Nanopteron solution of the Korteweg-de Vries equation
Jian-Yong Wang, Xiao-Yan Tang, S. Y. Lou, Xiao-Nan Gao, Man Jia
aa r X i v : . [ n li n . S I] M a y Nanopteron solution of the Korteweg-de Vriesequation
Jian-Yong Wang , , Xiao-Yan Tang , , S. Y. Lou , , , Xiao-Nan Gao , Man Jia Department of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai, 200240, China Department of Mathematics and Physics, Quzhou University, Quzhou 324000, China Institute of System Science, School of Information Science Technology, East China Normal University, Shanghai, 200241, China Faculty of Science, Ningbo University, Ningbo, 315211, China Shanghai Key Laboratory of Trustworthy Computing, East China Normal University, Shanghai 200062, China
Abstract:
The nanopteron, which is a permanent but weakly nonlocal soliton, has been an interestingtopic in numerical study for many decades. However, analytical solution of such a special soliton is rarelyconsidered. In this Letter, we study the explicit nanopteron solution of the Korteweg-de Vries (KdV)equation. Starting from the soliton-cnoidal wave solution of the KdV equation, the nanopteron structureis shown to exist. It is found that for the suitable choice of the wave parameters, the soliton core ofthe soliton-cnoidal wave trends to be a classical soliton of the KdV equation and the surrounded cnoidalperiodic wave appears as small amplitude sinusoidal variations on both sides of the main core. Someinteresting features of the wave propagation are revealed. In addition to the elastic interaction, it issurprising that the phase shift of the cnoidal periodic wave after the interaction with the soliton core isalways half of its wavelength, and this conclusion is universal to soliton-cnoidal wave interactions.
PACS numbers:
The Korteweg-de Vries (KdV) equation u t + Auu x + Bu xxx = 0 , (1)which was originally derived to describe the propagation of gravity waves in shallow water [1], is nowregarded as one of the most important systems in soliton theory. It arises as a fundamental model indiverse branches of physics, such as nonlinear optics, Bose-Einstein condensates and hydrodynamics [2].In particular, the KdV equation plays a significant role in the study of small but finite amplitude ionacoustic waves, magnetoacoustic waves, Alfv´en waves in plasma physics [3]. It has been reported in an1xperimental observation that dynamical properties of dust acoustic waves are found to agree quite well,particularly at low amplitudes and low Mach numbers, with the classical soliton solution of the KdVequation [4].Since the dramatic discovery of the particle-like behavior of the localized waves by Zabusky andKruskal in 1965 [5], there has been an unprecedented burst of research activities on solitons. Severaleffective methods, such as the inverse scattering transformation method [6], the Hirota bilinear formal-ism [7], the Darboux transformation (DT) [8] , the B¨acklund transformation (BT) [9], etc., have beendeveloped to find the multiple soliton solutions of the KdV equation and other integrable systems. Besidesmultiple soliton solutions, interactions between solitons and other types of nonlinear waves is anothertopic of great interest [10–14]. Recently, by combining the symmetry reduction method with the DT orBT related nonlocal symmetries, researchers have established the interaction solutions between solitonsand cnoidal periodic waves of the KdV equation [11] as well as the nonlinear Schr¨odinger equation [12].Meanwhile, hinted by these results, two equivalent simple direct methods, the truncated Painlev´e andthe generalized tanh function expansion approach, are developed to find interaction solutions betweensolitons and other types of nonlinear waves, such as cnoidal waves, Painlev´e waves, Airy waves and Besselwaves [13, 14].In this Letter, we report a new analytical solution of a special weakly nonlocal soliton of the KdVequation explicitly. Such a particular solution is called nanopteron. The concept of nanopteron wasoriginally introduced by Boyd when he was studying a weakly nonlocal soliton in the φ model numerically.It is a quasisoliton which almost satisfies the classical soliton, but fails because of small amplitudeoscillatory tails extending to infinity in space [15]. During the past decades, the nanopteron structurein both continuous and discrete systems has been studied extensively [15–29]. For instance, Hunter andScheurle have shown asymptotically that capillary-gravity water waves can be consistently modeled by asingularly perturbed KdV equation and solutions of this wave equation are of nanopteron type when theBond number is less than one third [18]. Actually, the investigation of interaction between a topologicalsoliton and a background small amplitude wave has been an important topic in condensed matter physicsfor more than three decades [19]. In particular, it was shown that a small amplitude oscillatory wavecan propagate transparently through a standing topological soliton with a phase shift. In addition, thenanopteron structure has also been investigated in plasma physics [30, 31]. Keane et al. studied theAlfv´en solitons in a fermionic quantum plasma numerically [31]. Starting from the governing equationsfor Hall magnetohydrodynamics including quantum corrections, a coupled Zakharov-type system wasderived and numerically solved for both time independent and dependent cases. The time-independentAlfv´en density soliton shares a similar form of a nanopteron structure as an approximately Gaussian peaksurrounded by smaller sinusoidal variations. Then taking the time-independent nanopteron solution asan initial condition, it was numerically confirmed that the shape of the Gaussian peak retains the sameprofile during its interaction with surrounded sinusoidal variations. Obviously, some of the above resultssuggest that the interaction between a soliton and a small amplitude background wave is elastic, otherwisethe moving waves will degenerate during their propagations. Consequently, it is rather meaningful and2ignificant to obtain an analytical solution describing such types of waves. Balancing the highest nonlinearity and dispersive terms in the KdV equation (1), we assume itssolution in the following generalized truncated tanh expansion u = u + u tanh( w ) + u tanh( w ) , (2)where u , u , u and w are functions of ( x, t ) to be determined later.Substituting eq. (2) into eq. (1) and vanishing all the coefficients of the different powers of tanh( w ),we obtain the system of six overdetermined equations that u , u , u and w need to satisfy. It is fortunateto find that three of these over-determined equations are consistent. From the coefficients of tanh( w ) ,tanh( w ) and tanh( w ) , we find that u , u and u can be solved as u = − Bw x A , (3) u = 12 Bw xx A , (4) u = BA (cid:18) w xx w x − w xxx w x + 8 w x (cid:19) − w t Aw x . (5)Consequently, the solution (2) can be reformed in terms of wu = BA h w xx tanh( w ) + w x tanh ( w )) − w t Bw x + 4 w xxx w x − w xx w x − w x i . (6)Then, from the coefficient of tanh( w ) , we obtain the associated compatibility condition of ww t w x + B (cid:18) w xxx w x − w xx w x − w x (cid:19) + λ = 0 , (7)where λ is a constant of integration. Finally, one can verify that the remaining two over-determinedequations obtained from vanishing the coefficients of tanh( w ) and tanh( w ) are identically satisfied byusing Eqs. (3), (4), (5) and (7).In order to find the interaction solution between a soliton and a cnoidal wave, we make the followingansatz for the solution of eq. (7) w = ξ + c arctanh( c sn ( η, m )) ,ξ = x − V tW , η = x − V tW , (8)where sn is the usual Jacobi elliptic sine function and the parameter m is known as its modulus. V and V are velocities of the soliton and its surrounded cnoidal wave, respectively. W and W are quantitiesrelated to the soliton width and the conidal wavelength, respectively.3ubstituting the ansatz (8) back into eqs. (6) and (7) and setting zero the coefficients of the differentpowers of Jacobi elliptic functions, we obtain a group of overdetermined equations of the wave param-eters { m, V , V , W , W , c , c , λ } . When solving these overdetermined equations, if we take the ellipticmodulus m , velocities V and V as arbitrary, a nontrivial solution of the other five wave parameters { W , W , c , c , λ } can be determined as W = s B (1 − m ) V − V , W = s B (1 − m ) V − V ,λ = ( m − V + (3 m + 1) V m − ,c = δ , c = m, δ = 1 . (9)By combining eqs. (6), (8) and (9), the explicit soliton-cnoidal wave solution of the KdV equationcan be obtained as u = 3( V − V )2 AG h ( m − G ) ( m −
1) tanh( w ) − δmS ( m − G ) tanh( w ) + m − i − ( m + 7) V − (3 m + 5) V A ( m − , (10)with G = 1 − m S + δmCD,S ≡ sn ( η, m ) , C ≡ cn ( η, m ) , D ≡ dn ( η, m ) . As pointed out in our previous paper [12], the soliton-cnoidal wave can be viewed as a dressed soliton,namely, a soliton is dressed by a cnoidal periodic wave. Consequently, the soliton-cnoidal wave can bedivided into two parts. By taking the limit tanh( w ) = ± C L = 3( V − V )(1 + δmS )( m − G ) AG + (5 − m ) V + (3 m − V A ( m − ,x − V t < , (11)and C R = 3( V − V )(1 − δmS )( m − G ) AG + (5 − m ) V + (3 m − V A ( m − .x − V t > . (12)Correspondingly, the soliton part of the wave is S L = u − C L , x − V t < , (13)4nd S R = u − C R , x − V t > . (14)To illustrate the dressed structure more clearly, let us look at some figures. Fig. 1(a) exhibits thesoliton-conidal wave structure of u determined by eq. (10) at t = 0. Fig. 1(b) and Fig. 1(c) reveal therelated structures of the cnoidal periodic wave and the soliton core of u , respectively. Obviously, thesuperposition of Fig. 1(b) and Fig. 1(c) is just Fig. 1(a). It is observed from Fig. 1(b) that apart fromthe soliton center, the solution rapidly tends to a cnoidal periodic wave. It is clear from Fig. 1(c) thatafter removing the periodic wave background from u , the left is just a soliton structure given by eqs. (13)and (14). Fig. 1(d) shows an elastic overtaking collision process between a soliton and a cnoidal wavewhere both are right-going and the soliton is traveling faster. It can be concluded from Fig. 1(d) thatdespite the cnoidal periodic wave is a delocalized structure, in the space-time evolution of the soliton-cnoidal wave, every peak of the conidal periodic wave elastically interacts with the soliton core except fora phase shift. To plot Fig. 1, the selection of the nonlinearity coefficient A and the dispersion coefficient B are given by A = 48619 √ √ ≃ . ,B = 388110739 √ √ ≃ . , (15)which is derived from the KdV equation describing the propagation of ion acoustic waves. - - -
50 0 50 100 1500.10.20.30.40.50.60.7 x u H a L - - -
50 0 50 100 1500.10.20.30.40.5 x C L + C R H b L - - -
50 0 50 100 1500.00.10.20.30.40.5 x S L + S R H c L Fig. 1 (a) The soliton-cnoidal wave structure of the KdV equation given by eq. (10). (b) The related cnoidal periodic wavestructure given by eqs. (11) and (12). (c) The related soliton structure given by eqs. (13) and (14). (d) The density plotof u for its space-time evolution. The parameters are m = 0 . V = 0 . V = 0 . δ = 1, A and B are given by eq. (15). t = 0,the conidal periodic wave peaks on the left side of the soliton core have interacted with the soliton core,while not the cnoidal periodic peaks on the right side. So there is a phase shift between them. Obviously,the conidal periodic wave peaks on the left and the right sides of the soliton core can be expressed by C L and C R , respectively. By using eqs. (11) and (12), it is surprising to find that the collision-inducedphase shift of the cnoidal periodic wave is given by∆ cn = 2 W K ( m ) = λ c , (16)where λ c is the wavelength of the cnoidal periodic wave, K ( m ) is the first kind of complete elliptic integral.This result can be easily verified. By substituting η = η + 2 K ( m ) into (11) and using the Jacobi ellipticidentities sn (2 K ( m ) , m ) = 0, cn (2 K ( m ) , m ) = −
1, and dn (2 K ( m ) , m ) = 1, we can directly demonstratethat C L ( η + 2 K ( m )) = C R ( η ). Similarly, by substituting η = η + 4 K ( m ) into eqs. (11)-(12) and usingthe Jacobi elliptic identities sn (4 K ( m ) , m ) = 0, cn (4 K ( m ) , m ) = 1, and dn (4 K ( m ) , m ) = 1, we can alsodemonstrate that C L ( η + 4 K ( m )) = C L ( η ), and C R ( η + 4 K ( m )) = C R ( η ). So, C L and C R are functionsof the period 4 K ( m ). Incidentally, the periods of the sn and cn functions are also 4 K ( m ), while theperiod of the dn function is 2 K ( m ). It is noted that the phase shift of the interaction wave in Fig. 1(b)can be computed from eq. (16) as 24.95, which coincides with the figure.The phase shift formula (16) tells that the phase shift of a cnoidal periodic wave after its interactionwith a soliton is always half of its wavelength. More significantly, this phase shift formula is universalto all the soliton-cnoidal wave solutions obtained in Refs. [11–13]. Unfortunately, it is still difficult tocalculate the phase shift of the soliton because of the mixture of the tanh and Jacobi elliptic functions.The existence of the Jacobi elliptic functions prevents us from calculating the difference of the phasesat two different time limits, approaching negative and positive infinities, respectively. Therefore, analternative method should be designed to overcome this difficulty.Due to the fact that the parameter m appears as not only the modulus of the Jacobi elliptic functionbut also its coefficient since c = m , the amplitude of the cnoidal periodic wave trends to thrive with m increasing. From Fig. 2(a), which is plotted to illustrate this phenomenon at t = 0, it can be observedthat as soon as the parameter m approaches 0 .
99, the amplitude of the cnoidal wave becomes comparableto the soliton core. Fig. 2(b) shows that the solution (10) exponentially approaches the cnoidal wave as x → ±∞ . We also notice from Fig. 2(c) that after the periodic wave peaks C L and C R are taken awayfrom the exact solution u , only a tall and slim soliton structure is revived. Fig. 2(d) reveals that thesoliton core and every peak of the cnoidal periodic wave can pass through each other transparently witha phase shift. 6 -
20 0 20 40 - x u H a L - -
20 0 20 40 - x C L + C R H b L - -
20 0 20 4001020304050 x S L + S R H c L Fig. 2 (a) The soliton-cnoidal wave structure with m = 0 .
99 at t = 0. (b) The related cnoidal periodic wave structure. (c)The related soliton structure. (d) The density plot of u for its space-time evolution. The parameters are m = 0 . V = 0 . V = 0, δ = 1, A and B are given by eq. (15). Before we proceed further, let us first review the classical soliton solution of the KdV equation.By using the usual tanh expansion method, the single soliton solution of the KdV equation (1) can beobtained as u = 1 AW [8 B + V W − B tanh ( ξ )] , ξ = x − V tW . (17)Imposing the boundary conditions u → , u ξ → , u ξξ → , as ξ → ±∞ , (18)the width of the soliton can be determined and the above solution becomes u = 3 VA sech (cid:18) x − V tW (cid:19) , W = r BV . (19)Obviously, there is an intimate connection between the usual tanh function expansion method andthe generalized tanh function expansion method. If we take w as a straight line solution, namely, w =( x − V t ) /W , the solution (6) reduces to the single soliton solution (17) obtained by the usual tanh functionexpansion method.Now, let us consider the asymptotic behavior of the soliton-cnoidal wave solution (10). Under the7ltra limit condtion m = 0 ( G = 1), V = V and V = − V , the wave parameters (9) degenerate to W = r BV , W = r BV ,λ = 3 V , c = δ , c = 0 , (20)and the soliton-cnoidal wave solution (10) reduces to the classical soliton solution (19). From eq. (20),it is interesting to notice that the substitution of the straight line solution w = ( x − V t ) /W into thecompatibility condition (7), the width of the soliton can also be determined as W = p B/V with λ = 3 V /
2. However, the deeper physical reason why the compatibility condition of w plays the similarrole as the boundary condition (18) still need further consideration. This interesting limit case hintsus to consider the asymptotic behavior of the soliton-cnoidal wave solution (10). Under the asymptoticcondition V = V , V = − V , and m →
0, we find that the soliton core profile goes to be the classical KdVsoliton while the surrounded conidal periodic wave becomes a small amplitude sinusoidal wave oscillatingaround zero. This wave profile, in which the classical soliton is dressed by small amplitude oscillations,is just the nanopteron structure proposed by Boyd. In this sense, the soliton-cnoidal wave solution (10)can also be named as a nanopteron solution for the sake of its asymptotic behavior.Under the limit m →
0, the function K ( m ) trends to π/
2. Thus, the collision-induced phase shift ofthe small amplitude background wave can be approximately taken as∆ sin = 2 W K ( m ) ≃ W π. (21) NanopteronClassical soliton60 80 100 120 140 - - -
50 0 50 100 1500.00.20.40.60.81.01.2 x u Fig. 3 A comparison of the classical soliton solution (19) to the nanopteron solution (10) at t = 0. The wave parametersare V = 0 . V = 0 . V = − . m = 0 . δ = 1, A and B are given by eq. (15). A comparison of the classical soliton to the nanopteron for m = 0 .
001 at t = 0 is given in Fig. 3,which demonstrates that the curves of two solutions coincide exactly with each other at a large spacescale. However, the inset on the right side of Fig. 3 shows that the oscillating tail is nonvanishing despiteof a tiny amplitude. When m becomes a little larger, the nanopteron tail grows up conspicuously. Fig.4(a) presents a comparison of classical soliton to the nanopteron structures for m = 0 .
02 and δ = ± t = 0. It is observed that the soliton core of the nanopteron is higher and slightly narrower than theclassical soliton when δ = −
1, while shorter and slightly wider when δ = −
1. Interestingly, the crests andtroughs of the surrounded sinusoidal waves of the nanopteron structures with δ = ± δ = 1.Fig. 4(b) shows that apart from the soliton center, the solution rapidly approaches a small amplitudesinusoidal wave oscillating around zero. From eq. (21), the phase shift of the small amplitude sinusoidalwave can be approximately calculated as W π ≃ .
77, which is in accordance with Fig. 4(b). Fig. 4(c)reveals that only a soliton S L + S R is left after the sinusoidal wave are ruled out from the exact solution u . Fig. 4(d) is a three-dimensional plot of the nanopteron solution with δ = 1. Nanopteron H ∆= L Nanopteron H ∆=- L Classical soliton - - -
50 0 50 100 1500.00.20.40.60.81.0 x u H a L - - -
50 0 50 100 150 - - x C L + C R H b L - - -
20 0 20 40 600.00.20.40.60.81.0 x S L + S R H c L Fig. 4 (a) A comparison of the classical soliton solution (19) to the nanopteron solution (10) with δ = ±
1. (b) The relatedsmall amplitude background wave with δ = 1. (c) The related soliton structure with δ = 1. (c) The three-dimensional plotof the nanopteron with δ = 1 for its space-time evolution. The other parameters are m = 0 . V = 0 . V = 0 . V = − . A and B are given by eq. (15). In this Letter, we present a new soliton-conidal wave solution of the KdV equation, which we alsoname as a nanopteron solution. Based on this solution, some interesting features are revealed. First,it has been observed that the soliton core preserves its shape and velocity during the collision withthe cnoidal periodic wave peaks. Second, from the dressed structure of the solution, it is found thatthe collision-induced phase shift of the cnoidal periodic wave is always half of its wavelength, which isbelieved to be universal to all the soliton-cnoidal interactions. Third, the nanopteron structure is realizedas a special limit case. It is found that for the suitable choice of the wave parameters, the soliton core ofthe soliton-cnoidal wave trends to be the classical KdV soliton and the surrounded cnoidal periodic waveappears as small amplitude sinusoidal variations on both sides of the main core.The explicit solution obtained in this letter can be applied in many physical scenarios. For in-stance, the nanopteron structure can be viewed as a perturbed classical soliton, and it may provide somecorrection to the classical soliton in both theoretical and experimental studies.9
Acknowledgments
The work was sponsored by the National Natural Science Foundations of China (Nos. 11275123,11175092, 11205092 and 10905038), Shanghai Knowledge Service Platform for Trustworthy Internet ofThings (No. ZF1213), Scientific Research Fund of Zhejiang Provincial Education Department underGrant No. Y201017148, and K. C. Wong Magna Fund in Ningbo University.
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