Scattering of Solitons of Modified KdV Equation with Self-consistent Sources

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1 Commun. Theor. Phys. Beijing, China 49 8 pp c Chinese Physical Society Vol. 49, No. 4, April 5, 8 Scattering o Solitons o Modiied KdV Equation with Sel-consistent Sources ZHANG Da-Jun and WU Hua Department o Mathematics, Shanghai University, Shanghai 444, China Received March, 7 Abstract This paper investigates in detail the dynamics o the modiied KdV equation with sel-consistent sources, including characteristics o one-soliton, scattering conditions and phase shits o two solitons, degenerate case o two solitons and ghost solitons, etc. Co-moving coordinate rames are employed in asymptotic analysis. PACS numbers:.3.ik, 5.45.Yv Key words: modiied kdv equation with sel-consistent source, soliton scattering Introduction Soliton equations with sel-consistent sources [ 6] can provide variety o dynamics o physical models due to the nonconstant velocities o solitary waves resulting rom sources. Thereore such systems are o much physical interest. Mathematically, many classical solving methods, such as inverse scattering transorm, Darboux transormation, Hirota method and Wronskian technique, have applied to such systems and many explicit exact solutions were derived. [7 8] In general, or the solitons without sources, they scatter elastically with phase-shits ater their interactions, and the method o asymptotic analysis in a new coordinate rame co-moving with one o solitons is usually the main tool to investigate soliton interactions. [9] Do the solitons with multi-sources still scatter as those without source or with a single source? How do the sources eect their interactions? This paper will ocus on investigating the scattering and source-eects o multi-solitons with multi-sources by means o asymptotic analysis. This is dierent rom Res. [] [6], where most equations were only attached with a single source and the dynamics considered was also or one-soliton with one source. In this paper, we will take the modiied KdV equation with sel-consistent sources mkdvescs as an example to investigate in detail the scattering o solitons with sources and see how the sources aect their dynamics. Asymptotic analysis will be employed as the main tool in our discussions. We irst review characteristics o one-soliton with a source. Then we discuss the conditions or two-soliton scattering and also obtain the ormulae or phase-shits. Some interesting dynamics o the mkdvescs are ound in our investigations, or example, the ghost solitons in the degenerate case o two solitons. The paper is organized as ollows. In Sec. we list the exact solutions in Hirota s orm to the mkdvescs. In Sec. 3 we discuss one-soliton characteristics and twosoliton scattering. mkdvescs and N-Soliton Solutions In this section let us list the known solutions in Hirota s orm o the mkdvescs. The mkdvescs is [7] N u t + u xxx + 6u u x + φ,j + φ,j x =, a j= φ,j x = λ j φ,j + uφ,j, φ,j x = uφ,j + λ j φ,j, j =,,...,N, b where {φ,j + φ,j x} are N sel-consistent sources. This equation has been solved via the inverse scattering transorm [7] and bilinear approach. [5] In Re. [5], through the dependent variable transormations u = i ln x, a φ,j = ḡj + g j, φ,j = i ḡj g j, j =,,...,N, b equation was written into its bilinear orm D t + Dx 3 N = 4i ḡ j g j, D x =, D x ḡ j = λ j g j, j= 3a 3b λ j R, j =,,...,N, 3c where i =, and ḡ j are the complex conjugates o and g j, and D is the well-known Hirota bilinear operator deined as [,] D m x D n t a b = x x m t t n ax, tbx, t x =x,t =t. Consequently, by means o Hirota s procedure, N-soliton solution to the mkdvescs can be derived and expressed The project supported by National Natural Science Foundation o China under Grant Nos. 377 and 67 and the Foundation o Shanghai Education Committee or Shanghai Prospective Excellent Young Teachers djzhang@sta.shu.edu.cn

2 8 ZHANG Da-Jun and WU Hua Vol. 49 through Eq. with [5] = { N exp µ j ξ j + π i + µ=, j= g h = β h t e ξ h + + N j=h+ µ=, j<l N µ j µ l A jl }, 4a { h µ h exp µ j ξ j + A jh + π i j= µ j ξ j + A jh π i j<l N, j,l h µ j µ l A jl }, 4b velocity o the soliton but not the shape, we can have a soliton travelling with variety o trajectories. One special case is that when < and β t 4k 3, equation 9 turns out to be zero, and we will have a stationary soliton. Figure a describes such a stationary soliton. The sources φ, and φ, in this case are also stationary and their shapes are given in Figs. b and c, respectively. λ j = k j, e A jl = where ξ j = k j x 4k 3 jt kj k l k j + k l, h =,,...N, β j zdz + ξ j, 5 k j and ξ j are real constants, β j t is an arbitrary nonnegative continuous unction o t deined on, +, and the sum over µ =, reers to each o the µ j =, or j =,,...,N. Mathematically, each β j t acts as one source and the number o sources cannot be more than the number o solitons, i.e., each soliton has at most one source. Some β j t equal to zero will result in a soliton without source and when all the {β j t} are zero equation 4a will degenerate to be or N-soliton solution o the mkdv equation. 3 Scattering o Solitons One advantage o N-soliton solution expressed in Hirota s orm, i.e., Eq. 4, is that it is convenient or investigating scattering o multi-solitons. In this section, we mainly discuss two-soliton scattering o the mkdvescs. Let us start rom one-soliton dynamics. 3. Dynamics o One Soliton When taking N = in Eq. 4, we get = + i e ξ, g = β t e ξ, and through Eq. we urther have u = sechξ, φ, = β t e ξ sechξ, φ, = β t e ξ sechξ, 6a 6b 7a 7b where ξ is given by Eq. 5. Equation 7a provides a soliton moving with a constant amplitude and top trace or velocity xt = 4k t + β zdz ξ, 8 x t = d dt xt = 4 + β t. 9 As the arbitrary non-negative unction β t plays the role o source and in eect it changes the top trace or Fig. Stationary one-soliton and sources o the mkd- VESCS. a A stationary soliton u given by Eq. 7 or β t 4k, 3 =.5 and ξ = ; b The corresponding source φ, given by Eq. 7; c The corresponding source φ, given by Eq. 7. Another case is to take β t = sint. In this case, the soliton swings periodically along the line x = [4k + / ]t ξ + / but its shape is not changed, as described in Fig. a, while igures b and

3 No. 4 Scattering o Solitons o Modiied KdV Equation with Sel-consistent Sources 8 c give graphics o the corresponding sources φ, and φ,. Fig. Moving one-soliton and sources o the mkd- VESCS. a A moving soliton u given by Eq. 7 or β t = sint, =.5 and ξ =.5, b The corresponding source φ, given by Eq. 7; c The corresponding source φ, given by Eq Scattering o Two Solitons and Asymptotic Analysis In general, or two normal solitons without sources, they scatter elastically with phase-shits during their interactions. [9] As we have shown that the β j t plays the role o source and it can result in a soliton travelling with variety o trajectories, we can also have variety o two-soliton interactions by choosing dierent sources. as The two-soliton solution o the mkdvescs is given and the corresponding sources are φ,j = ḡj + g j, where u = i ln, x φ,j = i = + i e ξ + e ξ ḡj g j, j =,, k k + e ξ +ξ, g = β t e ξ + i k e ξ +ξ, 3a + k g = β t e ξ i k e ξ +ξ, 3b + k and each ξ j is deined as Eq. 5. Equation together with Eq. describes interactions between two solitons with sources. Their velocities respectively are v = 4k + β t/ and v = 4k + β t/k. In what ollows let us respectively call these two solitons ξ -soliton and ξ -soliton or convenience. Besides, in this subsection we always suppose that k > >, 4 which guarantees k /k + to be always positive. Now let us investigate the condition o two-soliton scattering. Proposition There will be scattering between two solitons with sources deined by Eqs. and when they satisy β t β t + 4k k k ɛ <, t, +, 5 where ɛ is some positive number. In this case, the velocity o ξ -soliton is always larger than the one o ξ -soliton or the whole t-axis. Ater scattering, ξ -soliton will have a rightward phase shit /k ln[k /k + and ξ -soliton will have a letward phase shit / ln[k /k + ]. I, instead o Eq. 3, β t β t + 4k k k ɛ >, t, +, 6 then the soliton scattering also exists while ater scattering ξ -soliton will have a letward phase shit /k ln[k /k + / and ξ -soliton will have a rightward phase shit / ln[k /k + ]. Proo We start rom the case o k > >. The scattering o solitons can be investigated through extracting the initial and inal soliton states, and this can be done by means o asymptotic analysis. [9] We irst consider the ollowing coordinate rame co-moving with ξ -soliton, X = x 4k t β t dt + ξ, t. 7

4 8 ZHANG Da-Jun and WU Hua Vol. 49 In this rame ξ or X = ξ / always stays constant while [ β t ξ = k X + k β ] t + 4k k k dt + ξ k ξ, as t ±. 8 In act, by noticing condition 5, ξ can be rewritten as ξ = k X + k ɛ tt + ξ k ξ, 9 where ɛ t is some real unction related to t and not more than ɛ or t, +. Thus, in this rame, we have u sech ξ + ln k k, as t ; k + and u sechξ, as t +, respectively. These are two ξ -solitons with slightly dierent top traces. By comparing their top traces we ind ξ -soliton has a letward phase shit / ln[k /k + ]. In the same way, we consider another coordinate rame co-moving with ξ -soliton, Y = x 4kt β t dt + ξ, t. k k In this rame ξ or Y = ξ /k always stays constant while [ β t ξ = Y β ] t + 4k k k dt + ξ ξ ±, as t ±, 3 k and u respectively tends to two ξ -solitons u k sechξ, as t 4 u k sech ξ + ln k k, as t +, 5 k + which has a rightward phase shit /k ln[k / k + ]. Thus, this proposition is right or the case o k > >. Similarly, it is not diicult to check that the proposition is also valid or the condition 6 and the other three cases o k > >, i.e., k < <, k > >, and k < <. So we complete the proo. Figures 3 gives two graphics o two-soliton scattering. One is or the interaction between a stationary soliton and a moving soliton, and another is or the head-on collision between a positive amplitude soliton and a negative amplitude soliton. Proposition Equations 5 and 6 are two suicient but not necessary conditions or two-soliton scattering with phase shits. For more suicient conditions, equations 5 and 6 can respectively be replaced by [ β t β ] t +4k k k dt, as t ±, 6 [ β t β ] t + 4k k k dt ±, as t ±.7 Under these two new conditions we have the same results on scattering and phase shits as Proposition. Fig. 3 Two-soliton scattering o the mkdvescs. a Shape and motion o the -soliton solution u given by Eq. or β t 4k, 3 =., ξ =, β t 8k, 3 k =.35 and ξ =. b Shape and motion o the two-soliton solution u given by Eq. or β t, =.4, ξ =, β t = sint, k =.45, and ξ =. 3.3 Degenerate Case o = k and Ghost Solitons An interesting case in two-soliton interactions is o = k. In this case, = + i e ξ + e ξ, 8 g j = β j t; e ξ j, j =,, 9 and ξ j = x 4k 3 t β j zdz + ξ j, j =,. 3 This seems to be a one-soliton with a special source, but we would look at it as a degenerate two-soliton case. In act, back to Re. [5], rom the procedure by which we derive N-soliton solution with N sources through Hirota s method, we ind that each soliton can be attached with at most one source. That is why we call the solution generated by Eqs. 6 3 degenerate -soliton solution noting that it has two dierent source representatives g and g and thereore it cannot be one-soliton solution although it looks like but not one-soliton with a special source split into two terms. Figure 4a shows that two solitons with same amplitude but dierent top traces joint together, like a soliton

5 No. 4 Scattering o Solitons o Modiied KdV Equation with Sel-consistent Sources 83 suddenly knocking on a wall and changing direction. To see the reason behind such strange dynamics, we still extract the initial and inal soliton states. We consider the degenerate case = k o the interaction o ξ -soliton and ξ -soliton, where ξ j -solitons are described as in the previous subsection. Suppose that β j t satisy β t β tdt ±, as t ±, 3 and we consider the ollowing coordinate rame co-moving with ξ -soliton, X = x 4kt β t dt + ξ, t, 3 where ξ stays constant but ξ = X + β t β tdt + ξ ξ ±, as t ±. 33 Noting that u = i ln = 4 e ξ + e ξ X + e ξ + e ξ, 34 we can ind that { k sechξ, as t, u 35, as t +, which means ξ -soliton exists initially but disappears inally. Then we consider another rame co-moving with ξ -soliton, i.e., Y = x 4kt where ξ stays constant but ξ = Y β t dt + ξ, t, 36 β t β tdt + ξ ξ, as t ±. 37 In this case we { have, as t, u 38 sechξ, as t +, which means ξ -soliton does not exist initially but appears inally. Similar results can be obtained or β j t satisying β t β tdt, as t ±. 39 Figure 4 makes us associate such a degenerate case to the ghost solitons o the Hirota Satasuma equation. [9] In our case, the soliton u travels irst with an original source and then suddenly with another dierent source. Obviously, the ghost comes rom the eect o sources. Fig. 4 Degenerate case = k o two-soliton scattering. a Shape and motion o the solution u given by Eqs. and 8 or = k =.5, β t =, β t = sint, and ξ = ξ =. b Density plot o the solution u given by Eqs. and 8 or = k =.5, β t = 4k 3 +.5cos t, β t = 4k cos t, ξ = ξ =, x [ 8,8], t [ 5,] and plot range [.5,.5]. Grey area denotes zero value and bright strap denotes positive soliton. Figure 5 describes the interaction between degenerate two-soliton and a stationary soliton. 3.4 Two Solitons Travelling with Same Velocity For two-soliton o the mkdvescs, choosing suitable sources, they will move with same velocity. In this case, they travel in a parallel way, as shown in Fig. 6. This is dierent rom the same case o solitons o nonlinear Schrödinger equation where they will periodically interact each other. 4 Conclusion We have investigated the sources eect in soliton interactions o the mkdvescs. For one-soliton with a source, the source in eect changes velocity o the soliton but not the shape, and thus we can have a soliton travelling with variety o trajectories. For two-soliton with sources, they can also scatter with phase shits as those ordinary solitons without sources i the sources satisy some conditions. Asymptotic analysis in co-moving coordinate rames has been shown to be a powerul tool to analytically get those scattering dynamics. We also investigated some special cases o two-soliton interactions, particularly, the ghost solitons in the degenerate case o two solitons, which is new or the mkdvescs. In this case, the soliton u shows strange behaviors that it travels irst with an original source and then suddenly with another dierent source, or, looks like a soliton suddenly knocking on a wall and changing direction. Obviously, we can control the angles o incidence and relection by choos-

6 84 ZHANG Da-Jun and WU Hua ing suitable sources and control the knocking point by coordinate transers. We explained such ghost solitons also by means o asymptotic analysis. Although, in this paper we only ocused on the mkdvescs, we believe that Vol. 49 our discussions and results are general and can be generalized to other soliton equations with sel-consistent sources. Besides, such variety o dynamics o solitons with sources is also o much physical interest. Fig. 5 Scattering between degenerate two-soliton and a stationary soliton. a Shape and motion o the solution u given by Eqs. and 4 or N = 3, k = k =.5, β t =, β t = 8k3, ξ = ξ =, k3 =.55, β3 t = 4k33 and ξ3 = 6. b Density plot o a with larger ranges x [ 5, 35], t [ 3, 3] and plot range [.5,.5]. Grey area denotes zero value and bright straps denotes positive soliton. Fig. 6 Two solitons travelling with same velocity. a Shape and motion o the solution u given by Eqs. and or k =.5, β t =, ξ =, k =.4, β t = 4k k k and ξ =.9. b Shape and motion o the solution u 3 given by Eqs. and, or k =.8, β t = 4k + cos t/k, ξ =, k =.5, β t = 4k3 + cos t/k, and ξ =. Reerences [] [] [3] [4] [5] [6] [7] [8] [9] [] [] [] V.K. Mel nikov, Phys. Lett. A V.K. Mel nikov, Commun. Math. Phys V.K. Mel nikov, Phys. Lett. A V.K. Mel nikov, Commun. Math. Phys V.K. Mel nikov, Commun. Math. Phys V.K. Mel nikov, Inverse Probl Y.B. Zeng, W.X. Ma, and R.L. Lin, J. Math. Phys R.L. Lin, et al., Physica A Y.B. Zeng, et al., J. Math. Phys Y.B. Zeng, et al., J. Phys. A: Math. Gen T. Xiao, et al., J. Phys. A: Math. Gen Y.J. Shao and Y.B. Zeng, J. Phys. A: Math. Gen [3] [4] [5] [6] [7] [8] X.J. Liu, et al., J. Phys. A: Math. Gen T. Xiao, et al., J. Phys. A: Math. Gen D.J. Zhang, J. Phys. Soc. Jpn D.J. Zhang and D.Y. Chen, Physica A D.J. Zhang, Chaos, Solitons and Fractals S.F. Deng, D.Y. Chen, and D.J. Zhang, J. Phys. Soc. Jpn [9] J. Hietarinta, Scattering o Solitons and Dromions, in Scattering: Scattering and Inverse Scattering in Pure and Applied Science, eds. R. Pike and P. Sabatier, Academic Press, London, pp [] R. Hirota, Phys. Rev. Lett [] R. Hirota, The Direct Method in Soliton Theory, Cambridge University Press, Cambridge 4.