Evolution of Higher-Order Gray Hirota Solitary Waves

Transcription

1 Evolution of Higher-Order Gray Hirota Solitary Waves By S. M. Hoseini and T. R. Marchant The defocusing Hirota equation has dark and gray soliton solutions which are stable on a background of periodic waves of constant amplitude. In this paper, gray solitary wave evolution for a higher-order defocusing Hirota equation is examined. A direct analysis is used to identify families of higher-order gray Hirota solitary waves, which are embedded for certain parameter values. Soliton perturbation theory is used to determine the detailed behavior of an evolving higher-order gray Hirota solitary wave. An integral expression for the first-order correction to the wave is found and analytical expressions for the steady-state and transient components of the solitary wave tail are derived. A subtle and complex picture of the development of solitary wave tails emerges. It is found that solitary wave tails develop for two reasons, one is decay of the solitary wave caused by resonance, the second is corrections at first-order to the background wave. Strong agreement is found between the theoretical predictions of the perturbation theory and numerical solutions of the governing equations. 1. Introduction Dark solitons, which appear as a localized intensity dip on a stable traveling wave background, have been extensively studied by several authors. Kivshar and Luther-Davies [1] presents a historical overview of optical dark solitons Address for correspondence: S. M. Hoseini, Mathematics Department, Vali-e-Asr University, Rafsanjan, Iran; hoseini@uow.edu.au STUDIES IN APPLIED MATHEMATICS 121: C 2008 by the Massachusetts Institute of Technology

2 118 S. M. Hoseini and T. R. Marchant and the physical origins of the defocusing cubic nonlinear Schrödinger NLS) equations. By analyzing the modulational instability the differences between bright and dark solitons were discussed and they showed that small excitations of the nonzero background wave are stable for the defocusing NLS equation and unstable for focusing case. Similar stability considerations apply to the focusing and defocusing Hirota equations also. Li et al. [2], by applying inverse scattering transform IST) method, and Mahalingam and Porsezian [3], by using the Painlevé analysis and the Hirota bilinearization method constructed a generalized dark solitary wave solution of the higher-order NLS equation η t + i α 1 η xx + α 2 η 2 η ) + α 3 η xxx + α 4 η 2 η) x + α 5 η η 2 ) x = 0, 1) where the real coefficients, the α i, are determined by the physical model under consideration. Li et al. [2] also showed that the absolute value η of the dark two-soliton solution of 1) can be considered as the superposition of the absolute values of two interacting dark one-soliton solutions; the only effects of the collision are the coordinate shifts that the solitons suffer. The defocusing Hirota equation η t + 6 η 2 η x η xxx = 0, 2) is a special case of 1) and is integrable. It has a number of physical applications, such as ultrashort light pulses in the subpicosecond regime, whose duration is shorter than 100 femtoseconds, see Li et al. [2] and Mahalingam and Porsezian [3]. The defocusing Hirota equation also is part of the NLS hierarchy of integrable equations, and its soliton solution has a very similar form to the defocusing NLS soliton. This is similar to the relation between the focusing Hirota and NLS equations, see Hoseini and Marchant [4]. The higher-order defocusing Hirota equation η t + 6 η 2 η x η xxx + ɛ c 1 η 4 η x + c 2 η η x 2 ) x + c 3 η ηη xx ) x + c 4 η η x η xx + c 5 ηη x η xx + c 6η 5x ) = 0,ɛ 1, 3) is considered in this paper. When the higher-order coefficients are given by c 1, c 2, c 3, c 4, c 5, c 6 ) = 1, 1 3, 1 3, 1 3, 0, 1 ), 4) 30 then 3) is a member of the Hirota integrable hierarchy. The integrable hierarchy is obtained using the Lax hierarchy of the Hirota equations. Hence 3) represents a generalization of a member of the integrable hierarchy member 4). Direct perturbation theory for solitons requires that the complete set of the eigenfunctions for the linearized problem related to the nonlinear wave equation be determined. Yang [5] constructed this set for a large class of integrable nonlinear wave equations such as the Korteweg-de Vries KdV),

3 Evolution of Higher-Order Gray Hirota Solitary Waves 119 NLS and modified KdV equations. The same procedure can be exploited to find the eigenstates of the adjoint linearization operator. His finding shows that the eigenfunctions for these hierarchies are the squared Jost solutions. Similar results for focusing and defocusing Hirota equations have been obtained, for example, see Hoseini and Marchant [4] and Section 3 of this paper. Chen and Yang [6] developed direct soliton perturbation theory for the derivative NLS DNLS) and the modified NLS equations. Using the similarity between the KdV and DNLS hierarchies they showed that the eigenfunctions for linearized bright DNLS equation are the derivatives of the squared Jost solutions. This is contrast to the counterpart for NLS, Hirota and mkdv hierarchies, where the eigenfunctions are just the squared Jost solutions. Suppressing the secular terms, they also found the slow evolution of soliton parameters and the perturbation-induced radiation. Hoseini and Marchant [4] examined bright solitary wave interaction for a focusing version of the higher-order Hirota equation. A family of higher-order embedded solitons was found, using an asymptotic transformation. When embedded solitons do not exist, soliton perturbation theory was used to determine the details of an evolving solitary wave, to first-order. In particular, an integral expression was found for the first-order correction to the solitary wave profile. They also asymptotically analyzed the integral expression to derive an expression for the tail of the solitary wave. It was shown that for the right-moving solitary wave a steady-state tail forms, while for the left-moving wave, some transients propagate on the steady-state tail. The tail amplitudes for both right and left-moving profiles were confirmed numerically and the analytical expression for the tail amplitude was consistent with the results of the asymptotic transformation. Chen et al. [7] extended soliton perturbation theory to the NLS dark soliton, overcoming the difficulties caused by divergency of the perturbed soliton parameters. By applying the IST method, the non-localized continuous) eigenfunction of the linearization operator based on the squared Jost solution, and its adjoint state, were found by using an appropriate inner product, in a manner similar to that for bright solitons and the orthogonality relationships between the non-localized and localized eigenstates and their adjoint counterparts were established. Then, as an application of Jordan s lemma, it was proved that the first-order correction to the solitary wave profile can be expanded by the squared Jost solutions. Using the similarity between the first Lax pair of the defocusing Hirota and NLS equations, the process can be extended to 3), for more details, see Section 3 and Appendix A. The resonant interaction between solitary waves and linear radiation has been the subject of much recent research. This resonance can result from the linear phase velocity being the same as the soliton velocity or from the frequency of the solitary wave being embedded in the linear wave spectrum. This resonance normally leads to radiation loss and the formation of a tail

4 120 S. M. Hoseini and T. R. Marchant behind or in front of the solitary wave. Special cases where the tail vanishes and no radiation loss occurs are called embedded solitons, see Pelinovsky and Yang [8] or Champneys [9]. Yang and Akylas [10] considered a higher-order NLS equation 1) with α 4 = α 5 = 0). They found infinite families of embedded, double-humped solitons and plotted the soliton profiles by solving the governing ordinary differential equation ode) numerically. Their stability was also considered numerically. It was found that for an energy increasing perturbation the embedded solitons are stable while energy decreasing perturbations lead to decay of the soliton. Minzoni et al. [11] considered a higher-order NLS equation which also included the η xxx term in 1). They considered a double humped wave, joined by linear radiation and used a Lagrangian averaging technique to determine the details of the embedded soliton. They showed that the humped wave has an oscillatory one-sided stability. The focusing Hirota equation has a two-parameter bright soliton family, with arbitrary amplitude and velocity, which is embedded in the linear wave spectrum. Rodriguez et al. [12] explains why these embedded solitons do not emit radiation and proves their stability. Yang [13] considered solitary wave evolution for a higher-order Hirota equation of the form η t + 6 η 2 η x + η xxx + iɛ c 1 η 2 η + c 2 η η 2 ) x ), ɛ 1. 5) Using 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 odes, describing the evolution of the wave amplitude and frequency, were found. The defocusing Hirota equation 2) has a three-parameter gray soliton family, with arbitrary amplitude and wavenumbers for the envelope and the background wave. In contrast to the bright Hirota solitons, the gray Hirota solitons are embedded in the linear wave spectrum for certain parameter values only. In Section 2, the Hirota bilinearization method is modified to find the gray soliton solution of the defocusing Hirota equation 2) and a direct substitution method is used to find the higher-order gray Hirota solitary wave solution of 3). Two families of higher-order gray solitary waves are identified. In Section 3, soliton perturbation theory is used to derive the details of an evolving gray solitary wave at first-order. An integral expression for the first-order correction to the solitary wave is derived. Explicit expressions for the steady-state and transient components of the solitary wave tails are found, along with expressions for the location of the tails. Moreover, parameter ranges in which the gray Hirota solitons are embedded are determined. A subtle and complex picture of solitary wave evolution is found, where tails occur for two reasons, radiation shedding due to resonance and corrections to the background wave at first-order. An excellent comparison is found between the theoretical and numerical solutions for the first-order solitary wave correction. In Appendix A, the Lax pairs and

5 Evolution of Higher-Order Gray Hirota Solitary Waves 121 Jost solutions for the defocusing Hirota equation are presented, while Appendix B has details of the numerical scheme used to solve 3). 2. Gray Hirota solitary waves 2.1. The gray Hirota soliton Here, by using Hirota bilinearization method, the single soliton solution for the Hirota equation 2) is found. We consider the transformation η = e iδx bt) Gx, t)f 1 x, t), 6) where Gx, t) is complex and Fx, t) is real. By using the transformation 6) in 2), the bilinearized relations for G and F { Dt + 2δ 2 + b)d x 3iδDx 2 } D3 x G.F = 0, { 3D 2 x + δ 2 b) } 7) F.F = 6 G 2, are found, where the Hirota bilinear operator is defined as Dx m Dn t f.g = x ) m x 1 t ) n f x, t)gx 1, t 1 ) x1=x,t1=t. 8) t 1 The functions G and F are defined by G = g βg 1 ) and F = 1 + β f 1. 9) By substituting 9) in 7) and collecting the terms at order β 0, g 0 satisfies { Dt + 2δ 2 + b)d x 3iδDx 2 } D3 x g0.1 = 0, g 0 2 = 1 6 b δ2 ), 10) which can be solved as 6a iκ) 2 g 0 = where b = 6µ 2 + δ 2, µ 2 = a 2 + κ 2, 11) b δ 2 ) and a and κ are two free parameters. The coefficients of β and β 2 lead to { g0 D 3 x 2δ 2 D x bd x 3iδDx 2 D ) t) + g0x 3D 2 x + 2δ 2 + b + 6iδD x + g 0xx 3D x 3iδ) Dx 3 D t) g0.1 } f g 1 ) = 0, { 3D 2 x + δ 2 b) } f 1.1 = 3 g 0 2 ) g 1 + g1, { g0 D 3 x 2δ 2 D x bd x 3iδDx 2 D ) t) + g0x 3D 2 x + 2δ 2 + b + 6iδD x + g 0xx 3D x 3iδ) D 3 x D t) g0.1 } f 1.g 1 = 0, { 3D 2 x + δ 2 b) } f 1. f 1 = 6 g 0 2 g 1 2, 12)

6 122 S. M. Hoseini and T. R. Marchant which can be solved by assuming g 1 = a + iκ a iκ, Hence, f 1 = a + iκ a iκ e2κx vt), where v = 3δ 2 + 6a 2 6δa + 2κ 2. 13) η = a iκ µ eiϕ a + iκ tanh κθ), b = 6µ 2 + δ 2, where ϕ = δx bt) + ϕ 0, θ = x vt θ 0, 14) is the Hirota gray soliton solution. Note that 14) can be also derived using the IST approach, see Li et al. [2]. This is a gray soliton solution with three free parameters a, κ, δ). a is the minimum intensity of the wave, δ is the background wave number and κ is the wavenumber of the soliton envelope. The boundary values for 14) are a iκ)2 η e iϕ, as x, η µe iϕ, as x, µ hence the amplitude of the background traveling wave is µ The higher-order Hirota solitary wave The higher-order solitary wave solution of 3) can be found directly, as η = a iκ µ a + iκ tanh κθ)eiϕ + ɛ a iκ µ B 0 + ib 1 tanh κθ + B 2 tanh 2 κθ + ib 3 tanh 3 κθ)e iϕ, where θ = x v h t, ϕ = δx b h t) + ϕ 0, 15) and the various parameters are B 1 = 1 3 c 1 + c 2 + c 3 + c 4 + 2c 5 )κδa 1 12 c 1 + c 2 + c 3 + c 4 + c 5 )δ 2 κ c 1 c 2 + c 3 + 3c 4 + 3c 5 )κ c 1 2c 2 2c 3 + c 4 c 5 )κa 2, B 2 = 1 6 2c 2 c 3 c 4 + c 5 )κ 2 a 1 6 c 1 + c 2 + c 3 + c 4 + 3c 5 )δκ 2, B 3 = 1 6 κ3 c 1 + c 2 2c 3 2c 4 2c 5 ),

7 Evolution of Higher-Order Gray Hirota Solitary Waves 123 with the velocity v h and parameter b h given by v h = 3δ 2 + 6a 2 6δa + 2κ 2 + c 1 a c 1 + 2c 4 2c 3 + 2c 5 )κ 4 + 3c 2 c 5 6c 3 3c 4 )δ 2 a c 1 2c 4 3c 3 2c 5 )δ c 4 + c 1 c 2 + 3c 5 + 5c 3 )aδ 3 + c 4 c 1 c 5 + 4c 3 2c 2 )a 3 δ 2c 3 κ 2 a 2 + 2c 3 2c 5 )aκ 2 δ + 2c 3 + c 2 + c 5 c 4 )κ 2 δ a 6δ)B 0 )ɛ, b h = 6µ 2 + δ 2 + c 1 a 4 + 2c 4 2c 3 + c 1 + 2c 5 )κ 4 + c 4 + c 2 2c 3 c 5 )δ 2 a 2 16) c 1 3c 3 2c 5 2c 4 )δ 4 + 2c 3 + 2c 5 + 2c 1 + 2c 4 + 2c 2 )aκ 2 δ 6c 3 κ 2 a aB 0 + c 1 2c 4 3c 3 2c 5 )κ 2 δ 2 )ɛ. The parameter B 0 is arbitrary and represents a higher-order correction to the background wave. The higher-order gray one-soliton solution 15) 16) can only be derived if one of the algebraic relationships c 1 + 3c 3 + 2c 4 + 2c c 6 = 0, 17) 2κ 2 + 3κδ δ 2 = 0, 18) is satisfied. The condition 17), relating the higher-order coefficients, implies that a three-parameter family of higher-order asymptotic gray Hirota solitary waves exists. The condition 18) relates the three free parameters. Physically 18) implies that the soliton velocity is the same as the phase velocity of the background wave v = b from 14)), and implies the existence an additional two-parameter family of higher-order asymptotic gray Hirota solitary waves. 3. Perturbation theory and solitary wave evolution It was found in Section 2.2 that two and three-parameter families of higher-order asymptotic gray Hirota solitary waves exist if 17) or 18) is satisfied. In this section direct soliton perturbation theory, based on the Jost solution of the defocusing Hirota equation, is used to determine the details of the evolving solitary wave, to first-order. Analytical expressions are found for the tails of the solitary wave. Tails form for two reasons; the shedding of radiation due to resonance and first-order corrections to the background traveling wave.

8 124 S. M. Hoseini and T. R. Marchant Comparisons of the solitary wave tails are made with numerical solutions and an excellent agreement is found The perturbation solution Given the similarity between the dark Hirota and NLS solitons, the technique for analyzing them is closely related. We first investigate the linear stability of the background radiation for the Hirota equation, see Kivshar and Luther-Davies [1] for the application to the NLS equation. The traveling-wave solution of the defocusing Hirota equation 12) is given by η = ρ 0 e iδx iβt, β = 6ρ 2 0 δ + δ3, 19) where ρ 0 and δ are free parameters. To investigate the stability we substitute η = ρ 0 + ν)e iδx iβt+iϕ, 20) as a small excitation to 19) in the defocusing Hirota equation 2), where ν and derivatives of ϕ are assumed small. By taking ν, ϕ e ipt iqx, the dispersion relation q 3 p + 3qδ 2 + 6ρ0 2 q) 2 = 9q 2 δ 2 q 2 + 4ρ0 2 ), 21) is obtained. Solving 21) gives v p± = p q = q2 + 3δ 2 + 6ρ 2 0 ± 3δ q 2 + 4ρ 2 0, 22) v g± = p q = 3q2 + 3δ 2 + 6ρ0 2 ± 3δ q 2 + 4ρ0 2 ± 3δq 2, 23) q 2 + 4ρ0 2 as the phase velocities v p±, and the group velocities v g±, of the perturbed wave, respectively. As p is real, small excitations of the background wave 19) are stable. Note that for the focusing Hirota equation the excitation 20) is unstable. Letting q = 0, gives v m± = 6µ 2 + 3δ 2 ± 6δµ, 24) as the minimum phase velocities of the perturbed wave. Hence there are two different phase velocities for small perturbations propagating on the background wave. This is the same situation as the case of the defocusing NLS equation, see 2.14) of Kivshar and Luther-Davies [1], for which two phase velocities are also identified. Linear resonance occurs between the soliton and the linear radiation when it is possible for the soliton to travel at the same speed as the linear radiation. There are two possible phase velocities 22) for the perturbed wave, which have the minima 24). When the soliton velocity v from 13) is equal to the

9 Evolution of Higher-Order Gray Hirota Solitary Waves 125 smaller of the minimum phase velocities, then resonance is possible. Letting v = v m± gives 2κ 2 ± 3δµ + 3δa = 0, 25) as the condition for resonance to occur. Factorizing 25) gives δ = 2 a ± µ), 3 and hence condition [ 2 δ I = 3 a µ), 2 ] a + µ), 26) 3 implies linear resonance can not occur. This condition is also obtained from the asymptotic analysis in Section 3.2. When δ I Hirota solitons are not embedded. This can be clearly seen from the special case δ = 0 I, for which the soliton velocity v = 6a 2 + 2κ 2, which is less than the phase velocity v p = 6a 2 + 6κ 2 + q 2. Hence for the defocusing Hirota equation, gray embedded solitons only occur for δ I c. This is in contrast to the bright Hirota solitons, which are embedded for all parameter values. The single solitary wave solution 14) can be considered as an excitation on the background wave 19) and may be formulated as ψ 0 = a iκ µ a + iκ tanh κθ)eiδθ+ϕ), ϕ = 2δ 3aδ + δ 2 2κ 2 )t ϕ 0, θ = x vt θ 0, where v = 3a 2 + 2κ 2 + 3a δ) 2,µ 2 = a 2 + κ 2, 27) where a, κ, δ, ϕ 0 and θ 0 are all free parameters. The perturbed solution of the higher-order Hirota equation 3) is then written as ψ = e iϕ ηθ,t, T 1, T 2,...), θ = x t 0 v dt θ 0, ϕ = t 0 λ dt ϕ 0. 28) The soliton parameters κ, a, δ, θ 0, and ϕ 0 all vary on the slow time scales T n = ɛ n t, n = 1, 2,... The forms of the slow variation are chosen to eliminate the secular terms in the expansion. Some of these integrals are divergent, hence the coefficients of these divergent terms must be equal to zero, see, for example, Chen et al. [7]. This feature of the perturbation method for the dark soliton is different to the counterpart for the bright soliton. In our analysis below, these slow variations, which are not explicitly shown, are determined to let η 1 in 40) rest on a zero mean level. Hence, the mean level of the background 19) does not change in the presence of the perturbed terms in 3). The only effect of the higher-order terms on the traveling wave background 19) is a phase shift, at Oɛ), which is given by β = 6µ 2 δ + δ 3 + ɛδ c 1 µ 4 + c 2 δ 2 µ 2 2c 3 δ 2 µ 2 c 4 δ 2 µ 2 c 5 δ 2 µ 2 + c 6 δ 4). 29)

10 126 S. M. Hoseini and T. R. Marchant The main focus here is to find the first-order correction, η 1, to the solitary wave profile, which allows analytical expressions to be found for the tails of the solitary wave. By substituting 28) in 3) one obtains η t + iλη vη θ η θθθ + 6 η 2 η θ = ɛhη) ɛη T1 iηϕ 0T1 η θ θ 0T1 ), 30) where H represents the perturbed terms in 3). The solution ηθ, t, T 1, T 2,...) can be found by expanding as η = η 0 θ) + ɛη 1 θ,t, T 1, T 2,...) +..., 31) and substituting in 30), where the initial solution is η 0 θ) = a iκ) µ 1 a + iκ tanh κθ)e iδθ. At the zeroth-order, the equation is satisfied automatically, and at first-order, η 1t + iλη η 0 2 v)η 1θ η 1θθθ + rθ)η 1 + qθ)η1 = ω 1, where rθ) = 6η 0 η 0 θ, qθ) = 6η 0 η 0θ, ω 1 = Hη 0 ) η 0T1 + iη 0 ϕ 0T1 + η 0θ θ 0T1, 32) is obtained. The linearized Equation 32) can be written in the matrix form t + L)w 1 = H and is solved using the initial condition η 1 t=0 = 0, where ) Gθ) qθ) L = q θ) G, w 1 = η 1,η T θ) 1), H = ω1,ω1) T, Gθ) = θθθ + 6 η 0 2 v) θ + rθ) + iλ. 33) It is not difficult to show that the non-localized continuous) eigenstate of L is found as =S S = S 2 1, S2 2 )T, where S = S 1, S 2 ) T is a Jost solution of the Hirota equation. Thus the non-localized eigenstates can be found similar to Chen et al. [7] by a iκ) 2 w 2 = µ 4 α a + iκ) 2 ) µ 2 α a iκ + iκ sech κθe κθ ) 2 e iδ 2ω)θ, iµ 2 α 1 α a iκ) κa + iκ) sech κθe κθ ) 2 e iδ+2ω)θ 34) see Appendix A. The continuous eigenfunction of the adjoint operator L = σ 2 Lσ 2 is also needed, where ) 0 i σ 2 =, 35) i 0 is a Pauli matrix. It is easy to show that w 2 = a, b ) T is a continuous eigenfunction for L where w 2 = a, b) T is the eigenfunction for L in 33). The inner product

11 Evolution of Higher-Order Gray Hirota Solitary Waves 127 fθ), gθ) = fθ) T gθ) dθ, 36) is used to find nonzero inner products of the non-localized and localized eigenstates and their adjoint eigenstates. Note that the forms of two localized eigenfunctions can be determined explicitly by calculating w 2 α=a+iκ and dw 2 dα α=a+iκ. The completeness of the set of eigenstates can be proved in an analogous manner to the proof for the NLS counterpart, which has been presented in Yan et al. [14], using Green s function theory, and in Chen et al. [7], by using Jordan s lemma and in Huang et al. [15] by applying a generalized Marchenko equation. The first-order solution has the form + w 1 = gw 2 dα, where g = imα) ϑ 1 eiϑt ), ϑ = 8ω a iκ)2 kα) = 2π µ 2 ξ + a 32 δ ) ξ a), Mα) = and the auxiliary variables are defined as H, w 2, kα) ) α a iκ 2 1 µ 2 α 2 ), 37) α a + iκ ω = 1 2 α µ2 α 1 ), ξ = 1 2 α + µ2 α 1 ). 38) The quantity Mα), found using the residue theorem, is Mα) = iα2 µ 2 )a + iκ)α a + iκ) 2 [α + µ 6 α 5 )E 1 360µ πω µ 4 α 4 )E 2 + α 1 + µ 2 α 3 )E 3 + α 2 E 4 ] sinh κ where the various coefficients are given by E 1 = c 1 5c 2 8c 3 2c 4 2c 5 120c 6, E 2 = 5c 1 + 5c c c c 6 )δ 2c 1 10c c c 4 6c c 6 )a, E 3 = 31c c c 3 8c c c 6 )a 2 10c c 2 80c 3 50c c 5 600c 6 )aδ ), 39) + 45c 2 90c 3 45c c 5 900c 6 )δ c 1 15c c 3 )κ 2, E 4 = 80c 1 10c c c c c 6 )κ 2 δ 45c 2 90c 3 45c 4 45c 5 900c 6 )δ 3

12 128 S. M. Hoseini and T. R. Marchant + 90c 2 180c 3 90c 4 150c c 6 )δ 2 a + 64c c c 3 + 8c 4 + 8c c 6 )a 3 120c c c c 4 30c 5 )a 2 δ + 32c c c 3 16c c c 6 )aκ 2. And finally we obtain η 1 θ,t) = ia iκ) 2 µ 2 + M ϑ 1 e iϑt ) α a + iκ) 2 α a iκ + iκ sech κθe κθ ) 2 e iδ 2ω)θ dα, 40) as the first-order correction to the solitary wave. This is qualitatively similar to the second integral in the first-order correction to the bright Hirota solitary wave profile, see 36) in Hoseini and Marchant [4]. We also note the limit 1 e ±iϑt ) lim = it, 41) ϑ 0 ϑ thus the integrand of 40) is singular at ϑ = 0ast Asymptotic results and the solitary wave tail In contrast to bright Hirota solitons, gray Hirota solitons are right-moving only. The analysis below shows that the evolution of the solitary wave tails for the higher-order defocusing Hirota equation is subtle and complicated. For some parameter ranges, tails only appear in front of the solitary wave, whilst for other parameter choices tails appear in front of and behind the solitary wave. For higher-order bright Hirota solitary waves non-zero tails only occur when embedded asymptotic solitons do not exist and hence radiation is shed. For higher-order gray Hirota solitary waves tails also occur for another reason; the first-order correction of the background wave, well ahead and behind the solitary wave. The asymptotic analysis considered here is similar to that of Hoseini and Marchant [4], and references therein. The contribution to 40) for large time comes from the neighborhood of the singularities. Note that for δ I see the definition 26)), the leading-order contribution to 40) can be found by considering the singular points α =±µ, and for δ I c, the extra singular points α = a δ ± 1 2 9δ2 12aδ 4κ 2 ) 1 2 α ± c, 42) are needed. Hence when δ I, there are two singular points and when δ I c, there are four. We suppose µδ 0 and define the quantities

13 Evolution of Higher-Order Gray Hirota Solitary Waves 129 S ± = µ a 32 )) δ µ ± a), h ± tail = π a iκ) 2 ) M µ) µ a iκ 2. 43) 4 µ 2 S µ a + iκ The length of the solitary wave tails is 4S ±, while their respective amplitudes are h ± tail. It should be noted that when µδ < 0, the procedure is slightly modified by replacing h ± tail and S± in 43) by h tail and S. For δ I we begin our analysis by making the substitution α =±µ + z/t and obtain η 1 i 2π h± tail eiδθ + ) e 2izc p e 2izc p 4S ± ) dz, 44) z where c p = θ/t is an O1) constant. Well behind of the soliton, θ 1 and c p < 0. Using the integral e izp dz = iπsgnp), 45) z implies the integrals within 44) have the same value. Thus the cancellation of the two integrals implies η 1 0 well behind the solitary wave. Hence tails only form ahead of the solitary wave for δ I. Ahead of the wave θ 1), we choose two intervals, 0 < c p < 4S and 4S < c p < 4S +. Then the leading-order expression for 40) has the form η 1 { h tail + h+ tail) e iδθ, 0 < c p < 4S, h + tail eiδθ, 4S < c p < 4S +. 46) We note that for the special case δ = 0, S = S + ), the second tail vanishes and 46) yields a single tail which propagates ahead of the solitary wave. The length of the solitary wave tails 4S ±, is the difference between the soliton velocity 13) and the group velocity of the perturbations on the background wave 23), for q = 0. The two tails correspond to the two different group velocities in 23). Moreover, for δ I, both background group velocities are greater than the soliton velocity, thus tails only form ahead of the wave. The wavenumber of the tail is δ, the same as the background wave, thus it represents a higher-order correction to the background wave, and is not due to linear resonance. For δ I c a similar procedure for large time yields h + tail eiδθ + h ± c eiδ ± 1 )θ + T +, 0 < c p < 4S +, η 1 γµ a + iκ γµ a iκ ) 2 h tail eiδθ + T, 4S < c p < 0, 47)

14 130 S. M. Hoseini and T. R. Marchant where γ = sgnδµ). A new tail component is now generated ahead of the solitary wave. Its amplitude is h ± iδa iκ)π α ± c = c a iκ ) 2 10µ α c ±3 ± 2 2a) c ± 1 + 3c 3 + 2c 4 + 2c c 6 ) 1 ] [ 4S S + κ 2 3µ 2) α ± c + µ2 ± 2 π ± 2κ 2 + 3aδ δ 2 ) sinh 1 1 2κ where ± 1 = α±2 c µ 2, ± 2 = α±2 c + µ 2. 48) Note that in 47), h ± c α ± c 1 = 0, in α ± c ), 2 2a) < c p < 4S +, if a) < S +. 49) The fact that tails now occur in front of and behind the soliton is due to one of the group velocities 23) now being less than the soliton velocity, while the other is greater. Moreover the tail in front of the soliton now consists of components with three different wavenumbers, hence the long-time tail is oscillatory in nature. In contrast, behind the solitary wave the long-time tail will be flat, as it consists of a component with a single wavenumber. The two leading transient terms are calculated from contributions at points of stationary phase. The relevant phase of 40) is ϑ 1 = ϑ 2ωc p, which has the stationary point when ϑ 1 = dϑ 1 dα = Rα 4 = 0, where R = 3α 6 6δα 5 + 6δa c p + µ 2 4a 2) α 4 + µ 2 6δa c p + µ 2 4a 2) α 2 6δµ 4 α + 3µ 6. 50) Due to the high order of 50), the solutions are found numerically for constant values of a, k, δ, asc p is varied. It is not difficult to show that α 1 s µ 2 is a stationary point when α s is a stationary point. Numerical solutions of R show that for finite c p, which is analyzed here, there are just two roots for R. Using the method of stationary phase gives T ± = i2π) 1 2 t 1 2 a iκ) 2 µ 2 ϑ 1 α s) 1 2 Mαs )ϑ 1 α s )α s a + iκ) 2 [ α s a iκ) 2 e iϑ 1α s )t±sgnϑ 1 α s)) π 4 ) αs 1 µ 2 a iκ ) 2 e iϑ 1 α s )t±sgnϑ 1 α s)) π )] 4 e iδθ. 51) It is noteworthy that the tail component h ± c only occurs for δ I c.in Section 3.1 it was shown that this condition implied that the gray Hirota

15 Evolution of Higher-Order Gray Hirota Solitary Waves 131 solitons were embedded, thus the tail components h ± c are associated with radiation shedding due to linear resonance. The other tail components, with amplitudes h ± tail, have the same wavenumber, δ, as the background wave, thus are associated with corrections to the background wave at first-order. The components h ± c vanish if the algebraic conditions 17) or 18) are satisfied. Hence these conditions imply the existence of higher-order embedded gray Hirota solitary waves, if δ I c Numerical solutions and discussion Here the governing higher-order defocusing Hirota equation 3) is solved numerically and the results compared with the first-order solitary wave correction 40) for η 1. The main analytical features of the solitary tails, for different values of wavenumber δ, are also verified. For all the examples we choose ɛ = 0.05 and use the soliton parameters κ = a = 2 and µ = 2. The perturbation solution 40), is solved numerically by a higher-order quadrature scheme, and the higher-order Hirota equation 3), is solved using the fourth-order hybrid Runge Kutta finite-difference scheme, see Appendix B. The spatial and temporal gridspacings used are x = and t = The quantity ɛ 1 η η 0 is plotted from the numerical solution as it allows a comparison with the perturbation solution in the tail region. Also note that η 0 has the form 14) with b modified to include the higher-order phase corrections to the background radiation 29). For the first example, we choose δ = 0 I, thus the Hirota soliton is not embedded. Figure 1 shows the first-order correction η 1 versus θ at time t = 3. The higher-order coefficients used are c 1 = c 4 = 1 with all the other c i = 0. There is an excellent comparison between the numerical and perturbation solutions. The figure shows the surface elevation ahead the evolving solitary wave with the flat steady-state tail clearly visible. For this example the tail amplitude is given by h tail + ) h+ tail e iδθ = 2iκ5 aµ 9S + S 2c 2 + c 4 + c 3 c 5 ). 52) The amplitude of this tail, as predicted by the perturbation solution 40), is η 1 =0.89, which is the same as the analytical tail amplitude 52). The numerical tail amplitude is 0.924, a difference of about 3%. The maximum amplitude of the real part of the numerical tail is , which represents a small error compared with the analytical tail amplitude 52), which has a real zero part. The asymptotic theory implies that the length of the solitary wave tail is 4tS 24, which matches closely with the numerical results. For this example, where δ = 0, there is only one group velocity in 23), thus only one tail forms. Moreover, as 23) is greater than the soliton velocity, the tail only forms in front of the wave.

16 132 S. M. Hoseini and T. R. Marchant Figure 1. The first-order correction η 1 versus θ at t = 3 for wavenumber δ = 0. The parameters c 1 = c 4 = 1 and the other c i = 0. Shown is the perturbation solution 40) dashed curve) and numerical solution of 3) solid curve). This example does not satisfy 17), but for examples which do satisfy 17), the long time expression for η 1 is not the first-order correction to the higher-order solitary wave 15). This is because there are first-order corrections to the background wave, in front of and behind the wave, whilst the evolving wave for δ I) only has first-order corrections in front of the wave. The special case, where c 2 = c 4 = 1, c 5 = 1 and the other c i = 0, has no higher-order corrections to the background level in 15), thus η 1 = 0. In this case the perturbation theory predicts that no tails occur and its result is consistent with the higher-order solitary wave solution 15). Figure 2 shows the first-order correction η 1 versus θ at t = 3, for δ = 1 I, and c 1 = 1, with the other c i = 0. The other parameters are the same as for figure 1. Again, tails only appear in front of solitary wave. The figure shows the region in front of the moving solitary profile. For this example δ 0, thus the two distinct group velocities 23) result in two different tail regions occurring. The comparison between the numerical and perturbation solutions is again very good, with some very slight differences in amplitude and phase. The tail closest to the wave occurs in the region 0 <θ<4ts 13.45, with an analytical tail amplitude of The numerical tail amplitude is 0.375, which is a difference of about 3%. For the second tail, which occurs for tS <θ<4ts +

17 Evolution of Higher-Order Gray Hirota Solitary Waves 133 Figure 2. The first-order correction η 1 versus θ at t = 3 for wavenumber δ = 1. The parameters c 1 = 1 and the other c i = 0. Shown is the perturbation solution 40) dashed curve) and numerical solution of 3) solid curve) , the average taken of the numerical solution is 0.762, which is within 0.4% of the analytical steady-state tail amplitude of Oscillations on the second tail represent an undular bore, which links together the different amplitudes of the two tails, see, for example, Marchant and Smyth [16]. Lastly, we investigate the tail evolution for an example of δ I c for which the gray Hirota soliton is embedded. The parameters are the same as the Figure 2 except we choose δ = 1. Figure 3 shows the first-order solution η 1 versus θ at t = 3. Shown are the perturbation solution 40) and the quantity ɛ 1 η η 0 from the numerical solution of 3). Again, the numerical solution is very close to the perturbation solution 40), with slight phase and amplitude differences. The average taken of the numerical solution over the tail for θ 1, is , which is very close to the analytical tail amplitude of , a difference of about 0.6%. A similar average of the numerical solution, taken in front of the solitary wave, gives an amplitude of , which differs by about 1% from the analytical amplitude of Note that for this example, the h ± c contributions, resulting from linear resonance, are quite small with h+ c, h c of O10 2 ). For this example, the solution velocity lies between the two group velocities 23), thus tails occur behind and in front of the solitary wave. The figure

18 134 S. M. Hoseini and T. R. Marchant Figure 3. The first-order correction η 1 versus θ at t = 3 for wavenumber δ = 1. The parameters c 1 = 1 and the other c i = 0. Shown is the perturbation solution 40) dashed curve) and numerical solution of 3) solid curve). shows that the numerical tails occur between 40 θ 35. The asymptotic results 47) yields tail lengths of 4tS = 37.5 and 4tS , which are very close to the numerical values. In contrast to the flat tails, which occur for wavenumbers in δ I, the tails here are highly oscillatory. In front of the soliton, the superposition of the steady-state tail components of different wavenumbers with the transients 51) leads to a classical beating-like phenomena where the tail has multiple peaks and troughs of higher and lower amplitudes. For the tail behind the solitary wave the oscillations, which are analytically represented by T, form an undular bore, which links the mean level of zero, well behind the solitary wave, with tail amplitude of h + tail. The maximum amplitude crest to trough) of the oscillations behind the soliton is 0.33 at t = 3. As the oscillations decay like Ot 1 2 ), a relatively flat tail will not be reached until t = O10 6 ), when the amplitude of the oscillation will have decayed to less than 1% of this value. In summary, for a number of examples, the soliton perturbation theory, both analytical results and direct numerical solutions of 40) and numerical solutions of the governing pde 3) are in good agreement. Moreover, the results are consistent with the directly derived higher-order solitary wave 15) and the perturbated background wave 21).

19 Evolution of Higher-Order Gray Hirota Solitary Waves Conclusion Two families of higher-order gray Hirota solitary waves are identified which exist when 17) or 18) is satisfied), with these families being embedded in the linear wave spectrum for a certain parameter region, δ I c. Soliton perturbation theory is used to determine the evolution of the gray Hirota soliton for the higher-order defocusing Hirota equation, with an integral expression found for the first-order correction to the solitary wave profile. Asymptotic expansions, valid for large time, allows analytical expressions to be found for the solitary wave tails for the two main parameter regions. An extremely complicated picture of tail evolution emerges. In some cases, tails only form in front of the solitary wave, while in other cases they occur on both sides of the wave. Moreover, solitary wave tails form for two reasons; shedding of radiation due to linear resonance and corrections to the background wave at first-order. This paper shows that soliton perturbation theory is an extremely powerful technique for understanding solitary wave evolution for higher-order systems, which represent small corrections to integrable ones. In the future we plan to apply the method developed here to bright and gray solitary waves of other higher-order nearly integrable systems. Appendix A: Lax equations and Jost solutions for the Hirota equation In this Appendix the Lax pairs and Jost solutions for the defocusing Hirota equation 2) are presented. In Section 3.1 it is shown that the nonlocalized continuous) eigenfunction 34) of the linearized operator L 33) is just the squared Jost solution of the Hirota equation. It can be shown that the representation of Hirota equation in terms of the so-called Lax equations is S x = MS, S t = NS, where S = S 1 x, t), S 2 x, t)) T 53) and the Lax pair is M = i ξ 12 ) δ σ 3 + U, N = 4i ξ 1 ) 3 2 δ σ 3 4 ξ 1 ) 2 2 δ U + 2i ξ 12 ) δ U 2 + U x )σ 3 54) 2U 3 + U xx + UU x U x U, ) ) 0 η 1 0 where U = η, and σ 3 = 0 0 1

20 136 S. M. Hoseini and T. R. Marchant is a Pauli matrix. It is easy to verify that Hirota equation is tantamount to the zero curvature representation M t N x + [M, N] = 0 of the Lax equations 53). Now the affine auxiliary parameter α is introduced to make ω = ξ 2 µ 2 as a single-valued function. Hence the parameters ω and ξ are functions of α as in 38). As x i.e., η µe iδx ) the Jost solution of 53) approaches E + x,α) = e 1 2 iδxσ 3 1 iµα 1) iµα 1 e iωxσ 3, 55) 1 and as x the asymptotic solution can be chosen as E x,α) = e 1 2 iθ sσ 3 E + x,α), where, θ s, the phase difference between each end of the boundary values, is determined as η µe iθ s, as x. 56) In other words the relationships between the Jost solutions of the Lax pair 53) and the asymptotic solutions E + x, α) and E x, α) can be found as x,α) = ψx,α),ψx,α)) E + x,α)asx +, x,α) = φx,α), φx,α)) E x,α)asx. 57) Also noteworthy is that the asymptotic behavior of Jost solutions 55) are exactly the same as NLS equation counterparts. This equivalence results from the fact that the first Lax equation of 53) and 54) are common for the Hirota and NLS equations. For example, in the single-soliton case, i.e., µ 2 e iθ s = a iκ) 2 ), the explicit expressions of the Jost solutions can be found as and ψx,α) = φx,α) = iµα 1 κ α α 1 1 iκ α α1 α α1 α α1 e 1 2 iθ s + e 1 2 iθ s sech κxe κx ) e i 1 2 δ+ω)x sech κxe κx ) e i 1 2 δ ω)x iκ α α 1 iµα 1 α α 1 α α1 e 1 2 iθ s, e 1 2 iθ s sech κxe κx ) e i 1 2 δ ω)x κ α α 1 sech κxe κx ) e i 1 2 δ+ω)x φx,α) = iµα 1 φx,µ 2 α 1 ), ψx,α) = iµα 1 ψx,µ 2 α 1 ), where α 1 = a + iκ, see Chen et al. [7].,

21 Evolution of Higher-Order Gray Hirota Solitary Waves 137 Appendix B: The numerical scheme for the higher-order defocusing Hirota equation The numerical solutions for the higher-order defocusing Hirota equation 3) were obtained by using a hybrid scheme consisting of fourth-order centered finite differences in the spatial coordinate x and a fourth-order Runge Kutta method for the temporal coordinate t. The numerical scheme described below is stable for reasonable choices of the space and time discretization units x and t. Various straight finite-difference methods tested by the authors were found to be unstable for nearly all nonzero values of ɛ. Given that the solution at the time t i is η i, j = ηx j = j x, t i = i t), j = 1,...,M), 58) then the numerical solution at time t i+1 is given by η i+1, j = η i, j k a i, j + 2ki, b j + 2kc i, j + kd i, j) + Dx j ), j = 1,...,M), 59) where the various functions are k a i, j = tfη i, j), k c i, j = tf η i, j kb i, j k b i, j = tf η i, j + 1 ) 2 ka i, j, ), k d i, j = tf η i, j + k c i, j), 60) and Dx) is the damping function used at the boundaries, thus the boundaries do not affect the wave evolution being studied. The function f is the finite-differenced form of all the terms in 3) involving spatial derivatives, f p i, j ) = p i, j ɛc1 p i, j 2) A ɛc3 p 12 x 8 x 3 i, j 2) A 3 ɛc 2 + c 5 )p i, j 124 x 3 A 2 A 1 ɛc 3 + c 4 )p i, j 124 x 3 A 2 A 1 ɛc 2 p i, j 124 x 3 A 2 A 1 ɛc x 3 A 1 2 A 1 ɛc 5 6 x 5 A 4, 61)

22 138 S. M. Hoseini and T. R. Marchant where the fourth-order centered finite-difference formulas are given by A 1 = p i, j 2 8p i, j 1 + 8p i, j+1 p i, j+2, A 2 = p i, j p i, j 1 30p i, j + 16p i, j+1 p i, j+2, A 3 = p i, j 3 8p i, j p i, j 1 13p i, j+1 + 8p i, j+2 p i, j+3, A 4 = p i, j 4 9p i, j p i, j 2 29p i, j p i, j+1 26p i, j+2 + 9p i, j+3 p i, j+4. The function Dx j ) = sech 2 [ j 1) x 10 ] [ + sech 2 j M) x ], 62) 10 damps the boundary values to zero which stops the reflection of the small-amplitude dispersive radiation back into the solution domain. The accuracy of the numerical method is fourth-order in time and space, i.e., O t 4, x 4 ). References 1. Y. S. KIVSHAR and B. LUTHER-DAVIES, Dark optical solitons: Physics and applications, Phys. Rep. 298: ). 2. L. LI, Z. LI, Z. XU, G. ZHOU, and K. H. SPATSCHEK, Gray optical dips in the subpicosecond regime, Phys. Rev. E 66: ). 3. A. MAHALINGAM and K. PORSEZIAN, Propagation of dark solitons with higher-order effects in optical fibers, Phys. Rev. E 64: ). 4. S. M. HOSEINI and T. R. MARCHANT, Solitary wave interaction and evolution for a higher-order Hirota equation, Wave Motion 44: ). 5. J. YANG, Complete eigenfunctions of linearized integrable equations expanded around a soliton solution, J. Math. Phys. 41: ). 6. X.-J CHEN and J. YANG, Direct perturbation theory for solitons of the derivative nonlinear Schrödinger equation and the modified nonlinear Schrödinger equation, Phys. Rev. E 65: ). 7. X.-J. CHEN, Z.-D. CHEN, and N.-N. HUANG, A direct perturbation theory for dark solitons based on a complete set of the squared Jost solutions, J. Phys. A: Math. Gen. 31: ). 8. D. E. PELINOVSKY and J. YANG, A normal form for nonlinear resonance of embedded solitons, Proc. Roy. Soc. Lond. A 458: ). 9. A. R. CHAMPNEYS, B. A. MALOMED, J. YANG, and D. J. KAUP, Embedded solitons: solitary waves in resonance with the linear spectrum, Physica D 152: ). 10. J. YANG and T. R. AKYLAS, Continuous families of embedded solitons in the third-order nonlinear Schrödinger equation, Stud. Appl. Math. 111: ). 11. A. A. MINZONI, N. F. SMYTH, and A. L. WORTHY, A variational approach to the stability of an embedded NLS soliton at the edge of the continuum, Physica D 206: ).

23 Evolution of Higher-Order Gray Hirota Solitary Waves R. F. RODRIGUEZ, J. A. REYES, A. ESPINOSA-CERON, J. FUJIOKA, and B. A. MALOMED, Standard and embedded solitons in nematic optical fibers, Phys. Rev. E 68: ). 13. J. YANG, Stable embedded solitons, Phy. Rev. Lett. 91: ). 14. T. YAN, H. CAI, and N.-N. HUANG, Direct perturbation theory for nearly integrable nonlinear equation with application to dark-soliton solutions, J. Phys. A: Math. Gen. 39: ). 15. N.-N. HUANG, S. CHI, and X.-J. CHEN, Foundation of direct perturbation method for dark solitons, J. Phys. A: Math. Gen. 32: ). 16. T. R. MARCHANT and N. F. SMYTH, The initial boundary problem for the Korteweg-de Vries equation on the negative quarter-plane, Proc. R. Soc. Lond. A 458: ). VALI-E-ASR UNIVERSITY UNIVERSITY OF WOLLONGONG Received October 26, 2007)