On the Camassa–Holm equation and a direct method of solution. II. Soliton solutions

Allen Parker

Abstract

Previous attempts to find explicit analytic multisoliton solutions of the general Camassa–Holm (CH) equation have met with limited success. This study (which falls into two parts, designated II and III) extends the results of the prior work (I) in which a bilinear form of the CH equation was constructed and then solved for the solitary-wave solutions. It is shown that Hirota's bilinear transformation method can be used to derive exact multisoliton solutions of the equation in a systematic way. Here, analytic two-soliton solutions are obtained explicitly and their structure and dynamics are investigated in the different parameter regimes, including the limiting ‘two-peakon’ form. The solutions possess a non-standard representation that is characterized by an additional parameter, and the structure of this key parameter is examined. These results pave the way for constructing the hallmark N-soliton solutions of the CH equation in part III.

Keywords:

1. Introduction

The eponymous Camassa–Holm (CH) equationEmbedded Image(1.1)where u=u(x,t) and k is a (real) constant, has generated considerable interest over the last decade. The equation was first reported by Fuchssteiner & Fokas (1981) in the context of hereditary symmetries, but was later ‘rediscovered’ by Camassa & Holm (1993) as a model for shallow water waves (SWWs). Yet, the single most compelling feature of equation (1.1)—and the one that has commanded most attention in recent years—is its complete integrability for all values of k. A substantial body of work on the CH equation now exists which describes and affirms the many facets of its integrable or soliton character (e.g. Camassa et al. 1994; Fuchssteiner 1996; Beals et al. 1998; Schiff 1998; Fisher & Schiff 1999; Constantin 2001; Johnson 2003). In the absence of the right-hand side of (1.1), the CH equation reduces to another SWW model, the Benjamin–Bona–Mahony (BBM) equation (Benjamin et al. 1972)Embedded Image(1.2)

The BBM equation has been studied exhaustively and is deemed to be non-integrable (see Parker 1995; Dye & Parker 2000 and references). Thus, the CH equation (1.1) may be considered the integrable extension (or ‘completion’) of the BBM equation, and so fills a ‘gap’ among the completely integrable SWW equations in (1+1)-dimensions. Apart from the celebrated Korteweg–de Vries (KdV) equation, this class of nonlinear evolution equations also includes the well-known AKNS–SWW equation (Ablowitz et al. 1974)Embedded Image(1.3)that has particular relevance for the present work. (Here we intend Embedded Image with f(x,t)→0 sufficiently rapidly as x→+∞).

Much attention has been given to the special case k=0 of equation (1.1),Embedded Image(1.4)which we have dubbed elsewhere the reduced Camassa–Holm (RCH) equation (Parker 2004). Though of no relevance to water waves (for which k>0), the RCH equation has been found to model nonlinear dispersive waves in hyperelastic rods (Dai 1998a). Equation (1.4) has intriguing soliton solutions that are composed of a train of interacting peaked solitary waves called ‘peakons’ (Camassa & Holm 1993). These multipeakons are non-analytic solutions that have a ‘corner’ (i.e. a finite discontinuity in the slope) at their crests; yet, surprisingly, they still exhibit the ‘elastic’ collision property that betokens their soliton character (Camassa et al. 1994; Beals et al. 1999). It is perhaps less well-known that the RCH equation also admits analytic solitary waves whose bell-shaped profiles get progressively sharper at the peaks as they approach the peakon waveform (Dai 1998b; Parker 2004). This unusual behaviour, whereby a weak solution (peakon) is obtained as the limit of classical solutions, stems from the nonlinear dispersive term uuxxx that appears on the right-hand side of equation (1.1) (Li & Olver 1997).

In contrast, exact solutions of the general CH equation (1.1)—and, in particular, the multisoliton solutions that are considered the hallmark of any integrable system—have proved far harder to come by. At the time of writing, previous attempts to obtain analytic N-soliton solutions of the CH equation have met with limited success. Schiff (1998) used Bäcklund transformations to obtain the solitary wave (N=1) and an explicit (though incomplete) expression for the two-soliton solution. More recently, Johnson (2003) implemented the inverse scattering transform procedure for the CH equation that was developed by Constantin (2001) and found analytic N-soliton solutions as far as N=3. In order to construct these solutions, the author was forced to rely on good guesswork and extensive use of symbolic manipulation software. As they stand, the resulting expressions, particularly that for the three-soliton, have no obvious and recognizable structure that might be generalized to N=4, much less the generic N-soliton. Nevertheless, Johnson's key observation—one shared by Schiff (1998)—that any explicit representation of the soliton solutions of the CH equation (1.1) should be formulated parametrically, provides a clue to further progress. What is also clear from these, and other, attempts to solve the CH equation, is the need to transform (1.1) to another integrable system which is, in some sense, more amenable to exact solution (see also Fokas 1995; Camassa & Zenchuk 2001).

In a previous study (Parker 2004)—designated I—we proposed a direct procedure for solving the CH equation that is based on Hirota's bilinear transformation method (Hirota 1980). Using a reciprocal transformation that was first reported by Fuchssteiner (1996) and later exploited by Schiff (1998), we showed that equation (1.1) can be mapped onto a version of the AKNS–SWW equation (1.3). By recasting the latter equation into a bilinear form and solving this in the conventional manner, we were then able to derive analytic solitary waves of equation (1.1) for any value of the parameter k. In this second study II and the sequel III, our purpose is to obtain analytic multisoliton solutions of the general CH equation in a systematic and efficient manner, within the bilinear framework that was developed in I. In this way, we are able to circumvent the drawbacks of the inverse scattering transform approach that were encountered by Johnson (2003). Building on the results of I, we construct explicit expressions for the first few multisolitons and investigate their structure and wave dynamics. Examples of these solitons are given, and described, in various parameter regimes, including their limiting ‘multipeakon’ form. More specifically, in II we obtain the exact two-soliton solution of the CH equation and highlight the technical and computational difficulties that arise when implementing the bilinear methodology. Then, with the aid of these results, we proceed in part III to modify the direct procedure and demonstrate its efficacy for obtaining higher-order solitons. We find that, at each order N≥2, the analytic representation of the N-soliton requires an additional parameter which marks out its non-standard character. The structure of this ‘defining’ parameter is examined, and a procedure is presented for deriving the general N-soliton solution of the CH equation (1.1) for all k.

2. Bilinear form and solitary waves

For ease of reference, we reprise the main results of the companion study I that will be needed here and in III (the reader is referred to I for the detailed derivations and commentary). Without loss of generality, we may set k=κ2 in (1.1) and consider the CH equation in the formEmbedded Image(2.1)with κ>0. The special case of the RCH equation (1.4), with κ=0, is considered separately. By introducing the quantityEmbedded Image(2.2)we can recast the CH equation (2.1) into the conservation formEmbedded Image(2.3)which, in turn, permits us to define a coordinate transformation (x,t)→(y,t) byEmbedded Image(2.4)(No confusion arises and considerable clarity is gained, by using the same symbol ‘t’ for both time variables.) This transformation is well-defined provided r>0, and then its inverse is determined byEmbedded Image(2.5)

Crucially, (2.4) constitutes a reciprocal transformation (Kingston & Rogers 1982) which preserves the conservation law (2.3) in (y,t)-space. Under this transformation, equations (2.2) and (2.3) give rise to the completely integrable systemEmbedded Image(2.6)Embedded Image(2.7)which is the associated Camassa–Holm (ACH) equation studied by Schiff (1998). His estimable strategy was to solve for r(y,t) and use (2.6) to obtain a solution u(y,t) of the ACH equation. One then inverts (2.5) to get the coordinate transformation x(y,t) which yields a solution of the CH equation (2.1) in parametric form. Whereas Schiff used Bäcklund transformations to find solutions for r(y,t), we shall adopt an alternative approach based on Hirota's bilinear transform (Hirota 1980).

In I we showed that the equations (2.6) and (2.7) are equivalent to the systemEmbedded Image(2.8)Embedded Image(2.9)in which u(y,t) is replaced by the ‘potential’ Q(y,t). Eliminating r between these last two equations givesEmbedded Image(2.10)which, modulo a simple scaling, is just the AKNS–SWW equation (1.3). We now make a change of variableEmbedded Image(2.11)and, from (2.9), infer thatEmbedded Image(2.12)

The bilinear form of equation (2.10) is then obtained asEmbedded Image(2.13)Embedded Image(2.14)where Dy, Dt are the usual Hirota derivatives defined by (Hirota 1980)Embedded Image(2.15)The auxiliary variable τ that appears in (2.13) and (2.14) is merely a device to effect the bilinearization of (2.10) in terms of the D-operators—it plays no role in the ultimate form of the solutions.

A solitary wave of the CH equation is obtained by taking (Parker 2004)Embedded Image(2.16)where p, ω, σ and η are real parameters. This solves the bilinear equations (2.13) and (2.14) provided ω(p) satisfies the dispersion relationEmbedded Image(2.17)and σ=−p3. Introducing the wave speedEmbedded Image(2.18)in (y,t)-space, we can write the phase variable in (2.16) asEmbedded Image(2.19)where the τ-dependence has been absorbed in the arbitrary phase y0. We then obtainEmbedded Image(2.20)andEmbedded Image(2.21)

Exceptionally for the solitary wave, there is a simple relation between r and u,Embedded Image(2.22)which enables the integrals in (2.5) to be evaluated asEmbedded Image(2.23)with the wave speed Embedded Image in (x,t)-space given byEmbedded Image(2.24)Taken together, equations (2.21) and (2.23) are a parametric representation (in θ) for the analytic solitary wave u(x,t) of the CH equation (2.1) with κ>0. Since r and u are invariant under the parity transformation p→−p, we may assume that p>0. Moreover, to ensure that r>0 for all (y,t), we requireEmbedded Image(2.25)and, from (2.24),Embedded Image(2.26)Therefore, solitary waves propagate in the positive x-direction at supercritical speed. Their smooth, bell-shaped profiles have an amplitudeEmbedded Image(2.27)that is always less than the value of the wave speed Embedded Image (cf. the peakon wave below).

To obtain an analytic solitary wave for the RCH equation (κ=0), we use the transformationEmbedded Image(2.28)which reduces the equation (2.1) to equation (1.4). If we apply this to the solitary wave (2.21) and (2.23), we getEmbedded Image(2.29)where the coordinate transformation Embedded Image is given byEmbedded Image(2.30)This elevated solitary wave propagates to the right with supercritical speed V>3κ2, at the non-zero ambient height κ2. We emphasize that Embedded Image represents a two-parameter family (κ,p) of solutions of the RCH equation (1.4) in which κ>0 is now just a mathematical parameter. For each such solitary wave, there exists a dual antisolitary wave (of depression) that travels to the left at the same speed V. This follows directly by noting the invariance of (1.4) under the inversion mapEmbedded Image(2.31)These analytic solitary waves of the RCH equation were found in implicit form by Dai (1998b) and obtained explicitly by us in I. We remark that smooth antisolitary waves do not arise for the full CH equation since (2.31) is not a symmetry of (2.1).

For the special case κ=0, most interest has centred on the intriguing peakon wave (Camassa & Holm 1993)Embedded Image(2.32)This solution can be recovered from the analytic CH solitary wave, with κ>0, by taking the peakon limit (Parker 2004)Embedded Image(2.33)The peakon is a non-analytic (weak) solution of the RCH equation (1.4) that has a corner at its crest. In accord with (2.31), it can propagate to the right (peakon) or left (antipeakon) with an arbitrary speed Embedded Image that is equal to its amplitude (cf. (2.27)). But peaked solutions are not restricted to the RCH equation: as a direct consequence of (2.28), the general CH equation (2.1) also admits a non-analytic peakon solitary wave (Parker 2004)Embedded Image

3. Two-soliton solution of the Camassa–Holm equation

It is as well to pause here to summarize our direct procedure for finding exact solutions of the CH equation. We first select a suitable ansatz f(y,t) with which to solve the bilinear equations (2.13) and (2.14). We then substitute the resulting Hirota function f into (2.12) to get r(y,t). A solution u(y,t) of the ACH equation now follows by inserting r into equation (2.6). Finally, with both r(y,t) and u(y,t) known, we integrate (2.5) to obtain the coordinate transformation x(y,t). This yields a solution u(x,t) of the CH equation (2.1) in parametric form (in terms of y). To get an idea of the technical problems that we can expect to meet when solving for higher-order solitons, we begin by implementing the method as it stands.

Constructing analytic N-soliton solutions of bilinear equations of the type (2.13) and (2.14) follows a well-rehearsed and systematic procedure (Matsuno 1984). For N=2, we takeEmbedded Image(3.1)where pi, ωi, σi, ηi and A12 are real constants. (We have suppressed the auxiliary variable τ in f(y,t) since it will eventually be absorbed into the arbitrary phase constants ηi.) To proceed, we shall require the fundamental bilinear identityEmbedded Image(3.2)where Embedded Image is a general bilinear operator and we write F(p)=F(p, ω, σ). Then, with f given by (3.1), it is straightforward to show thatEmbedded Image(3.3)whenever F is even and F(0)=0. It follows that f solves the general bilinear equation Ff.f=0 provided thatEmbedded Image

With the aid of these results, and after a some routine algebra, we find that the ansatz (3.1) is a solution of the bilinear form (2.13) and (2.14) ifEmbedded Image(3.4)ωi(pi) satisfies the dispersion relationEmbedded Image(3.5)andEmbedded Image(3.6)The wave speeds in (y,t)-space are thenEmbedded Image(3.7)which permit us to write the phase variables in (3.1) asEmbedded Image(3.8)where the τ-dependence has been subsumed by the arbitrary phase constants αi. Not surprisingly, the expressions for ωi and ci replicate (2.17) and (2.18) for the solitary wave. This ensures that the Hirota function (3.1) represents a solution that is composed of two distinct ‘solitary’ waves and, crucially, will confer the same ‘two-soliton’ structure on the resulting solution of the CH equation.

We next substitute f into (2.12) to get r(y,t); but, even at order N=2, proceeding directly with the calculation involves some lengthy and tedious algebra. However, considerable computational efficiency and simplification result if we recast (2.12) in bilinear terms using the D-operators (2.15): this givesEmbedded Image(3.9)Then, making use of (3.3) with F=DyDt and f given by (3.1), we readily deduceEmbedded Image(3.10)whereEmbedded Image(3.11)This solution for can be reformulated in a myriad of ways; reassuringly, couched in terms of hyperbolic functions, it can be made to agree with the solution for r that was reported by Schiff (1998). However, we contend that the expression (3.10) is a more natural one for, as we shall see, it provides a motif for the analytic form of the CH solitons. Taken by itself, (3.10) represents a two-soliton solution of equations (2.6) and (2.7) (and the equivalent integrable system (2.8) and (2.9)). It describes the elastic interaction of two sech2 solitary waves of the form (2.20); typically, the only relic of the collision are the phase shifts experienced by the individual pulses. Here, they are determined by the parameter A12, equation (3.4), which is the signature of solitons of KdV type (Ablowitz & Segur 1981).

Since the solutions of the CH equation obtained by our direct method ultimately depend on r(y,t) (through equations (2.5) and (2.6)), it will pay us to examine this solution further. The parity invariance of (3.10) is best demonstrated by transforming p1→−p1, say, in (3.1) and (3.4), and then invoking the gauge invariance of (2.12) (and the symmetry in p1, p2). As this property is shared by all the solutions reported here (and in III), we will henceforth assume that pi>0 for all i, without further comment and loss of generality. We must also ensure the positivity of r(y, t) so that our solutions are analytic and the reciprocal mapping (2.4) is 1–1: this evidently requiresEmbedded Image(3.12)and, from (3.7),Embedded Image(3.13)

The two-soliton solution u(y, t) of the ACH equation now follows by inserting r(y, t) into (2.6). However, this apparently innocuous, but pivotal, computation is fraught with difficulty unless one proceeds with care. Even so, the algebra is still too extensive to report here and we will spare the reader the gory details. Rather, it is sufficient to record the key features of the calculation and draw the lessons necessary to progress to higher order. (Nothing is lost by this for, as we stated earlier, our intention is to get a handle on the difficulties that lie ahead.) Instead of (3.10), we work with f itself: accordingly, we substitute for r directly from (2.12) into (2.6), to obtainEmbedded Image(3.14)where W(f, fy, ft,…) is a trilinear expression in f and its derivatives. Notice that (3.14) already admits substantial simplification since the denominator only involves f3 (reduced from f6). Further advantage is gained by rewriting W using the bilinear formalism (we omit the details): then, after inserting (3.1), we deduce the compact resultEmbedded Image(3.15)whereEmbedded Image(3.16)and A12, ωi and θi are given by (3.4), (3.5) and (3.8), respectively. This solution can be reformulated in a number of ways and, in particular, can be made to agree with the two-soliton results reported by both Schiff (1998) and Johnson (2003). As anticipated, u(y,t) bears comparison with the structure of its counterpart r(y,t) in (3.10). A notable feature of (3.15) is the complement of exponential terms in the numerator that match exactly those in (3.10) and which, significantly, derive from the bilinear expression DyDtf.f in (3.9). We shall say more on this later; but for now, we simply note that (3.15) suggests a possible template for the structure of higher-order solitons.

The two-soliton solution u(y,t), equations (3.15) and (3.16), describes the collision of two smooth elevated ACH solitary waves of the form (2.21). Many different waveforms are possible depending on the choice of κ and wavenumbers p1, p2. Figure 1 shows a typical two-soliton (with κ=0.6): the taller, narrower wave (p1=1.5) catches and collides with the shorter, slower pulse (p2=1). In this instance, the two solitons coalesce to form a single symmetrical bell-shaped wave (at t=1.45) before re-emerging in characteristic fashion with their profiles intact. Post-interaction, the taller wave is (always) shifted forwards and the shorter wave backwards, by an amount that is again fixed by the parameter A12 (see equation (4.4)). By way of contrast, figure 2 depicts an ACH two-soliton in the peakon regime (2.33) where each solitary wave has a ‘bullet-nosed’ profile. This solution forms a symmetric double peak at the height of the interaction (t=−5.7) before separating into its component solitary waves.

Figure 1

Two-soliton solution u(y, t) of the ACH equation with κ=0.6: the taller solitary wave (p1=1.5) catches and passes through the shorter soliton (p2=1).

Figure 2

A ‘bullet-nosed’ ACH two-soliton solution in the near peakon regime: κ=0.08, κp1=0.9952, κp2=0.992.

To complete the two-soliton solution u(x,t) of the CH equation, it only remains to find the coordinate transformation x(y,t) using (2.5). Unfortunately, there is no simple relation between r and u, akin to (2.22) for the solitary wave, that might facilitate the integration. Nevertheless, the corresponding result (2.23) for the solitary wave—taken in conjunction with the soliton reduction principle (Parker 2000)—suggests how we might proceed. Accordingly, we define the further parameters,Embedded Image(3.17)Then, following some careful manipulation, we can evaluate the first of the integrals in (2.5) asEmbedded Image(3.18)where the arbitrary ‘constant’ of integration α must be considered a function of time. However, after differentiating (3.18) with respect to t, we find that the second of the equations in (2.5) is satisfied only if α is a constant.

Equations (3.15), (3.16) and (3.18), together with (3.1), (3.10) and (3.11), give an explicit representation (albeit parametrically in terms of y) for the analytic two-soliton solution of the CH equation (2.1) for any κ>0. It is a straightforward exercise (though one best performed using symbolic software) to verify that it is indeed an exact solution of (2.1). Figure 3 shows the typical CH two-soliton u(x,t) that is generated by the associated ACH waveform u(y,t) in figure 1. The effect of the coordinate transformation (3.18) is evident: the classical bell-shaped ACH solitary waves undergo significant distortion, resulting in the more ‘pinched’ profiles of their CH counterparts in figure 3. The deformation is even more pronounced for the two-soliton solution that is shown in figure 4. The bullet-nosed peaks of the corresponding ACH solution in figure 2 have been reshaped to produce the sharply pointed crests of a near two-peakon solution of the CH equation. In this peakon regime (κ→0, κpi→1, i=1,2), the smooth peak of each CH solitary wave seeks to reproduce the corner at the crest of a true peakon (equation (2.32)). In both these examples, it is evident that the transformation (3.18) transfers the soliton properties of the ACH solution u(y,t) to its CH cousin u(x,t). Indeed, it is easy to show (see below) that the latter waveform separates into a pair of CH solitary waves of the form (2.21) and (2.23), locally at spatial infinity (as t→±∞). The constituent solitons—which are identified by the phase variables (3.8)—propagate to the right with the ‘correct’ supercritical speeds Embedded Image and amplitudes that are given by equations (2.24) and (2.27) (with ppi, i=1, 2). We shall investigate below the wave dynamics of the CH two-soliton solutions and their post-collision phase shifts in particular.

Figure 3

A typical CH two-soliton solution u(x,t) generated from the ACH solution in figure 1 (κ=0.6, p1=1.5, p2=1).

Figure 4

A near two-peakon solution of the CH equation corresponding to the ACH two-soliton in figure 2 (with κ=0.08, κp1=0.9952, κp2=0.992).

It is worth dwelling on the coordinate transformation (3.18) that refashions the classical bell-shaped ACH two-soliton u(y,t) into its more sharply peaked CH counterpart u(x,t). Figure 5a,b show the transformations that map the ACH solitons in figure 1 (at t=−5) and figure 2 (at t=−26) onto the corresponding CH solutions in figures 3 and 4, respectively. In each case, the mapping is almost entirely linear, except for the two kinks that mark the positions of the wave crests. Thus, away from the peaks, there is little change to the shape of the waveform. In the outer regions, the linear asymptotes (longer dashes) areEmbedded Image(3.19)where, for the purpose of illustration, we have taken p1>p2 with the taller pulse to the left of the shorter one. In the inner region between the two-solitons, the asymptote (short dashing) is given byEmbedded Image(3.20)These lines all have slope m=1/κ that steepens rapidly as we approach the peakon limit κ→0 (figure 5b). Evidently, the significant distortion occurs in the proximity of the peaks, within the two kinked domains that are bounded by the three asymptotes. There, the transformation (3.18) is again approximately linear, but now the gradients Embedded Image (i=1,2) become progressively more shallow as we near the peakon regime (figure 5b). We omit the details and simply note that, in contrast to (3.19) and (3.20), the linear asymptotes for the two kinks are time dependent. This just means that the kinks propagate in unison with the two soliton pulses along the fixed ‘tramlines’ formed by the asymptotes in figure 5.

Figure 5

The coordinate transformation x(y,t) that maps the ACH two-soliton u(y,t) to its CH counterpart u(x,t): (a) for the solutions shown in figures 1 and 3 at t=−5; (b) for the solutions shown in figures 2 and 4 at t=−26.

4. Two-soliton wave dynamics

We now consider the wave dynamics of the CH two-soliton. Specifically, we wish to examine the asymptotic form of the solution u(x,t) as t→±∞. This will confirm its soliton character and allow us to compute the phase shifts that are considered an imprimatur of soliton interactions. Rather than use of the ACH solution (3.15), it is expedient to work with the Hirota form (3.1). The calculations are altogether routine and we leave the interested reader to pursue the details. For the sake of argument, we assume that p1>p2>0, so that θ1 identifies the taller and faster of the two wave pulses. Then, with θ1 kept fixed, we find thatEmbedded Image(4.1)where we have introduced the shifted variables Embedded Image in (y, t)-space. Now, if we follow (2.23) and (2.24) and write Embedded Image, with Embedded Image and Embedded Image arbitrary (i=1, 2), then the coordinate transformations (3.18) that correspond to (4.1) are, respectively,Embedded Image(4.2)Embedded Image(4.3)(Remark: for all these asymptotic approximations, we have used the gauge invariance of (2.12) to remove any residual exponential factors.) The calculations with θ2 fixed produce precisely analogous results (one has only to interchange the subscripts 1 and 2 and then swap the two expressions in (4.1) and those in equations (4.2) and (4.3)). Comparing (4.1), and its counterparts for fixed θ2, with (2.16), shows that the two-soliton solution (3.1) is composed of the same two solitary waves as t→−∞ and t→+∞, except for the phase shiftsEmbedded Image(4.4)These results apply (locally at spatial infinity |y|→∞) to both of the derived solutions r(y, t), equation (3.10) and the ACH two-soliton u(y, t), equation (3.15). In each case, the net phase shift in y-spaceEmbedded Image(4.5)is always backwards; i.e. the shorter of the solitary waves undergoes the greater post-collision displacement (all these in accord with classical KdV theory).

Similarly, comparison of (4.2) and (4.3)—and the corresponding results for fixed θ2—with (2.23), shows that the CH two-soliton u(x,t) is (locally at x→±∞) the sum of the same pair of solitary waves (as t→±∞). The phase shifts follow directly asEmbedded Image(4.6)where the parameters ki=κpi, i=1,2, have been introduced in order to emphasize the functional dependence Δi(k1,k2). The resultant displacement of the CH solitons is thereforeEmbedded Image(4.7)

Evidently, the shifts Δ1, Δ2 and Δ12 in x-space are far more intricate than their counterparts in y-space, equations (4.4) and (4.5). For our choice of wavenumbers p1>p2, the (k1,k2)-space is given by 1>k1>k2>0 (equation (3.12)). In this parameter space, the phase shifts can be shown to have the following properties:

  1. Δ1>0: post-collision, the taller, faster soliton is always shifted forwards;

  2. Δ2 can be positive, negative or zero: following interaction, the shorter, slower pulse may be displaced forwards, backwards or experience no shift at all (cf. δ2);

  3. Δ12 can take positive, negative or zero values: the smaller pulse may therefore be shifted backwards a distance that is less than, greater than or equal to that of the taller wave (cf. (4.5)).

For Δ1, there is little more to be said. Figure 6a shows the sign of Δ2 for the chosen parameter space k1>k2 (upper half of quadrant). Similarly, figure 6b shows the sign of Δ12, but also includes the symmetrical parameter space when k1<k2 (lower half quadrant). The curves along which Δ2 and Δ12 can be zero are clearly visible: we see that both curves approach the limit (k1,k2)→(1, 1) and so embrace solitons that lie in the near two-peakon regime. The phase shifts can be seen in figures 7 and 8 which picture the evolution of the CH two-soliton in (x,t)-space. Figure 7 shows an almost two-peakon interaction (κ=0.11, k1=0.98, k2=0.95) in which the shorter pulse experiences no displacement (Δ2=0). Figure 8 illustrates the case Δ2>0, when the smaller solitary wave is shifted forwards as a result of the collision. Notice that, in both these examples, the taller soliton is shifted forwards (in accord with (i)), but in figure 8 this displacement is much smaller than that of the shorter pulse.

Figure 6

Sign of the phase shifts for the CH two-soliton interaction. (a) Δ2: the displacement of the shorter pulse (in the restricted parameter space 1>k1>k2>0); (b) Δ12: net shift of the two soliton pulses (for all 1>k1, k2>0).

Figure 7

Evolution of a near two-peakon solution of the CH equation (κ=0.11, k1=0.98, k2=0.95): the shorter pulse suffers no post-collision phase shift (Δ2=0).

Figure 8

A CH two-soliton collision in which the shorter solitary wave is shifted forwards (Δ2>0).

5. Two-soliton solutions of the reduced Camassa–Holm equation κ=0

We consider the reduction κ=0 of the CH equation (2.1) for which, as far as we know, no analytic multisoliton solutions are known (other than the solitary wave (2.29)). It is well known that the RCH equation (1.4) admits non-classical solitons in the form of N-peakons (Camassa & Holm 1993). These intriguing solutions consist of a superposition of N distinct peakon solitary waves, (2.32), whose mutual interactions are well understood (Camassa et al. 1994; Beals et al. 1999). To obtain an analytic two-soliton solution of the RCH equation we make use of the transformation (2.28): applied to the two-soliton solution (3.15), we getEmbedded Image(5.1)where the coordinate transformation x(y,t) follows from (3.18) asEmbedded Image(5.2)and f and r are given by (3.1) and (3.10), respectively. Equations (5.1) and (5.2) give an explicit (parametric) representation (in y) for the classical two-soliton solution of the RCH equation (where κ>0 is now just a mathematical parameter). It describes the elastic interaction of two elevated pulses which (asymptotically as t→±∞) can be identified with a pair of smooth RCH solitary waves (of the form (2.29) and (2.30)). Moreover, the inversion map (2.31) shows that, for each such solution, there is a dual anti-two-soliton wave of depression. These analytic solutions of the reduced CH equation (1.4) appear to be new. Figure 9 pictures the typical RCH two-soliton propagating to the right, along with its leftward travelling antisoliton cousin.

Figure 9

A rightward propagating two-soliton solution of the RCH equation (κ=0) and its leftward moving dual antisoliton.

The two-soliton in figure 4 suggests that it should be possible to recover the two-peakon solution of the RCH equation in the limit (cf. (2.33))Embedded Image(5.3)However, the limiting process (5.3) is quite subtle (as will become apparent) and we shall present only its main features (leaving the interested reader to work through the details). Remarkably, the N-peakon solution of equation (1.4) is a superposition of peakons (Camassa & Holm 1993),Embedded Image(5.4)which interact ‘elastically’ in the manner characteristic of all solitons. In view of (2.32), we may identify the component peakon waves by the generalized amplitudes Embedded Image of the peaks centred at Embedded Image (which evolve according to a canonical Hamiltonian system). However, even for the two-peakon, the explicit formulae for these parameters are extremely intricate (Camassa et al. 1994; Beals et al. 1999). Unfortunately, the two-peakon solution no longer admits the simple linear time dependence of the analytic CH solitons (through the phase variables (3.8)). This complicates the limit (5.3) since, in marked contrast to the latter solutions, there is a post-interaction exchange of identities p1p2 between the two peakons. This unusual property of the multipeakons appears to have been overlooked in the literature and has implications for understanding the two-peakon dynamics (particularly those of the peakon–antipeakon collision).

To see how to implement the two-peakon limit, we need to gauge the effect that (5.3) has on the coordinate transformation (3.18); this is illustrated in figure 10 for ‘near peakon’ solutions with Embedded Image, Embedded Image. As we approach the limit—from κ=0.01 (with κp1=0.9999, κp2=0.99975) in figure 10a, to κ=0.00005 in figure 10b (which generates a solution that is indiscernable from a true two-peakon wave)—the mapping approximates ever more closely to a function composed of two ‘vertical’ steps. Hence, as we proceed to the limit, the transformation (3.18) may be replaced almost everywhere in the x-domain by the three asymptotes (dashed lines) that are given by (3.19) and (3.20). The exceptional points lie in the immediate vicinity of the peaks that are identified by the two ‘horizontal’ treads. But even in the y-domain the extent of these two regions diminishes rapidly with the distance between the outer tramlines (≈−8κ ln κ as κ→0). There is one caveat: in each case, figure 10 shows the taller, faster pulse to the left of the shorter one. Post-interaction, when the taller wave is now ahead, the asymptote (3.20) for the inner region between the two peaks must be replaced byEmbedded Image(5.5)

Figure 10

The ‘two-step’ coordinate transformation x(y,t) in the almost two-peakon limit (κ→0, κpi→1, i=1, 2): (a) κ=0.01(κp1=0.9999, κp2=0.99975); (b) κ=0.00005.

Armed with the relations (3.19), (3.20) and (5.5) and noting that in the limit (5.3) we haveEmbedded Imageone can now eliminate the parameter y in favour of x in the limiting form of (3.15). This is entirely straightforward and yields u(x,t) as a sum of two exponentials to recover the two-peakon solution that is given by setting N=2 in equation (5.4).

The phase shifts for the two-peakon interaction follow directly from equations (4.6) and (4.7) by letting ki→1, i=1, 2, and yieldsEmbedded ImageThese formulae agree with those reported by Camassa et al. (1994) and Beals et al. (1999). It should be emphasized that the peakon wave speeds Embedded Image are now arbitrary and, unlike the analytic CH two-soliton, may take positive and negative values. This permits a two-peakon solution of the RCH equation (1.4) that represents the head-on collision between a peakon and antipeakon (see Camassa et al. (1994) and Beals et al. (1999) for an analysis and description of this type of interaction). Contrary to the impression given in the literature, multipeakon solutions are not restricted to the RCH equation (1.4) (κ=0). Indeed, one need only apply the inverse of (2.28) to (5.4) to obtain a piecewise analytic N-peakon solution of the CH equation (2.1) for any κ>0. These weak solutions, which may feature pairwise peakon–antipeakon interactions, appear to have gone unnoticed until now.

6. Some concluding comments

Before we proceed to obtain further soliton solutions of the CH equation (2.1) in part III of this study—with the ultimate intention of finding the general N-soliton form—it will serve us to reflect on the preceding results for the two-soliton. Let us return to the ACH two-soliton (3.15): we have already noted its structural resemblance to the solution for r(y,t), equation (3.10). The other notable feature of the expression (3.15) is the additional parameter b12, equation (3.16), which is required to formulate u(y,t) explicitly. We stress that b12 is a ‘new’ parameter since it cannot be expressed solely in terms of A12, equation (3.4) (cf. the pre-eminent role of this interaction parameter in classical KdV theory). It is instructive to examine this key parameter a little more carefully: to do so, we again revert to the bilinear formalism that has, thus far, served us so well. In the previous study I, we observed that the ACH solitary wave (2.21) can be reformulated asEmbedded Image(6.1)which, on inserting (3.9), involves only the Hirota function f(y,t), equation (2.16). Couched in bilinear terms, this compact expression has an identifiable structure that can inform our analysis of the higher-order solitons. Thus, if we compare the ACH two-soliton (3.15) with (6.1) and recall the earlier comparison with r(y,t) in (3.9) and (3.10), then we are compelled to consider the bilinear expression Embedded Image. With f now given by (3.1) and making use of (3.3), we find thatEmbedded Image(6.2)whereEmbedded Image(6.3)

The similarity between (6.2) and the numerator in (3.15) is striking and cannot be easily ignored: the two expressions are identical save for the one change of sign in μ12, (6.3), compared with b12, (3.16). Consequently, the bilinear formulation (6.1) provides a generic structure for both the solitary wave and two-soliton, except for the (slight) perturbation of the symmetric term Embedded Image in (6.2) that is needed to obtain the latter solution. This is no coincidence: indeed, we will find that this generic form extends to the N-soliton solution. We shall pursue this further in part III; for now, we simply note that the expression (6.2) may be used to extract the coefficient μ12 which can then serve as a template for the key parameter b12. One more comment is pertinent here: the fact that (6.1) leads to the exact expression only for the solitary wave, betrays its exceptional status among the N-solitons, one that is already apparent from the special relation (2.22).

There is one final observation that we wish to make. All the dynamical properties and peakon limiting procedures for the two-soliton solution of the CH equation generalize quite readily to the higher-order multisolitons. One simply applies the results obtained above to the pairwise interactions between constituent solitary waves. Consequently, we need not revisit these features for the soliton solutions that will be discussed in the concluding part III of the study.

Acknowledgments

Note added in revision. We are grateful to a referee for bringing to our attention a recent paper by Li & Zhang (2004). In this study, the authors adapt and refine the solution procedure of Johnson (2003) for finding soliton solutions of the Camassa–Holm equation. By using the Darboux transformation (for the KdV equation), they solve equations (2.8) and (2.9) for the potential function Q rather than for r, as we have done. Following Johnson's scheme, they then proceed to obtain explicit expressions for the solitary wave and two-soliton solution of the CH equation (2.1) (for κ>0 only).

Footnotes

    • Received November 30, 2004.
    • Accepted June 17, 2005.

References

View Abstract