It is shown that the N-loop soliton solution to the short-pulse equation may be decomposed exactly into N separate soliton elements by using a Moloney-Hodnett type decomposition. For the case N = 2, the decomposition is used to calculate the phase shift of each soliton caused by its interaction with the other one. Corrections are made to some previous results in the literatur

Parkes, E.J.

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Strathprints Institutional RepositoryParkes, E.J. (2010) A note on loop-soliton solutions of the short-pulse equation. Physics Letters A,374. pp. 4321-4323. ISSN 0375-9601Strathprints is designed to allow users to access the research output of the University of Strathclyde.Copyright c© and Moral Rights for the papers on this site are retained by the individual authorsand/or other copyright owners. You may not engage in further distribution of the material for anyprofitmaking activities or any commercial gain. You may freely distribute both the url (http://strathprints.strath.ac.uk/) and the content of this paper for research or study, educational, ornot-for-profit purposes without prior permission or charge.Any correspondence concerning this service should be sent to Strathprints administrator:mailto:strathprints@strath.ac.ukhttp://strathprints.strath.ac.uk/A note on loop-soliton solutions of theshort-pulse equationE. J. Parkes aaDepartment of Mathematics and Statistics, University of Strathclyde, LivingstoneTower, Richmond Street, Glasgow G1 1XH, UKAbstractIt is shown that the 푁 -loop soliton solution to the short-pulse equation may bedecomposed exactly into 푁 separate soliton elements by using a Moloney–Hodnetttype decomposition. For the case 푁 = 2, the decomposition is used to calculate thephase shift of each soliton caused by its interaction with the other one. Correctionsare made to some previous results in the literature.Key words: short-pulse equation; sine-Gordon equation; 푁 -loop soliton;Moloney–Hodnett decomposition; phase shift.PACS: 02.30.Jr, 02.30.Ik, 05.45.Yv.1 IntroductionThe short-pulse equation (SPE), namely푢푥푡 = 푢+16(푢3)푥푥 , (1.1)models the propagation of ultra-short light pulses in silica optical ﬁbres [1]. (TheSPE is also known as the cubic Rabelo equation [2].) In recent years, various aspectsof the SPE have received attention in the literature. A useful summary of some ofthis work is given in [3]. Here we focus on the 푁 -loop soliton solution to the SPE.Such solutions may be found by making use of 푁 -soliton solutions of equationsrelated to the SPE. Our aim is to complement results on this aspect of the SPE asgiven in [4–7].In [4,5] it was shown that the SPE is related to a system of coupled nonlineardispersionless equations (CNDE) that are a special case of the system studied in [8].Email address: e.j.parkes@strath.ac.uk (E. J. Parkes).Preprint submitted to Physics Letters A 11 August 2010Also, as mentioned in [2,6,7,9–11], the SPE is related to the sine-Gordon equation(SGE)푧푦휏 = sin 푧 (1.2)by the transformation푢(푥, 푡) = 푧휏 (푦, 휏), 푥 = 푤(푦, 휏), 푡 = 휏, (1.3)where푤푦 = cos 푧, 푤휏 = −12푧2휏 . (1.4)In [12], we found some periodic and solitary travelling-waves solutions of the SPE(1.1) by direct integration. In [4], the two-loop soliton solution to the SPE wasfound via the two-loop soliton solution to the CNDE given in [8]. In the followingpapers, solutions to the SPE were found via solutions to the SGE: [6] (one- andtwo-loop solitons, single breather), [5] (푁 -loop solitons), [9] (travelling waves), [7](푁 -loop solitons, multi-breathers) and [11] (one- and two-phase periodic). Hirota’sD-operator method was used in [4,5,7,8].We complement the work in [4–7] as follows. In Section 2 we show that the 푁 -loopsoliton solution to the SPE can be decomposed exactly into 푁 separate solitonelements by using a Moloney–Hodnett type decomposition; this is in contrast tothe approximate decomposition in [4]. In Section 3 we use our decomposition tocalculate the phase shifts in the case 푁 = 2. We also point out errors in thecorresponding phase-shift calculation given in [4].2 The decomposition of the 푁-soliton solutionAs noted in [13,14], the original route to the 푁 -soliton solution of the SGE by useof Hirota’s method was via the bilinear transformation푧 = 4 tan−1(퐺/퐹 ), (2.1)where 퐹 and 퐺 are real functions of 휉푗 (푗 = 1, 2, . . . , 푁),휉푗 = 휉ˆ푗 + 휉0푗, where 휉ˆ푗 = 푝푗 푦 +휏푝푗, (2.2)the 푝푗 are arbitrary non-zero constants, and the 휉0푗 are arbitrary constants. Forour purposes it is more convenient to use the equivalent bi-logarithmic bilineartransformation as given in [15,16,7], namely푧 = 2푖 ln(푓 ∗/푓), (2.3)2where 푓 := 퐹 + 푖퐺 and * denotes the complex conjugate. In view of (1.3) and (2.3),Matsuno [7] deduced that the 푁 -soliton solution of the SPE may be found via thebi-logarithmic bilinear transformation푢(푥, 푡) := 푈(푦, 휏) = 2푖[ln(푓 ∗/푓)]휏 . (2.4)He also made the perceptive and important observation that the relation 푤푦 = cos 푧in (1.4) can be integrated to obtain푥 := 푤(푦, 휏) = 푦 − 2[ln(푓 ∗푓)]휏 + 푥0, (2.5)where 푥0 is an arbitrary constant.Hirota [14] showed that the logarithmic bilinear transformation appropriate for theKorteweg–de Vries (KdV) equation푢푡 + 6푢푢푥 + 푢푥푥푥 = 0 (2.6)is푢 = 2(ln퐹 )푥푥. (2.7)In [17], Moloney and Hodnett showed how (2.7) may be used to decompose the푁 -soliton solution of the KdV equation into 푁 separate soliton elements. Thisprocedure has also been used to decompose the 푁 -loop soliton solution to theVakhnenko equation [18,19]. Here we show how Moloney and Hodnett’s proceduremay be adapted and applied to (2.4) in order to decompose the 푁 -soliton solutionof the SPE into 푁 separate soliton elements.Firstly, we note that from (2.2) – (2.4)푈 =푁∑푗=1∂푧∂휉푗∂휉푗∂휏=푁∑푗=1푈푗 , (2.8)where푈푗 =2푖푝푗∂∂휉푗[(ln 푓)∗ − (ln 푓)] . (2.9)Secondly, we note that when Hirota’s method is used [7], it turns out that 푓 dependson the 휉푗 (푗 = 1, 2, . . . , 푁) via exp(휉푗), and for each value of 푗 it is possible to write푓 in the form푓 = 푓푗 [1 + 푞푗 exp(휉푗)], (2.10)where 푓푗 and 푞푗 do not involve exp(휉푗), i.e. they are independent of 휉푗 . On combining(2.9) and (2.10), and using the identity2푒2휃1 + 푒2휃= 1 + tanh(휃), (2.11)3we obtain푈푗 =2푝푗ℑ픪[tanh(푔푗2)], (2.12)whereexp(푔푗) = 푞푗 exp(휉푗). (2.13)Similarly, from (2.5), we obtain푥 = 푦 −푁∑푗=12푝푗{1 +ℜ픢[tanh(푔푗2)]}+ 푥0. (2.14)The 푢푗(푥, 푡) = 푈푗(푦, 휏) given by (2.12), (2.14) and 푡 = 휏 are the required separatesoliton elements.For reference purposes, we give the one-soliton solution. When 푁 = 1, 푞1 = 푖 andthen from (2.12) and (2.14) we obtain푢 := 푈1 =2푝1sech(휉1) (2.15)and푥 = 푦 − 2푝1{1 + tanh(휉1)}+ 푥0, (2.16)respectively. Equation (2.16) may be written in the form휂 := 푥+1푝21푡 =휉1푝1− 2푝1tanh(휉1) + 휂0, (2.17)where 휂0 is an arbitrary constant. (2.15) and (2.17) are a solution in parametric formfor 푢 as a function of 휂 via the parameter 휉1. This solution represents a loop solitonmoving in the negative 푥-direction with speed 1/푝21. The solution corresponds to(13) in [6], (3.5) in [12], and (3.2a) and (3.3) in [7]. (There are misprints in (3.2a)and in the text after (3.3) in [7].)Now we illustrate the decomposition that we have presented by considering thetwo-loop soliton solution. When 푁 = 2,푓 = 1 + 푖(푒휉1 + 푒휉2)− 푒휉1+휉2−2훿, (2.18)where푒−2훿 =(푝1 − 푝2푝1 + 푝2)2, (2.19)so that the 푞푗 in (2.13) are given by푞1 =푖− 푒휉2−2훿1 + 푖푒휉2, 푞2 =푖− 푒휉1−2훿1 + 푖푒휉1. (2.20)Now 푈 = 푈1 + 푈2, where 푈1 and 푈2 are given by (2.12) and (2.13).4Equivalent solutions, but in undecomposed form, were derived in [6] and [7]. Hirota’smethod was used in [7] but not in [6]; in the latter, solutions for the SPE were derivedfrom the kink–kink and kink–antikink solutions of the SGE as given in [20].In [4], the SPE was transformed into the CNDE by using a transformation similar tothe one used in [18,19] to transform the Vakhnenko equation into a more convenientform. Then the CNDE was put into Hirota form and the cases 푁 = 1 and 푁 = 2considered in detail. For 푁 = 2, the solution was expressed as an approximateMoloney–Hodnett decomposition. This is in contrast to our decomposition which isexact.3 The phase-shift calculation for the case 푁 = 2We now consider the phase-shift calculation for the case 푁 = 2. First we give a cal-culation that makes use of our decomposition. Then we indicate why the calculationin [4] is incorrect.In order to discuss the phase shift of each soliton due to its interaction with theother one, it is convenient to introduce the variables 휂푗 (푗 = 1, 2) deﬁned by휂푗 := 푥+1푝2푗푡 =휉ˆ푗푝푗−2∑푗=12푝푗{1 +ℜ픢[tanh(푔푗2)]}+ 푥0 (3.1)and to note the relationship푝2휉ˆ1 − 푝1휉ˆ2 =(푝22 − 푝21푝1푝2)휏. (3.2)Also, without loss of generality, we choose 휉푗0 = 훿 (푗 = 1, 2) in (2.2).First we consider the case 푝2 > 푝1 > 0 which corresponds to a loop–loop interaction.For this case our derivation is similar to the procedure given in Sections 3.2 and3.3 in [7]; the main diﬀerence is that our starting point is the decomposed solution.From (3.2) and (2.20) we deduce that, with 휉ˆ1 ﬁxed, 휉ˆ2 → ∞ and 푞1 → 푖푒−2훿 as휏 → −∞, and that 휉ˆ2 → −∞ and 푞1 → 푖 as 휏 → ∞. It follows from (2.12) and(3.1) that, as 푡→ ∓∞,푢1 → 2푝1sech(휉ˆ1 ∓ 훿), (3.3)휂1 → 휉ˆ1푝1− 2푝1[1 + tanh(휉ˆ1 ∓ 훿)]− 2푝2(1± 1) + 푥0. (3.4)Hence, for 푡 → ∓∞, 푢1 is a function of 휂1 via the parameter 휉ˆ1. From (2.15), as푡 → ∓∞, the crest of the soliton is located where 휉ˆ1 = ±훿 in the 푦-푡 plane; from5(3.4), this corresponds to휂1 = ± 훿푝1− 2푝1− 2푝2(1± 1) + 푥0 (3.5)in the 푥-푡 plane. As the soliton is propagating in the negative 푥-direction (withspeed 1/푝21), it makes sense to deﬁne the phase shift asΔ1 := 휂1(푡→ −∞)− 휂1(푡→∞) = 2훿∣푝1∣ −4∣푝2∣ . (3.6)A similar calculation in which 휉ˆ2 is held ﬁxed gives the following results correspond-ing to equations (3.3) – (3.6), respectively:푢2 → 2푝2sech(휉ˆ2 ± 훿), (3.7)휂2 → 휉ˆ2푝2− 2푝1(1∓ 1)− 2푝2[1 + tanh(휉ˆ2 ± 훿)] + 푥0, (3.8)휂2 = ∓ 훿푝2− 2푝1(1∓ 1)− 2푝2+ 푥0, (3.9)Δ2 := 휂2(푡→ −∞)− 휂2(푡→∞) = − 2훿∣푝2∣ +4∣푝1∣ . (3.10)(3.6) and (3.10) agree with (3.13a,b) in [7]. (The above calculation may be general-ized to the case 푁 > 2; the result agrees with (3.10) in [7].) It is straightforward toshow that (3.6) and (3.10) also hold for the antiloop–antiloop interaction for which−푝2 > −푝1 > 0. A similar calculation yields the phase shifts for the antiloop–loopinteraction for which 푝2 > −푝1 > 0, and for the loop–antiloop interaction for which−푝2 > 푝1 > 0, namelyΔ1 = − 2훿∣푝1∣ −4∣푝2∣ , Δ2 =2훿∣푝2∣ +4∣푝1∣ . (3.11)Before commenting on the phase-shift calculation for 푁 = 2 in [4], it is useful tolook at the one-soliton solution in [4]. In the notation of [4], this solution is푈(휂) = 2√1 + 푐1− 푐 sech(휂), (3.12)푥 =12(휎 + 휏)− 2√1 + 푐1− 푐 tanh(휂), (3.13)푡 =12(휎 − 휏), (3.14)where 휂 = 푘(휎− 푐휏)+훽, 푘 = 1/√1− 푐2 , 훽 is an arbitrary constant, 푐 is a constantsuch that ∣푐∣ < 1, and 휎 and 휏 are new independent variables. To express thissolution in a convenient form for interpretation in the 푥-푡 plane, the solution pair 푈6and 푥+푣푡 should be parameterized in terms of the single variable 휂 only, where theconstant 푣 is chosen suitably. We ﬁnd that 푣 has to be given by 푣 = (1+ 푐)/(1− 푐)and then푥+ 푣푡 =√1 + 푐1− 푐 휂 − 2√1 + 푐1− 푐 tanh(휂) (3.15)with 푣 > 0. Equations (3.12) and (3.15) are equivalent to (2.15) and (2.17), respec-tively.In [4], the two-loop solution is given as an approximate Moloney–Hodnett decompo-sition; this decomposition becomes exact only in the asymptotic limits 푡→ ∓∞. Inorder to investigate the phase shifts, the authors in [4] considered 푥+ 푣푗푡 (푗 = 1, 2)with 푣푗 = 1/푐푗 . (No doubt this choice of 푣푗 was inﬂuenced by the corresponding stepin [18,19] for the two-loop solution to the Vakhnenko equation in which 푣푗 = 1/푐푗 isindeed the correct expression for 푣푗 .) However, as indicated by our comments on theone-loop soliton solution, 푣푗 should be taken to be 푣푗 = (1+ 푐푗)/(1− 푐푗). This errorin [4] led to an incorrect calculation for the phase shifts, and the erroneous claimthat the asymptotic speeds of the two solitons in the negative 푥-direction are 1/푐1and 1/푐2, respectively; the respective correct speeds are (1 + 푐푗)/(1− 푐푗) (푗 = 1, 2).4 Concluding commentsWe have made observations and corrections regarding various aspects of the calcu-lation of loop soliton solutions to the SPE, the aim being to complement the workin [4–7]. As far as we are aware, the Moloney–Hodnett decomposition associatedwith the bi-logarithmic bilinear transformation (2.4), and presented in Section 2,is new, as are the expressions for the phase-shifts for the interaction between twoantiloops and between a loop and an antiloop as given in Section 3.References[1] T. Scha¨fer, C.E. Wayne, Propagation of ultra-short optical pulses in cubic nonlinearmedia, Physica D 196 (2004) 90–105.[2] A. Sakovich, S. Sakovich, On transformations of the Rabelo equations, SIGMA 3(2007) 086 (8pp).[3] Y. Shen, F. Williams, N. Whitaker, P.G. Kevrekidis, A. Saxena, D. J. Frantzeskakis,On some single-hump solutions of the short-pulse equation and their periodicgeneralizations, Phys. Lett. A 374 (2010) 2964–2967.[4] V.K. Kuetche, T.B. Bouetou, T.C. Kofane, On two-loop soliton solution of theSchafer–Wayne short-pulse equation using Hirota’s method and Hodnett–Moloneyapproach, J. Phys. Soc. Jpn. 76 (2007) 024004 (7pp).7[5] V.K. Kuetche, T.B. Bouetou, T.C. Kofane, On exact 푁 -loop soliton solution tononlinear coupled dispersionless evolution equations, Phys. Lett. A 372 (2008) 665–669.[6] A. Sakovich, S. Sakovich, Solitary wave solutions of the short pulse equation, J. Phys.A Math. Gen. 39 (2006) L361–367.[7] Y. Matsuno, Multiloop soliton and multibreather solutions of the short pulse modelequation, J. Phys. Soc. Jpn. 76 (2007) 084003 (6pp).[8] H. Kakuhata, K. Konno, Loop soliton solutions of string interacting with externalﬁeld, J. Phys. Soc. Jpn. 68 (1999) 757–762.[9] Z. Fu, Z. Chen, L. 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A note on loop-soliton solutions of the short-pulse equation

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A note on loop-soliton solutions of the short-pulse equation

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