−2iaθ + iπ1/2
[κ + i(ξs± − a)]
(33)
2
± ± Γ1 (ξs± )ei[(ϑ(ξs )+ξs cp )t∓(π/4)] , ± 1/2 ± (3t|ξs |) ϑ(ξs )
θ −1,
t → ∞.
(34)
The tail (34) consists of the steady-state result (31) plus the two leading-order transient terms, which are contributions from the points of stationary phase (one term for each of the roots (33)). The points of stationary phase (33) only exist when cp < (3a2 − κ2 ). For left-moving waves 3a2 > κ2 , and the slowly decaying transient terms, of O(t −(1/2) ), occur along the whole of the tail (for all cp < 0). For right-moving waves κ2 > 3a2 , and O(t −(1/2) ) transient terms only occur on the far edge of the tail, between −(κ2 + 9a2 ) < cp < −(κ2 − 3a2 ). For the near tail of a right-moving wave the transient terms decay more quickly than those in (34) and the flat steady-state tail will be quickly approached. 2.2. Numerical solutions and discussion Fig. 1 shows the first-order correction |η1 | versus θ at time t = 70. The parameters are κ = 1, a = 0.1, = 0.1, c1 = 1 and all the other ci = 0. The perturbation solution (27), was solved numerically using a high-order quadrature scheme. The higher order Hirota equation (2) was solved numerically using an implicit finite-difference scheme. The initial condition used in the numerical scheme was a Hirota soliton. The quantity −1 |ψ| is plotted from the numerical solution as it allows a comparison with the perturbation solution in the tail region. As κ2 > 3a2 this is an example of a right-moving wave. There is an excellent comparison between the numerical and perturbation solutions. The figure shows the surface elevation behind the evolving solitary wave with the flat steadystate tail clearly visible. The amplitude of this tail, as predicted by the perturbation solution (27), is |η1 | = 0.1140, which is the same, to at least 4dp, as the analytical expression (29) for the steady-state tail amplitude. The numerical results show that the tail has amplitude |h| = 0.1185, a difference of about 4%. The figure shows that the tail amplitude drops from its steady-state value, to near zero at θ ≈ −75. The asymptotic theory implies that the length of the solitary wave tail is (κ2 + 9a2 )t = 76.3, which is close to numerical tail length, of about 80. The tail will be flat for θ > −(κ2 − 3a2 )t = −67.9, as no slowly decaying transients exist on the near tail. There is an oscillation right at the end of the tail, so again the asymptotic result is consistent with the perturbation and numerical solutions. Note that near θ = 0 there is a significant difference between the numerical and perturbation solutions. This is due to the fact that the numerical solution includes the O (1) soliton profile while the theoretical solution, for η1 , does not.
Fig. 1. The first-order correction |η1 | versus θ at t = 70 for a right-moving wave. Shown is the perturbation solution (27) (solid curve) and numerical solution of (2) (dashed curve).
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Fig. 2. The first-order correction |η1 | versus θ at t = 150 for a left-moving wave. Shown is the perturbation solution (27) (solid curve) and numerical solution of (2) (dashed curve).
Fig. 2 shows the first-order correction |η1 | versus θ at time t = 150. The parameters are the same as for (Fig. 1), except that a = 0.8. Shown are the perturbation solution (27) and the quantity −1 |ψ| from the numerical solution of (2). The figure shows a portion of the region (−130 < θ < −80) in front of the left-moving solitary wave. The comparison between the numerical and perturbation solutions is very good, with some slight differences in phase and amplitude noticeable. Also the asymptotic expression (34) is the same (to 4dp) as the perturbation solution (27) over this region of the tail, confirming the validity of the asymptotic expression (34). In contrast to the flat tail for the right-moving wave, here the tail is highly oscillatory. The superposition of the steady-state tail wavelength (of λ = π/a ≈ 3.93) with the two different transient wavelengths leads to a classical beating-like phenomena where the tail has multiple peaks and troughs of higher and lower amplitudes. The steadystate tail amplitude is 7.60 × 10−2 . An average taken of the numerical solution, over the range −200 < θ < −50, is 7.74 × 10−2 , which is within 2% of the analytical steady-state tail amplitude. The maximum amplitude (crest to trough) of the tail oscillations is 4.56 × 10−2 at t = 150. As the oscillations decay like O(t −1/2 ), a relatively flat tail will not be reached until t = O(106 ), when the amplitude of the oscillations will have decayed to less than 1% of this value. Physically, the occurrence of slowly decaying transients on the tail of the left-moving wave, is related to the fact the wave is moving in the same direction as its shed linear radiation, while the right-moving wave moves in the opposite direction. In summary the perturbation theory results, both direct numerical solutions of (27) and the further asymptotic results derived from (27) are in good agreement with each other and numerical solutions of the governing pde (2) for examples of right- and left-moving solitary waves. 3. Conclusion Soliton perturbation theory has been applied to (2) with an integral expression found for the first-order correction to the solitary wave profile. An asymptotic expansion, valid for large time, allows a simple analytical expression to be found for the solitary wave tail. The theoretical results clearly explain the differences which occur for right- and left-moving solitary waves. A flat tail occurs for the right-moving case while the tail for the left-moving wave is highly oscillatory, due to slowly decaying transient terms. Acknowledgement The authors would like to thank an anonymous referee for some useful and constructive comments.
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References [2] K.B. Dysthe, E. Mjolhus, H.L. Pecseli, L. Stenflo, Langmuir solitons in magnetised plasmas, Plasma Phys. 20 (1978) 1087–1099. [3] H.A. Erbay, Nonlinear transverse waves in a generalized elastic solid and the complex modified Korteweg-de Vries equation’, Phys. Scripta 58 (1998) 9–14. [4] R. Grimshaw, Slowly varying waves I. Korteweg-de Vries equation, Proc. R. Soc. Lond. A 386 (1979) 359–375. [5] R. Grimshaw, Slowly varying waves. II. Nonlinear Schrödinger equation, Proc. Roy. Soc. Lond. A 386 (1979) 377–388. [6] R.H.J. Grimshaw, H. Mitsudera, Slowly varying waves of the perturbed Korteweg-de Vries equation revisited, Stud. Appl. Math. 90 (1993) 75–89. [7] R.L. Herman, A direct approach to studying soliton perturbations, J. Phys. A: Math. Gen. 23 (1990) 2327–2362. [8] R. Hirota, Exact envelope-soliton solutions of a nonlinear wave equation, J. Math. Phys. 14 (1973) 805–809. [9] T. Kano, Normal form of nonlinear Schrödinger equation, J. Phys. Soc. Jpn. 58 (1989) 4322–4328. [10] Y.S. Kivshar, B.A. Malomed, Dynamics of solitons in nearly integrable systems, Rev. Mod. Phys. 61 (1989) 763–820. [11] Y. Kodama, A.V. Mikhailov, Obstacles to asymptotic integrability, in: I.M. Gelfand, A. Fokas (Eds.), Algebraic Aspects of Integrable Systems, Birkhauser, 1997, pp. 173–204. [12] T.R. Marchant, N.F. Smyth, Soliton interaction for the extended Korteweg-de Vries equation, IMA J. Appl. Math. 56 (1996) 157–176. [13] D.E. Pelinovsky, J. Yang, A normal form for nonlinear resonance of embedded solitons, Proc. R. Soc. Lond. A 458 (2002) 1469–1497. [14] R.F. Rodriguez, J.A. Reyes, A. Espinosa-Ceron, J. Fujioka, B.A. Malomed, Standard and embedded solitons in nematic optical fibers, Phys. Rev. E 68 (2003), 036606-1–036606-14. [15] J. Yang, Complete eigenfunctions of linearized integrable equation expanded around a soliton solution, J. Math. Phys. 41 (2000) 6614–6638. [16] J. Yang, Stable embedded solitons, Phys. Rev. Lett. 91 (2003), 143903–1-4. [17] J. Yang, D.J. Kaup, Stability and evolution of solitary waves in perturbed generalized nonlinear Schrödinger equations, SIAM J. Appl. Math. 60 (2000) 967–989. [18] V.E. Zakharov, A.B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Soviet Phys. JETP 34 (1971) 62–69.
Mathematics and Computers in Simulation 80 (2009) 770–778
Soliton perturbation theory for a higher order Hirota equation S.M. Hoseini a , T.R. Marchant b,∗ b
a Mathematics Department, Vali-e-Asr University, Rafsanjan, Iran School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, 2522 NSW, Australia
Available online 2 September 2009
Abstract Solitary wave evolution for a higher order Hirota equation is examined. For the higher order Hirota equation resonance between the solitary waves and linear radiation causes radiation loss. Soliton perturbation theory is used to determine the details of the evolving wave and its tail. An analytical expression for the solitary wave tail is derived and compared to numerical solutions. An excellent comparison between numerical and theoretical solutions is obtained for both right- and left-moving waves. Also, a two-parameter family of higher order asymptotic embedded solitons is identified. © 2009 IMACS. Published by Elsevier B.V. All rights reserved. Keywords: Higher order Hirota equation; Embedded solitons; Soliton perturbation theory; Solitary wave tails
1. Introduction The Hirota equation is an integrable equation which has a number of physical applications, such as the propagation of optical pulses in nematic liquid crystal waveguides, see Rodriguez et al. [14], and for a certain parameter regime, femtosecond pulse propagation in optical fibres, see Yang [16]. The Hirota equation ψt + 6|ψ|2 ψx + ψxxx = 0,
(1)
is integrable, see Hirota [8] for a derivation of the N-soliton solution. The Hirota equation is closely related to both the Nonlinear Schrödinger (NLS) and modified Korteweg-de Vries (mKdV) equations, as it is a complex generalisation of the mKdV equation and it is part of the NLS hierarchy of integrable equations The Hirota equation (1) has a two-parameter soliton family, with arbitrary amplitude and velocity, which are embedded in the linear wave spectrum. Rodriguez et al. [14] explains why these embedded solitons do not emit radiation and proves their stability. The Hirota equation is also called the complex modified Korteweg-de Vries (cmKdV) equation although this name is shared with a non-integrable variant, in which the nonlinear term in (1) is replaced by (|ψ|2 ψ)x . The non-integrable cmKdV equation has physical applications such as Langmuir solitons in a plasma, see Dysthe et al. [2], and transverse waves in a elastic solid, see Erbay [3]. The analysis of how soliton solutions of integrable equations behave under integrable and non-integrable perturbations has a long and rich history. Grimshaw [4,5] considered slowly varying solitary waves solutions of the Korteweg-de Vries (KdV) and NLS equations, using a multiple scales expansion. He found expressions for the variation in amplitude, ∗
Corresponding author. E-mail addresses: [email protected] (S.M. Hoseini), tim [email protected] (T.R. Marchant).
0378-4754/$36.00 © 2009 IMACS. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.matcom.2009.08.012
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771
correction to the wavespeed and the trailing tail. The connection with conservation laws was also explored. Grimshaw and Mitsudera [6] revisited the perturbed KdV problem and found results for a more general form of the perturbation. Kivshar and Malomed [10] comprehensively surveyed the use of perturbation theory for nearly integrable equations, using inverse scattering and other techniques. They considered multiple applications for the KdV, NLS, sine-Gordon and Landau-Lifshitz equations with both conservative and dissipative perturbation terms. Some other important examples of soliton perturbation theory in the literature include Herman [7], for applications to perturbed KdV, NLS and mKdV equations, or Yang and Kaup [17] for an application to a generalised NLS equation. A higher order Hirota equation of the form ψt + 6|ψ|2 ψx + ψxxx + H(ψ) = 0, H(ψ) = c1
|ψ|4 ψ
2 x + c2 (ψ|ψx | )x
1,
∗ +c ψ , + c3 ψ∗ (ψψxx )x + c4 ψ∗ ψx ψxx + c5 ψψx ψxx 6 5x
(2)
is considered in this paper, where starred quantities are complex conjugates and is a measure of the importance of the higher order terms. If the higher order coefficients are given by 2 2 2 2 , (3) (c1 , c2 , c3 , c4 , c5 , c6 ) = 1, , , , 0, 3 3 3 15 then (2) is a member of the NLS integrable hierarchy, see Kano [9]. Hence (2) represents a generalisation of the integrable hierarchy (3). Similar generalisations of the integrable hierarchy for the KdV, NLS and mKdV equations has been the focus of much theoretical interest, due to the concept of asymptotic integrability and the usefulness of asymptotic transformations in obtaining analytical results, see, for example, Kodama and Mikhailov [11] and Marchant and Smyth [12]. Yang [16] considered solitary wave evolution for a higher order Hirota equation of the form ψt + 6|ψ|2 ψx + ψxxx + i(c1 |ψ|2 ψ + c2 ψ(|ψ|2 )x ),
1.
(4)
Using soliton perturbation theory he found that there exists a one-parameter family, of arbitrary amplitude, of embedded solitons. An expression for the amplitude of the solitary wave tail and ordinary differential equations (odes), describing the evolution of the wave amplitude and frequency, were found. In this paper we examine solitary wave evolution for the higher order Hirota equation (2). In general, resonance between the linear radiation and the solitary wave causes radiation loss and the formation of a solitary wave tail. In Section 2 soliton perturbation theory is used to derive the details of the solitary wave tail, for (2). An analytical expression is found for the tail and an excellent comparison is found between the theoretical and numerical solutions. Moreover, the perturbation theory shows that the tail vanishes when c1 − 3c3 − 2c4 − 2c5 + 20c6 = 0,
(5)
is satisfied and confirms the existence of the two-parameter family of higher order asymptotic embedded solitons. 2. Soliton perturbation theory There are many applications of soliton perturbation theory in the literature. A popular approach is the direct method, which will be used here. This is based on the complete set of eigenfunctions for the linearised equation, expanded about the single soliton solution. Suppression of the secular growth then allows the slow variations in the soliton parameters to be determined. See Hereman [7] or Yang [15,16] for details. In this section the evolving solitary wave tail is determined, to first-order. In particular, an analytical expression is found for the amplitude of the solitary wave tail, which allows the special cases in which no tail occurs to be identified. These special cases correspond to a two-parameter family of higher order asymptotic embedded solitons. The Hirota soliton is ψ0 = κei(aθ+ϕ) sech κθ, where v =
κ2
− 3a2 , λ
ϕ = λt − ϕ0 , =
−2(a2
+ κ2 )a,
θ = x − vt − θ0 ,
(6)
where κ, a, θ0 and ϕ0 are free parameters. κ is the amplitude, v the velocity, a and λ are phase properties, while θ0 and φ0 describe the initial location of the wave and its initial phase.
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For the higher order Hirota equation (2) the perturbed solution can be written as t t v dt − θ0 , ϕ = λ dt − ϕ0 . ψ = eiϕ η(θ, t, T1 , T2 , . . .), θ = x − 0
(7)
0
By substituting (7) in the higher order Hirota equation (2) we obtain ηt + iλη − vηθ + ηθθθ + 6|η|2 ηθ = −H(η) − (ηT1 − iηϕ0T1 − ηθ θ0T1 ) + O(2 ).
(8)
The solution η(θ, t) can be expanded as a perturbation series η = η0 (θ) + η1 (θ, t, T1 , T2 , . . .) + . . . ,
(9)
which is substituted into (8). The method describes the slow variation of the soliton parameters κ, a, θ0 and ϕ0 and the development of the (small) tail behind the solitary wave, due to the perturbation terms in H. The main interest here is the development of the solitary wave tail and the fact that certain combinations of the terms in H lead to no tail being generated at all. At O() the equation η1t + iλη1 + (6|η0 |2 − v)η1θ + η1θθθ + r(θ)η1 + q(θ)η∗1 = ω1 ,
where r(θ) = 6η∗0 η0θ , q(θ) = 6η0 η0θ , η0 = ψ0 e−iϕ = κeiaθ sech κθ, ω1 = −H(η0 ) − η0T1 + iη0 ϕ0T1 + η0θ θ0T1 , is found. The linear equation (10) can be written in the matrix form (∂t + L)w1 = H where ω1 η1 G(θ) q(θ) , H= , , w1 = L= η∗1 ω1∗ q∗ (θ) G∗ (θ)
(10)
(11)
G(θ) = ∂θθθ + (6|η0 − v)∂θ + r(θ) + iλ. |2
The technique used to solve (10) is similar to that applied to the perturbed KdV and NLS equations. The solution is expanded as the set of eigenfunctions for the linearisation operator. These eigenfunctions are just the squared eigenstates of the Zakharov–Shabat system with the soliton potential, see Yang [15] and Zakharov and Shabat [18]. We define the inner product between functions f (θ) and g(θ) as ∞ f (θ), g(θ) = f (θ)T g(θ) dθ, (12) −∞
and define L† , the adjoint of L, as −G∗ (θ) q∗ (θ) † , L = q(θ) −G(θ)
(13)
Note that T † g − vf T g]|∞ f, Lg = [f T gθθ − fθT gθ + fθθ −∞ + L f, g.
The non-localized (continuous) eigenstates of L are −κ2 sech2 κθei(2a−ξ)θ , w2 = (κ tanh κθ + iξ − ia)2 e−iξθ −(κ tanh κθ − iξ + ia)2 e+iξθ w3 = , κ2 sech2 κθe−i(2a−ξ)θ
(14)
(15)
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and the localized (discrete) eigenstates of L are −eiaθ −η0 = κ sech κθ, w4 = η∗0 e−iaθ η0θ (−κ tanh κθ + ai)eiaθ w5 = = κ sech κθ, η∗0θ (−κ tanh κθ − ai)e−iaθ (−3a(1 − κθ tanh κθ) − iκ2 θ)eiaθ w6 = sech κθ, (−3a(1 − κθ tanh κθ) + iκ2 θ)e−iaθ (−2iaκ2 θ + (κ2 + 3a2 )(1 − κθ tanh κθ))eiaθ w7 = sech κθ, (2iaκ2 θ + (κ2 + 3a2 )(1 − κθ tanh κθ))e−iaθ
773
(16)
where Lw4 = Lw5 = 0, Lw6 =
+ 9a2 )w4 , Lw7 = −2κ(κ2 + 9a2 )w5 , Lw2 = iϑw2 , Lw3 = −iϑw3 ,
2iκ(κ2
(17)
and the parameter ϑ is defined as ϑ = (ξ + 2a)k1 ,
k1 = κ2 + (ξ − a)2 .
(18)
L†
The eigenstates of can be related to the corresponding eigenstate of L. If an eigenstate of L has the vector form † † T (a, b) then the corresponding eigenstate of L† is (−a∗ , b∗ )T . Note that w6 , w7 , w6 and w7 are called generalised eigenstates, as they satisfy the modified eigenvalue equations, in (17). The continuous and discrete eigenstates (both true and generalised) form a complete set in L2 , in which the solitary wave solution (20) can be expanded, see Yang and Kaup [17]. The only non-zero inner products of the (bounded) eigenstates and adjoint (bounded) eigenstates are †
†
w4 , w6 = 6a = −w6 , w4 , †
†
†
†
w5 , w6 = −2i(κ2 + 3a2 ) = −w6 , w5 , w4 , w7 = −2(κ2 + 3a2 ) = −w7 , w4 , †
†
w5 , w7 = −2ia(κ2 + 3a2 ) = −w7 , w5 , †
(19)
w2 (θ; ξ), w2 (θ; ξ ) = 2πk12 δ(ξ − ξ ), †
w3 (θ; ξ), w3 (θ; ξ ) = −2πk12 δ(ξ − ξ ), where δ is the Dirac-delta function. The eigenfunctions (16) and the inner products (19) are used to find expansion coefficients of the first-order solitary wave solutions of (2). Note that in the case a = 0 (and (2) becomes a higher order mKdV equation) Yang [15] found the equivalent expressions to the eigenfunctions (16) and the adjoints. Let us start by expanding the first-order solitary wave solution of (11) ∞ w1 = [g1 (t; ξ)w2 + g2 (t; ξ)w3 ] dξ + h1 (t)w4 + h2 (t)w5 + h3 (t)w6 + h4 (t)w7 , (20) −∞
and also right hand site of (11), ∞ H= [Γ1 (ξ)w2 + Γ2 (ξ)w3 ] dξ + p1 w4 + p2 w5 + p3 w6 + p4 w7 , −∞
(21)
as a linear combination of all the eigenstates. Then it is not difficult to show that the coefficients pj , Γ1 (ξ) and Γ2 (ξ) can be determined by applying inner product of the adjoint eigenfunctions L† on (21) and then using orthogonality
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relations (19) †
†
†
†
†
†
†
†
†
†
†
†
†
†
†
†
†
†
H, w7 w5 , w6 − H, w6 w5 , w7
p1 =
†
†
w4 , w7 w5 , w6 − w4 , w6 w5 , w7 †
†
H, w7 w4 , w6 − H, w6 w4 , w7
p2 =
†
†
w5 , w7 w4 , w6 − w5 , w6 w4 , w7 †
†
†
w6 , w5 w7 , w4 − w6 , w4 w7 , w5 †
†
†
w7 , w5 w6 , w4 − w7 , w4 w6 , w5 H, w2 , 2πk12
,
†
†
Γ1 (ξ) =
(22)
,
†
H, w5 w6 , w4 − H, w4 w6 , w5
p4 =
,
†
H, w5 w7 , w4 − H, w4 w7 , w5
p3 =
,
Γ2 (ξ) = −
H, w3 . 2πk12
If the coefficients hi and pi are non-zero then secular growth occurs in η1 . From (20), the conditions to eliminate the secular terms are †
†
†
†
H, w4 = H, w5 = H, w6 = H, w7 = 0.
(23)
as these imply that the hi and pi are all zero. Alternatively, the Fredholm alternative for the linear equation (11) directly implies the solvability conditions (23). Thus, the first-order correction (20) can be written as an integral of the continuous eigenstates of L. We note that this fact is a general feature of the direct perturbation approach. Applying the secularity conditions (23) gives the solitary wave parameters as v= θ0 = ϕ0 =
λ = −2(κ2 + a2 )a, κ2 − 3a2 , 1 1 5c6 a4 + 10c6 + c5 + c2 − 2c3 − c4 κ2 a2 + (5c2 − c4 − 34c3 − c5 + 8c1 + 105c6 )κ4 t, 3 45 4 1 a −4c6 a4 − c5 κ2 a2 + (420c6 + 32c1 − 24c4 − 96c3 − 44c5 )κ4 t. 3 45 (24)
These represent first-order corrections to the velocity and the initial position and phase of the solitary wave. The first-order solution has the form ∞ iΓ1 (ξ) w1 = (g1 w2 + g1∗ w3 ) dξ, where g1 = − (25) (1 − e−iϑt ). ϑ −∞ The quantity Γ1 (ξ) (i.e. Γ2 (ξ) = Γ1 (ξ)), found using the residue theorem, is 360κ6 Γ1 (ξ) = i(45κ3 (A1 + A2 ) − κ2 (15A3 + 9A4 )(ξ − a) + 5κA1 (ξ − a)2 − A4 (ξ − a)3 ) sech
πξ−a 2 κ
, (26)
where the various coefficients are A1 =
(−c1 + c2 + 10c3 + 4c4 − 120c6 )κ5 a,
A2 =
(−c2 + 2c3 + c4 + c5 − 20c6 )κ3 a3 + (−c2 − 6c3 − 3c4 + c5 + 100c6 )κ5 a,
A3 =
(3c2 − 6c3 − 3c4 + c5 + 60c6 )κ4 a2 + (3c2 + 2c3 + c4 + c5 − 60c6 )κ6 ,
A4 =
(c1 − 5c2 − 8c3 − 2c4 − 2c5 + 120c6 )κ6 ,
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Rearranging (25) gives ∞ ∞ 1 − eiϑt 1 − e−iϑt 2 η1 (θ, t) = (κ tanh κθ − i(ξ − a))2 eiξθ dξ − κ sech2 κθe−i(ξ−2a)θ dξ, iΓ1 (ξ) iΓ1 (ξ) ϑ ϑ −∞ −∞ (27) as an integral expression for the first-order correction to the solitary wave profile. We also note that (1 − e±iϑt ) = ∓it, ϑ→0 ϑ lim
(28)
so the integrand of (27) is singular at ϑ = 0 (ξ = −2a) in the limit of large time, as t → ∞. 2.1. The solitary wave tail As radiation is shed from the solitary wave, it propagates in the negative x-direction, according to cg = −3k2 , the group velocity of the linearised version of the Hirota equation. Hence for right-moving waves, κ2 > 3a2 , the radiation forms a tail behind the evolving solitary wave. For left-moving waves, 3a2 > κ2 , and the radiation and the wave move in the same direction, with the radiation forming a front ahead of the solitary wave. For convenience we call the region, θ −1, the solitary wave tail for both right- and left-moving solitary waves. The asymptotic analysis considered here is similar to that of Herman [7] (see his Appendix B) and Pelinovsky and Yang [13] (see their Section 4). The Riemann–Lebesgue lemma implies that the integral (27) will decay to zero, as t → ∞, except near any singular points. Hence for large time, the leading-order behaviour of (27) can be found by considering contributions near ξ = −2a. We substitute the transformation ξ = −2a + z/t into (27) and obtain ∞ iz(γ(k2 +9a2 )+cp ) ∞ eizcp e (κ(±1) + i3a)2 −2iaθ η1 (θ, t)∼ie Γ1 (−2a) dz − dz , |θ| 1, t → ∞, 2 2 γ(κ + 9a ) z −∞ z −∞ (29) where the ± in the expression refers to the regions well in front (+) and behind (−) the solitary wave. Also we let θ/t = cp , an O(1) constant. Also, the integral ∞ izp e dz = iπsgn(p), (30) −∞ z is needed to evaluate (29). Well in front of the soliton, cp > 0, so both integrals within (29) have the same value. So the cancellation of the two integrals implies η∼0 well ahead of the solitary wave. This is physically consistent with the fact that the group velocity of the linear radiation is negative. Behind the wave we choose −(k2 + 9a2 ) < cp < 0. Then the integrals within (29) have the opposite sign and 1 iaπ(κ2 + a2 ) 10 κ − 3ia 3πa (c1 − 3c3 − 2c4 − 2c5 + 20c6 ) sech , κ + 3ia 2κ
η1 ∼he−2iaθ ,
where h =
(31) θ −1,
t → ∞,
is obtained as the steady-state tail of the solitary wave at first-order. The tail (31) is qualitatively similar to the expression found by Yang [16] for the tail of an evolving solitary wave governed by (4). For the higher order Hirota equation (4) it was found that a one-parameter family of embedded solitons occurred for fixed values of the wavenumber a. For (2) the tail (31) is zero if (5) is satisfied and a two-parameter family of asymptotic embedded solitons occurs. Note that this two-parameter family only exists up to first-order, as the second-order tail, at O(2 ), may be non-zero. The tail amplitude (31) is also zero if a = 0, hence a one-parameter family of embedded solitons (the real mKdV soliton family) also exists. This family are part of the one-parameter family of embedded solitons found by Yang [16]. The leading-order transient terms in the tail region will occur due to contributions from (27) at points of stationary phase. The relevant phase of (27) is ϑ1 = ϑ + ξcp = (ξ + 2a)(κ2 + (ξ − a)2 ) + ξcp ,
(32)
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and points of stationary phase, which occur when dϑ1 /dξ = 0, are 1 ± ξs = ± − (κ2 − 3a2 + cp ). 3 Using the method of stationary phase gives η1 ∼he−2iaθ + iπ1/2
[κ + i(ξs± − a)]
(33)
2
± ± Γ1 (ξs± )ei[(ϑ(ξs )+ξs cp )t∓(π/4)] , ± 1/2 ± (3t|ξs |) ϑ(ξs )
θ −1,
t → ∞.
(34)
The tail (34) consists of the steady-state result (31) plus the two leading-order transient terms, which are contributions from the points of stationary phase (one term for each of the roots (33)). The points of stationary phase (33) only exist when cp < (3a2 − κ2 ). For left-moving waves 3a2 > κ2 , and the slowly decaying transient terms, of O(t −(1/2) ), occur along the whole of the tail (for all cp < 0). For right-moving waves κ2 > 3a2 , and O(t −(1/2) ) transient terms only occur on the far edge of the tail, between −(κ2 + 9a2 ) < cp < −(κ2 − 3a2 ). For the near tail of a right-moving wave the transient terms decay more quickly than those in (34) and the flat steady-state tail will be quickly approached. 2.2. Numerical solutions and discussion Fig. 1 shows the first-order correction |η1 | versus θ at time t = 70. The parameters are κ = 1, a = 0.1, = 0.1, c1 = 1 and all the other ci = 0. The perturbation solution (27), was solved numerically using a high-order quadrature scheme. The higher order Hirota equation (2) was solved numerically using an implicit finite-difference scheme. The initial condition used in the numerical scheme was a Hirota soliton. The quantity −1 |ψ| is plotted from the numerical solution as it allows a comparison with the perturbation solution in the tail region. As κ2 > 3a2 this is an example of a right-moving wave. There is an excellent comparison between the numerical and perturbation solutions. The figure shows the surface elevation behind the evolving solitary wave with the flat steadystate tail clearly visible. The amplitude of this tail, as predicted by the perturbation solution (27), is |η1 | = 0.1140, which is the same, to at least 4dp, as the analytical expression (29) for the steady-state tail amplitude. The numerical results show that the tail has amplitude |h| = 0.1185, a difference of about 4%. The figure shows that the tail amplitude drops from its steady-state value, to near zero at θ ≈ −75. The asymptotic theory implies that the length of the solitary wave tail is (κ2 + 9a2 )t = 76.3, which is close to numerical tail length, of about 80. The tail will be flat for θ > −(κ2 − 3a2 )t = −67.9, as no slowly decaying transients exist on the near tail. There is an oscillation right at the end of the tail, so again the asymptotic result is consistent with the perturbation and numerical solutions. Note that near θ = 0 there is a significant difference between the numerical and perturbation solutions. This is due to the fact that the numerical solution includes the O (1) soliton profile while the theoretical solution, for η1 , does not.
Fig. 1. The first-order correction |η1 | versus θ at t = 70 for a right-moving wave. Shown is the perturbation solution (27) (solid curve) and numerical solution of (2) (dashed curve).
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Fig. 2. The first-order correction |η1 | versus θ at t = 150 for a left-moving wave. Shown is the perturbation solution (27) (solid curve) and numerical solution of (2) (dashed curve).
Fig. 2 shows the first-order correction |η1 | versus θ at time t = 150. The parameters are the same as for (Fig. 1), except that a = 0.8. Shown are the perturbation solution (27) and the quantity −1 |ψ| from the numerical solution of (2). The figure shows a portion of the region (−130 < θ < −80) in front of the left-moving solitary wave. The comparison between the numerical and perturbation solutions is very good, with some slight differences in phase and amplitude noticeable. Also the asymptotic expression (34) is the same (to 4dp) as the perturbation solution (27) over this region of the tail, confirming the validity of the asymptotic expression (34). In contrast to the flat tail for the right-moving wave, here the tail is highly oscillatory. The superposition of the steady-state tail wavelength (of λ = π/a ≈ 3.93) with the two different transient wavelengths leads to a classical beating-like phenomena where the tail has multiple peaks and troughs of higher and lower amplitudes. The steadystate tail amplitude is 7.60 × 10−2 . An average taken of the numerical solution, over the range −200 < θ < −50, is 7.74 × 10−2 , which is within 2% of the analytical steady-state tail amplitude. The maximum amplitude (crest to trough) of the tail oscillations is 4.56 × 10−2 at t = 150. As the oscillations decay like O(t −1/2 ), a relatively flat tail will not be reached until t = O(106 ), when the amplitude of the oscillations will have decayed to less than 1% of this value. Physically, the occurrence of slowly decaying transients on the tail of the left-moving wave, is related to the fact the wave is moving in the same direction as its shed linear radiation, while the right-moving wave moves in the opposite direction. In summary the perturbation theory results, both direct numerical solutions of (27) and the further asymptotic results derived from (27) are in good agreement with each other and numerical solutions of the governing pde (2) for examples of right- and left-moving solitary waves. 3. Conclusion Soliton perturbation theory has been applied to (2) with an integral expression found for the first-order correction to the solitary wave profile. An asymptotic expansion, valid for large time, allows a simple analytical expression to be found for the solitary wave tail. The theoretical results clearly explain the differences which occur for right- and left-moving solitary waves. A flat tail occurs for the right-moving case while the tail for the left-moving wave is highly oscillatory, due to slowly decaying transient terms. Acknowledgement The authors would like to thank an anonymous referee for some useful and constructive comments.
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