## Abstract

In the concluding study (designated III), we modify the direct bilinear transformation method for solving the Camassa–Holm (CH) equation that was set down in part II of this work. We demonstrate its efficacy for finding *analytic* multisoliton solutions of the equation and give explicit expressions for the first few solitons. It is shown that, at each order *N*, the *N*-soliton has a non-standard representation that is characterized by an ‘extra’ parameter. The stucture of this parameter is investigated and a procedure for constructing the general *N*-soliton solution of the CH equation is presented.

## 1. Introduction

The Camassa–Holm (CH) shallow water wave (SWW) equation (Camassa & Holm 1993),(1.1)has been the subject of intensive study over the last decade. As a consequence, the credentials of the CH equation as a completely integrable system are now well established (see Parker 2004). The constant parameter *κ* in (1.1) may be assumed to be nonnegative without any loss of generality. The case *κ*=0 of equation (1.1),(1.2)which we have dubbed the *reduced Camassa–Holm* (RCH) equation (Parker 2004), merits separate and special attention. In two previous studies—which were designated I and II (Parker 2004, 2005)—it was shown that Hirota's bilinear transformation method (Hirota 1980) can be used to derive analytic soliton solutions of equations (1.1) and (1.2). These papers were concerned specifically with their solitary waves and two-soliton solutions. In the concluding study III, we shall extend the results of the aforementioned works to higher-order solitons, with the ultimate aim of obtaining the general analytic *N*-soliton solution of equation (1.1) for all values of *κ*≥0.

In order to make further progress, we need to consider the lessons that can be drawn from the analysis and results for the two-soliton solution of the CH equation that were given in II (§3). Although we were able to accomplish the calculations for the two-soliton solution entirely ‘by hand’, it is clear that this approach will not serve for higher-order *N*-solitons (not least because the computational burden increases dramatically, even at order *N*=3). Of course, we may use mathematical software (such as Mathematica) to aid our calculations; yet, the work of Johnson (2003) offers a cautionary note against an undue reliance on symbolic manipulation tools. Fortunately, we can mitigate their use by refining our direct bilinear procedure so as to expedite the computations.

Before proceeding, a final prefatory remark is in order. In what follows, we will simply state any previously established equations and results without further explanation and leave the reader to refer back to the earlier studies I and II for their derivation and commentary.

## 2. A modification of the direct method

To see how we should modify our direct method for solving the CH equation (1.1), let us consider the equivalent *associated Camassa–Holm* (ACH) equation (see II, §3),(2.1)(2.2)where *r*(*y*,*t*)>0 is defined by and the coordinates (*x*,*t*) and (*y*,*t*) are related by the reciprocal transformation(2.3)Under the change of dependent variable(2.4)the ACH equation has the bilinear form (Parker 2004)(2.5)(2.6)where D_{y}, D_{t} are the Hirota derivatives (Hirota 1980). It follows that, once a solution *f* of (2.5) and (2.6) has been found, we can derive *r* and *u* directly from equations (2.4) and (2.1), respectively. After inserting these functions into (2.3) to get the coordinate transformation *x*=*x*(*y*,*t*), we finally obtain a solution *u*(*x*,*t*) of the CH equation (1.1) in parametric form (in *y*).

Now, if (2.4) is inserted into the pivotal calculation (2.1), we obtain the ACH solution in the greatly simplified form(2.7)where *W*(*f*, *f*_{y}, *f*_{t}, …) is a trilinear expression in *f* and its derivatives. To find an analytic two-soliton, we take(2.8)which solves the bilinear equations (2.5) and (2.6) provided that(2.9)and(2.10)The solution for *r*(*y*,*t*) follows by substituting *f* into (2.4), or better, into its bilinear equivalent(2.11)Either way, the computation is quite straightforward and yields (Parker 2004)(2.12)where(2.13)and *c*_{i} are the wave speeds in (*y*,*t*)-space,(2.14)It is convenient to recast the phase variables in (2.8) as(2.15)where the *τ*-dependence has been absorbed into the arbitrary phase constants *α*_{i}. The requirement that *r*>0 imposes the restrictions(2.16)which, in turn, ensures that the reciprocal coordinate transformation (2.3) is 1–1.

The two-soliton solution of the ACH equation is found by inserting (2.8) in (2.7) and gives (Parker 2004)(2.17)where(2.18)The corresponding two-soliton *u*(*x*,*t*) of the CH equation follows by integrating equation (2.3) to yield the inverse coordinate transformation(2.19)where(2.20)

Thus, equations (2.17) and (2.19) provide an explicit representation (albeit parametrically in *y*) for the analytic two-soliton of the general CH equation (1.1) with *κ*>0.

For our part, the crucial observation is that another factor of *f* has been removed in the final solution (2.17) when compared with the generic expression (2.7). This further simplificaton is no coincidence and its extension to *N*≥3 has important consequences for the efficacy of our direct method. For if we are to find the general *N*-soliton solution of the CH equation, then we must surely seek its simplest possible form. As such, the compact expression (2.17) for *N*=2 provides a template for formulating the *N*-soliton that brings with it a significant reduction in computation as *N* increases (and the Hirota ansatz *f* grows in complexity). However, the simplified two-soliton form (2.17) was obtained in part II by making judicious use of the dispersion relations (2.10). In fact, this reduction is quite general since the trilinear form *W*(*y*,*t*) in (2.7) can be factorized with the aid of the bilinear equations (2.5) and (2.6). After some careful manipulation of the latter equations, we find that(2.21)where the function *R*(*y*,*t*) can be expressed in bilinear terms as(2.22)

Equation (2.21) replicates the structure of the two-soliton (2.17) and gives the solution *u*(*y*,*t*) of the ACH equation in the simplest possible form. We reiterate that the result is true for any solution *f* of the bilinear equations (2.5) and (2.6) and is therefore not contingent on any dispersion laws. Significantly, the expression for *R* depends on the auxiliary variable *τ* which implies that the reduction (2.21) is made possible only through the bilinear formalism! If we now insert (2.4) into (2.21), then *u* is couched solely in terms of the Hirota function *f*. Consequently, the calculation (2.21) can be considered a routine exercise which can be best performed using mathematical software. In what follows, we will employ Mathematica for this purpose, without further comment. With this modification in place, we are now in a position to construct further multisoliton solutions of the CH equation (1.1).

## 3. Further soliton solutions of the Camassa–Holm equation

### (a) The three-soliton solution

For the three-soliton, we choose the standard ansatz (Matsuno 1984)(3.1)where , are the usual phase variables (and the notation *i*<*j* means that the sum is taken over the ordered pairs of (1,2), (1,3) and (2,3)). Then, making repeated use of the bilinear identity(3.2)where *F*(D_{y}, D_{t}, D_{τ}) is a general bilinear operator and *F*(** p**)=

*F*(

*p*,

*ω*,

*σ*), we find that (3.1) is a solution of the bilinear form (2.5) and (2.6) if(3.3)and

*ω*

_{i},

*σ*

_{i}(

*i*=1, 2, 3) satisfy the dispersion laws given in (2.10). In view of (2.22), the parameters

*σ*

_{i}are now needed to compute

*u*(

*y*,

*t*) (although, as before, the

*τ*-dependence does not appear in the final form of the solution). The function

*r*(

*y*,

*t*) follows by substituting (3.1) into (2.4), or alternatively (2.11), and gives(3.4)where(3.5)

*ν*

_{ij}generalizes

*ν*

_{12}in (2.13) and(3.6)

The symbol 〈*i*〉 here means that the summation is strictly over the three cyclic permutations of (1 2 3). The expression for *ν*_{123} may be simplified by substituting for *A*_{ij} and *c*_{i} from (3.3) and (2.14) (just as for *ν*_{12}). However, the formula (3.6) is preferred here because it emphasizes that the parameter is entirely predictable and so readily lends itself to generalization. Indeed, as *r* is of no interest in itself (though it is a solution of equations (2.1) and (2.2)), we need only note that it is positive under (2.16), as required (hereafter, we shall omit the detailed result for *r* and use equation (2.4) when we come to compute *u*(*y*,*t*) in (2.21)).

To obtain the three-soliton solution *u*(*y*,*t*) of the ACH equation, we first compute *R*(*y*,*t*). Substituting for *f* from (3.1) into (2.22), we find that(3.7)The parameters *b*_{13} and *b*_{23} in (3.7) generalize (2.18) and are conveniently written(3.8)with(3.9)The coefficient *b*_{123} of the symmetric term in (3.7) can then be formulated as (cf. equation (3.8))(3.10)where(3.11)

The symbol ≪ ≫ denotes the sum over all distinct products of the wave numbers that are obtained from the permutations (*i j k*) of (1 2 3).

The three-soliton solution *u*(*y*,*t*) of the ACH equation follows by substituting *f*, *r* and *R* from (3.1), (3.4) and (3.7) in equation (2.21). Reassuringly, the solution can be shown to coincide with the result that was reported by Johnson (2003). It describes the familiar elastic interaction of three smooth ACH ‘solitary’ waves in (*y*,*t*)-space. Although we have no wish to dwell especially on the ACH three-soliton, it merits some comments. The functional form of *R*(*y*,*t*), equation (3.7), mimics precisely that of *M*(*y*,*t*) in (3.5)—this is clearly no accident. It is all the more remarkable for the fact that the 16 terms in (3.7) have been reduced from the more than *one thousand* that originally make up the right-hand side of (2.22)! The other notable feature of (3.7) is the additional parameter *b*_{123}, equation (3.10), that is needed to formulate *R* explicitly. The ‘extra’ parameter appears at each order *N* and is a characteristic of the CH solitons; it will be examined in more detail later.

The final step in the procedure is to find the coordinate transformation from (*y*,*t*) to (*x*,*t*) that is given by integrating (2.3). Yet, the complexity of both *r*(*y*,*t*) and *u*(*y*,*t*) would seem to make this a formidable, if not intractable task, even with the aid of symbolic software (see Johnson (2003) for a discussion). Fortunately, we have the advantage of the two-soliton calculation and soliton reduction principle (Parker 2000) to guide us. The latter tells us that the transformation for *N*=3 should recover the corresponding result for *N*=2, equation (2.19), whenever the three-soliton solution is reduced to a two-soliton. As we did for the two-soliton (II, §3), we choose to evaluate the first of the two integrals in (2.3). With the aid of the parameters *a*_{i},*b*_{i} (*i*=1, 2, 3), equation (2.20), we are able to write the integrand as partial fractions to obtain(3.12)

(3.13)

(3.14)

If we now differentiate (3.12) with respect to *t*, then we find that the second equation in (2.3) is satisfied provided *α* is a constant (so that we can more easily generalize our results, we prefer to use here—rather than the cyclic sum —to denote the sum over (*i*,*j*)=(1,2), (1,3), (2,3), where it is understood that the third subscript *k* takes the remaining value of the triplet (1 2 3)).

This completes the analytic three-soliton solution *u*(*x*,*t*) of the CH equation (1.1) for *κ*>0. Its explicit parametric representation (in *y*) is given by equations (2.21) and (3.12) (taken together with the expressions for *r*, *f*, *R*, *P* and *Q* detailed above). The solution describes a train of three elevated CH ‘solitary’ waves travelling to the right with ‘correct’ supercritical (asymptotic) speeds (see I, §5). Post-interaction, the individual solitons reassert their shapes and speeds in characteristic fashion, with only the cumulative phase shifts betraying their mutual collisions. The choice of *κ* and wavenumbers *p*_{i} (*i*=1, 2, 3) gives rise to a rich diversity of waveforms; we give just two examples of what is possible. Figure 1 shows a more typical CH three soliton (with *κ*=0.4, *p*_{1}=2.1, *p*_{2}=1.9, *p*_{3}=1.6), whereas figure 2 pictures an approximate three-peakon solution (obtained as *κ*→0, *κp*_{i}→1, *i*=1, 2, 3). Figures 3 and 4 show the same two solutions as space-time plots in which the net phase shift sustained by each wave pulse can be seen. The results are quite different: in the former, the tallest soliton is shifted forwards, whilst the shortest is moved backwards and the middle one remains static. However, in figure 4, both the tallest and shortest pulses of the near three-peakon solution are displaced forwards, whereas the intermediate wave is shifted backwards.

Finally, we turn to the special case *κ*=0 of the equation (1.1). To date, the only multisoliton solutions that have been found for the reduced equation (1.2) are the piecewise analytic *N*-peakons (Camassa & Holm 1993). However, if we transform the solution (2.21) and (3.12) under the mapping(3.15)that reduces equation (1.1) to (1.2), then we obtain the *analytic* three-soliton solution(3.16)of the RCH equation. Here *r*, *f*, *R*, *P*, *Q* are unchanged from above and *κ* is now a free parameter, independent of the equation. In addition, if we apply the inversion mapping(3.17)to (3.16), then we deduce the mirror image *anti*-three-soliton solution of equation (1.2) which propagates leftwards. These analytic multisolitons of the RCH equation were not previously known. We remark that no such anti-solitons exist for the full CH-equation (1.1) with *κ*>0.

### (b) The four-soliton solution

So as to confirm the efficacy of our direct method, we present here the four-soliton solution of the CH equation. As this solution is obtained by the exact same procedure that is described in §3*a*, we will content ourselves with stating the results. The starting point is the Hirota function (Matsuno 1984)(3.18)where the notation is the natural extension of that used in (3.1). Using (3.2), we find that (3.18) is a solution of the bilinear equations (2.5) and (2.6) provided that *ω*_{i}, *σ*_{i} and *A*_{ij} are given by (2.10) and (3.3), respectively. The four-soliton solution can then be expressed, *parametrically* in *y*, as(3.19)where(3.20)(3.21)(3.22)and(3.23)with(3.24)

The parameters *b*_{ijk}, *C*_{ijk} that enter into equations (3.20) and (3.24) are the obvious extensions of *b*_{123}, *C*_{123} defined in (3.10) and (3.11), respectively (the symbol that is used in (3.20) denotes the sum over the cyclic permutations of (*ijk*) for the four choices 1≤*i*<*j*<*k*≤4.) If desired, the function *r*(*y*,*t*) may be found explicitly using (2.4) or (2.11); just as for the lower-order solitons, its has the generic form (3.4) in which *M*(*y*,*t*) now mimics *R*(*y*,*t*), equation (3.20).

It is evident that the expression for the CH four-soliton *u*(*x*,*t*) generalizes that for the three-soliton solution given in §3*a*. As anticipated, the ‘new’ feature is the coefficient *b*_{1234} of the symmetric term in *R*(*y*,*t*); we shall examine the structure of this key parameter in §4. A measure of the efficiency of the procedure can be gained by noting that the remaining *sixty-five* terms in equation (3.20) are derived from the more than *twelve thousand* that result from the bilinear terms in (2.22). We give examples of the four-soliton in figures 5–8. Figures 5 and 7 picture the typical smooth-peaked four-soliton solution of the CH equation with *κ*=0.4. By way of contrast, figures 6 and 8 have *κ*=0.1 and capture the elastic interactions of two near-peakon waves and a pair of shorter, bell-shaped solitary-wave pulses. The cumulative phase shifts sustained by the constituent solitons are clearly visible in figures 7 and 8. An analytic four-soliton solution of the RCH equation (1.2) (*κ*=0) follows directly by inserting the above expressions *r*, *f*, *R*, *P* and *Q* for *N*=4 into the generic representation (3.16). Its dual *anti*-four-soliton solution is obtained in the usual way by using (3.17); both these solutions are reported here for the first time.

## 4. *N*-soliton solution of the Camassa–Holm equation

Now that we have demonstrated the efficacy of our direct method, we turn to the main purpose of this study; namely, the construction of an analytic *N*-soliton solution of the general CH equation (1.1). Following our well-rehearsed procedure, we first solve the bilinear form of the CH equation. But, an *N*-soliton solution of the bilinear equations (2.5) and (2.6) can be generated by using the celebrated Hirota ansatz (Hirota 1980)(4.1)where the *τ*-dependence in *f*(*y*,*t*) has been suppressed for the reasons stated earlier. Then, (4.1) is a solution of (2.5) and (2.6) if the interaction coefficients *A*_{ij}, 1≤*i*<*j*≤*N*, are given by (3.3) and *ω*_{i}(*p*_{i}), *σ*_{i}(*p*_{i}) are the dispersion laws (2.10).

The *N*-soliton solution *u*(*y*,*t*) of the ACH equation is obtained by substituting *f* from (4.1) into equations (2.4) and (2.22) to get *r*(*y*,*t*) and *R*(*y*,*t*) and then inserting these functions into (2.21). The calculations are essentially routine and, in principle at least, one can compute the ACH *N*-soliton to any desired order (using suitable mathematical software as appropriate). We shall return to the formulation of *r* and, more particularly, to that of the key function *R*, later in this section. It only remains to determine the coordinate mapping *x*(*y*,*t*) that is found by integrating (2.3). However, the transformation is best obtained (essentially by induction) using the soliton reduction principle (Parker 2000) as, not surprisingly, it simply generalizes the transformations for 1≤*N*≤4. Thus, with *κ*>0, we find that(4.2)where(4.3)and *Q*(*y*,*t*) replicates *P*(*y*,*t*) but has the parameters *a*_{i} and *b*_{i} interchanged (cf. equations (3.21) and (3.22)). The summation notations used in (4.3) are the self-evident extensions of their earlier definitions. Although the expression (4.3) may appear excessively intricate, it has the advantage of generalizing the results for *N*=3 and 4 in a canonical way. We will consider an alternative formulation for *P* and *Q* below.

This completes the *N*-soliton solution of the general CH equation (1.1) for any *κ*>0. Its explicit parametric representation (in *y*) is given by (2.21) and (4.2) and is valid in the usual parameter space that is defined by (2.16). The analytic *N*-soliton and dual anti-*N*-soliton solutions of the reduced CH equation (1.2), with *κ*=0, are obtained in the familiar way by using the transformation (3.15) followed by the inversion (3.17). These solutions have not been reported before now.

Let us now re-examine the structure of the CH solitons in the light of our results. We recall that, in every case, the key function *R*(*y*,*t*) in equation (2.21) mimics *M*(*y*,*t*) in the general expression (3.4) for *r*(*y*,*t*); this is surely not coincidental. But equation (2.11) shows that *M* is determined by the bilinear expression D_{y}D_{t}*f*.*f* which, in turn, has the same form as . This lends credence to our assertion (in II, §6) that the ACH *N*-soliton solution *u*(*y*,*t*), equation (2.21), possesses the generic structure (though not the precise analytic form, except in the case of the solitary wave *N*=1)(4.4)Put another way, the functional form of *R*(*y*,*t*) is *entirely predictable* since it is made up of the same (exponential) terms as . This provides an alternative means of formulating the *N*-soliton solution while simultaneously serving as a useful check on our solutions.

To illustrate, let us revisit the three-soliton solution: with *f* given by (3.1), we obtain (cf. equations (6.2) and (6.3) of II, §6)(4.5)where(4.6)and(4.7)with(4.8)

These results mimic precisely the expressions for *R*(*y*,*t*), *b*_{ij} and *b*_{123} in equations (3.7), (3.8) and (3.10), respectively. It follows that we can use the generic structure (2.21) to formulate the three-soliton solution in a different way. To proceed, we first write down a putative expression for *R*(*y*,*t*), using the readily computed form (4.5) as a template and introducing arbitrary coefficents as necessary. Then the soliton reduction procedure (Parker 2000) immediately determines all the unknown constants except for the coefficent *b*_{123} of the symmetric term . In fact, a glance back at (3.7) shows that this reduction amounts to replacing each *μ*_{ij} by the *known* parameter *b*_{ij} in the bracketed expression of (4.5). To complete the three-soliton solution *u*(*y*,*t*), we have only to find the key parameter *b*_{123}; unfortunately, it cannot be determined by the soliton reduction principle alone. Nevertheless, equations (3.10) and (4.7) show that *b*_{123} and *μ*_{123} differ only in their numerators and suggests we use *E*_{123} as a template for *C*_{123}. Accordingly, we introduce unknown coefficients into the expression (4.8) which are then determined by using the *two*-soliton reductions for *b*_{123} that are given by(4.9)

The first of these identities (and the symmetry in *p*_{i}) requires that *E*_{ij}→*C*_{ij} in (4.8) and gives the first summation in (3.11); the second reduction is sufficient to fix all the remaining coefficients of *C*_{123}. Reassuringly, this reproduces the ACH three-soliton solution *u*(*y*,*t*) that we obtained in §3*a*.

The entire exercise is easily repeated for *N*=4 and yields the four-soliton solution found in §3*b* by the direct method. In principle, we can use this same iterative procedure to obtain the *N*-soliton to any order. One first constructs a putative ACH *N*-soliton *u*(*y*,*t*), equation (2.21), which mimics the generic form (4.4) in which *f* is the Hirota function (4.1). Following application of the soliton reduction procedure (it is sufficient to reduce *u* to a (*N*−1)-soliton asymptotically as *t*→±∞), the only remaining unknown is the coefficient *b*_{12…N} of the symmetric term in the expression for *R*(*y*,*t*) in (2.21). This ‘new’ parameter may be formulated as (cf. (2.18), (3.10) and (3.23))(4.10)where *D*_{ij} is defined in (3.9) and *C*_{12…N} remains to be found. But *b*_{12…N} is entirely predictable from the corresponding coefficient(4.11)of in the bilinear expression . Thus, for *N*=3 and 4, we haveThese expressions are easily generalized to give *μ*_{12…N} so that *E*_{12…N} in (4.11) is known. Using the latter as a template—and invoking reductions for *b*_{12…N} akin to (4.9)—enables us to determine *C*_{12…N}. In effect, deriving the ACH *N*-soliton *u*(*y*,*t*) by this alternative iterative procedure amounts to finding just the *one* parameter *C*_{12…N}. With the general coordinate transformation *x*(*y*,*t*) given by equations (4.2) and (4.3), the analytic *N*-soliton solution *u*(*x*,*t*) of the CH equation (1.1) follows for *all* values of *κ*>0 and those for the RCH equation (1.2) by virtue of the transformations (3.15) and (3.17).

## 5. Discussion

In this study—which concludes the previous works I and II (Parker 2004, 2005)—we have proposed a direct method for obtaining the *analytic N*-soliton solutions of the CH equation (1.1) for *any κ*. This fills an erstwhile gap in the manifold properties that establish the complete integrability of the CH equation. For *κ*>0, we extended the prior results of Schiff (1998) and Johnson (2003) beyond *N*=3; the four-soliton solution is given in explicit form for the first time (§3*b*). Moreover, our bilinear formulation has exhibited the characteristic properties and predictable structure of these solitons. For the case *κ*=0, only *non-analytic* multipeakon solutions (Camasa and Holm 1993) and analytic solitary and anti-solitary waves (Dai 1998; Parker 2004) were previously known. The explicit analytic *N*-solitons and dual *anti*-*N*-solitons of the RCH equation (1.2) that we report here are therefore new.

Our direct method makes use of Hirota's bilinear transformation theory (Hirota 1980) which remains a powerful tool for finding exact solutions of nonlinear evolution equations (Hirota 2004). Yet, we hope to have convinced the reader of more: that, in the case of the CH equation (1.1), the bilinear framework provides the ‘natural’ setting for eliciting the soliton solutions. On reflection, it is apparent that we have invoked the bilinear formalism at every opportunity in our derivations. In particular, it led us to the simplification (2.21) and (2.22) that embodies the generic representation of the ACH solution *u*(*y*,*t*). This crucial result, coupled with the Hirota function (4.1) and coordinate transformation (4.2), yields the explicit *N*-soliton solution *u*(*x*,*t*) of the general CH equation (1.1) with *κ*>0 (albeit in *parametric* form in terms of *y*). But this reformulaton of the solution has further significance for the efficacy of the method: the computational saving that ensues from (2.21) cannot be overstated. Consequently, we need only resort to mathematical software (such as Mathematica) to perform *routine* (but otherwise tedious) calculations. This is no idle remark, but is intended to draw attention to the limitations of symbolic computation, especially when searching for general analytic results.

The analytic form of the first few CH solitons compelled us to consider the bilinear expression (4.4) which captures the generic structure of *u*(*y*,*t*) given in (2.21). As they stand, both results are quite general; yet, remarkably, when *f*(*y*,*t*) is the Hirota ansatz (4.1), they lead to the same functional form! More precisely, for the *N*-soliton solutions, the bilinear expression (2.2) for *R*(*y*,*t*) simplifies dramatically so as to duplicate the conciseness of . We can find no satisfactory explanation for this reduction (though it surely no coincidence), except to note that it is mediated by the dispersion laws . In other words, the auxiliary variable ‘*τ*’ plays a far more significant role here than at first envisaged (it being merely a clever device for casting the bilinear form of the CH equation, (2.5) and (2.6), in terms of the Hirota D-operators). By adopting (4.4) as a template for *u*(*y*,*t*), we developed an alternative procedure for deriving the *N*-soliton solutions. The method proceeds by iteration on the lower-order solitons and, although computationally less efficient, can serve to verify our solutions. More significantly, perhaps, it provides an insight into the structure of the CH solitons and particularly that of the key parameter *b*_{12…N}.

The additional parameter *b*_{12…N} that enters at each order *N*, points to the unusual character of the CH *N*-solitons. This adds to the panoply of non-standard features that endow the CH equation with a richer phenomenology than its classical SWW cousins. Among these features, we wish to mention the Painlevé property for the CH equation. If we apply the standard Painlevé analysis for PDEs (Weiss *et al*. 1983) to equation (1.1), we find that its general solution admits a series expansion of the form(5.1)about a (non-characteristic) singular manifold *ϕ*(*x*,*t*)=0, where *ϕ*, *u*_{j} are analytic functions and *u*_{0}, *u*_{2} are arbitrary. Since, the solution (5.1) contains algebraic singularities, it follows that the CH equation possesses only the ‘weak’ Painlevé property (Ramani *et al*. 1982). This confirms the limitations of the Painlevé PDE test as a predictor of complete integrablity. However, under the reciprocal transformation (2.3), we showed (Parker 2004) that the CH equation is equivalent to the integrable SWW equation of Ablowitz *et al*. (1974) which *does* pass the full Painlevé (P) test. This lends further support to the contention (Clarkson *et al*. 1989) that one should extend the P-test to related PDEs if it fails for the original equation. We remark that the general series (5.1) agrees with the reduced Painlevé expansion that was reported by Gilson and Pickering (1995), after one substitutes the (Kruskal) ansatz *ϕ*=*x*+*ψ*(*t*).

The Painlevé series (5.1) warrants some further comment. For an integrable equation that possesses the regular Painlevé property, the leading term of its (single-valued) P-series often suggests the form of a suitable solution ansatz. It is particularly helpful for finding an appropriate transformation that will lead to the Hirota form of a NEE (Gibbon *et al*. 1985). In this respect, the P-series expansion (5.1) for the CH equation is no exception, notwithstanding its singular nature. To see this, one integrates the conserved form of the CH equation (see II, §2),(5.2)to get(5.3)where we have made use of the reciprocal transformation (2.3) to obtain the last result. By comparing (5.3) with (5.1), we can identify the singular manifold function *ϕ*=*y*, *ϕ*_{x}=*r*. Hence, the leading order term of the P-series identifies the Liouville and reciprocal transformations that, crucially, map the CH equation to the KdV hierarchy. In this instance, however, (5.3) does not provide the Hirota transformation that would lead directly to a bilinear form of the CH equation in the original (*x*,*t*) coordinates. Not surprisingly, the connection between the Painlevé analysis and Hirota's method that was identified by Gibbon *et al*. (1985), is lost here owing to the singular nature of the Painlevé expansion (5.1).

There is one final result that we wish to record. The coordinate transformation (4.2) that maps the ACH *N*-soliton *u*(*y*,*t*) to its CH counterpart *u*(*x*,*t*) can be reformulated as follows:(5.4)where (cf. (4.3))(5.5)and has the expression (5.5) except that *m*_{i}=*b*_{i}/*a*_{i} is replaced by 1/*m*_{i}(*i*=1, …, *N*). (The arbitrary constant in (5.4) has been redefined by allowing .) But, if we now compare (5.5) with the Hirota ansatz (4.1), it is evident that and can be rewritten as(5.6)

(5.7)

Then, upon inserting (5.4) into the reciprocal coordinate transformation (2.3), we deduce(5.8)which bears comparison with (4.4). In short, the analytic *N*-soliton solution of the general CH equation (*κ*>0) can be found explicitly from equations (5.4), (5.6), (5.7) and (5.8). In this guise, we are brought full-circle back to the celebrated Hirota ‘*N*-soliton’ formula (4.1)! Accordingly, we have acquired yet a third technique for obtaining the CH *N*-solitons which, incidentally, has confirmed once more the analytic form of the solutions for *N*=1, 2, 3, 4.

**Note added in revision.** We are indebted to a referee for pointing out the article by Li (2005) which appeared after the submission of our parts II and III. The paper presents an alternative approach to finding analytic multisoliton solutions of the CH equation and extends the previous results of Li and Zhang (2004). The author shows how to construct (by Darboux transformation) the *N*-soliton solution and derives explicit expressions for the one-, two- and three-solitons of the CH equation (1.1) with *κ*>0. The special case *κ*=0 of the RCH (1.2) is not considered.

## Acknowledgements

The author particularly wishes to acknowledge the prior studies of Adrian Constantin, Robin Johnson and Jeremy Schiff, whose contributions on the Camassa–Holm equation have been the inspiration for our own work. We should also like to thank the referees for their helpful comments.

## Footnotes

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

- © 2005 The Royal Society