A selfadaptive moving mesh method for the CamassaHolm equation
Abstract
A selfadaptive moving mesh method is proposed for the numerical simulations of the CamassaHolm equation. It is an integrable scheme in the sense that it possesses the exact soliton solution. It is named a selfadaptive moving mesh method, because the nonuniform mesh is driven and adapted automatically by the solution. Once the nonuniform mesh is evolved, the solution is determined by solving a tridiagonal linear system. Due to these two superior features of the method, several test problems give very satisfactory results even if by using a small number of grid points.
July 12, 2019
keywords:
The CamassaHolm equation, integrable semidiscretization, peakon and cupson solutions, selfadaptive moving mesh methodMsc:
65M06, 35Q58, 37K401 Introduction
Since its discovery CH (), the CamassaHolm (CH) equation
(1) 
has attracted considerable interest because it describes unidirectional propagation of shallow water waves on a flat bottom. It also appeared in a mathematical search of recursion operators connected with the integrable partial differential equations FF (). By virtue of asymptotic procedures, the CH equation was reconfirmed as a valid approximation to the governing equation for shallow water waves Johnson02 (); Constantin08 (). The CH equation also arises as a model for water waves moving over an underlying shear flow Johnson03 (), in the study of a certain nonNewtonian fluids Busuioc (), and as a model for nonlinear waves in cylindrical hyperelastic rods Dai (). The CH equation is completely integrable (see CH () for the Lax pair formulation and Constantin (); Constantin2 () for the inverse scattering transform), and it has various exact solutions such as solitons, peakons, and cuspons. When , the CH equation admits peakon solutions which are represented by piecewise functions CH (); CHH (); Beals (). When , cusped soliton (cuspon) solutions, as well as smooth soliton solutions, were found by several authors. Schiff (); Kraenkel2 (); Johnson (); Li (); Parker (); Parker2 (); DaiLi (); Matsuno ().
Several numerical schemes have been proposed for the CH equation in the literature. These include a pseudospectral method Kalisch (), finite difference schemes Holden1 (); Coclite1 (), a finite volume method Artebrant (), finite element methods Chiwang (); Matsuo (); Matsuo2 (), multisymplectic methods Cohen (), and a particle method in terms of characteristics based on the multipeakon solution Camassa1 (); Lee1 (); Lee2 (); Lee3 (); Holden2 (). We comment that the schemes in Holden1 (); Coclite1 () and in Cohen () can handle peakonantipeakon interactions. However, it still remains a challenging problem for the numerical integration of the CH equation due to the singularities of cuspon and peakon solutions.
In the present paper, we will study an integrable difference scheme for the CH equation (1) based on an integrable semidiscrete CH equation proposed by the authors Ohta (). The scheme consists of an algebraic equation for the solution and the nonuniform mesh for a fixed time, and a time evolution equation for the mesh. Since the mesh is automatically driven and adapted by the solution, we name it a selfadaptive moving mesh method hereafter.
As a matter of fact, Harten and Hyman has proposed a selfadjusting grid method for onedimensional hyperbolic problems HH83 (). Since then, there has been significant progress in developing adaptive mesh methods for PDEs Miller (); Dorfi (); Brackbill (); Huang (); Stockie (); TaoTang (). These methods have been successfully applied to a variety of physical and engineering problems with singular or nearly singular solutions developed in fairly localized regions, such as shock waves, boundary layers, detonation waves, etc. Recently, an adaptive unwinding method was proposed for the CH equation Artebrant (). The method is high resolution and stable. However, in order to achieve a good accuracy, a large number of grid points () has to be used. In addition, the designed method is only suitable for the single peakon propagation and peakonpeakon interactions, not for the peakonantipeakon interaction. As shown subsequently, the selfadaptive moving mesh method gives accurate results by using a small number of grid points () for some challenging test problems.
The remainder of this paper is organized as follows. In Section 2, we present the selfadaptive moving mesh method and show it is consistent with the CH equation as the mesh size approaches to zero. Two time advancing methods in implementing the selfadaptive moving mesh method are presented in Section 3. In Section 4, several numerical experiments, including the propagations of “peakon” and “cuspon” solutions, cusponcuspon and solitoncuspon collisions, are shown. The concluding remarks are addressed in Section 5.
2 A selfadaptive moving mesh method for the CamassaHolm equation
It is shown in Ohta () that the CH equation can be derived from the bilinear equations of a deformation of the modified KP hierarchy
(2) 
through the hodograph transformation
(3) 
and the dependent variable transformation
Here , and are Hirota’s Doperator defined as
It is proved in Ohta () that the bilinear equations (2) admit a determinant solution , , , where is a Casoratitype determinant of any size. By discretizing the direction with an uniform mesh size , the following bilinear equations
(4) 
admits Casoratitype determinant solution with discrete index which is presented afterwards. Starting from Eq.(4), a semidiscrete CH equation
(5) 
was proposed (see the details in Ohta ()). Here the solution is approximated by at the grid points (). The mesh is a discrete analogue of the hodograph transformation from the domain with uniform mesh size to domain. As is seen, it is nonuniform and timedependent.
The semidiscrete CH equation (5) can be rewritten as
(6) 
by introducing a forward difference operator and an average operator and
In the present paper, Eq. (5) or Eq. (6) is used as a numerical scheme for the CH equation (1). It is shown to be integrable in Ohta () in the sense that it possesses soliton solution which, in the continuous limit, approaches soliton solution of the CH equation. The soliton solution is of the form
(7) 
with
where
Next, let us show that in the continuous limit, (), the proposed scheme is consistent with the CH equation. To this end, the equation (6) is rewritten as
By taking logarithmic derivative of the first equation, we get
Thus, we have
The dependent variable is a function of and , and we regard them as a function of and , where is the space coordinate of the th lattice point and is the time, defined by
Then in the continuous limit, (), we have
and
Further, from
we have
where the origin of space coordinate is taken so that cancels . Then the above semidiscrete CH equation converges to the CH equation
i.e.
(8) 
Setting , we obtain the the CH equation (1).
Note that, in our previous paper Ohta (), we put which gives an alternative form of the CH equation
(9) 
It is shown that they are equivalent under the scaling transformation , . In the present paper, for the convenience in comparing of our results with other papers Johnson (); Li (); Parker (); Parker2 (); DaiLi (); Matsuno (); MatsunoPeakon (), we set .
3 Implementation of the selfadaptive moving mesh method
In this Section, we will discuss how to implement the selfadaptive moving mesh method in actual computations. Generally, given an arbitrary initial condition , the initial nonuniform mesh can be obtained by solving the nonlinear algebraic equations by Newton’s iteration method. However, for the propagation or interaction of solitons or cuspons, which are challenging problems numerically, the initial condition can be calculated by (7) from and by putting , which are obtainable from the corresponding determinant solutions. The initial nonuniform mesh can also be calculated by Ohta ()
(10) 
On the other hand, once the nonuniform mesh is known, the solution can be easily obtained by solving a tridiagonal linear system based on the first equation of the scheme.
(11) 
where
and
In regard to the evolution of , we propose two time advancing methods. The first is the modified forward Euler method, where we assume remains unchanged in one time step. Integrating once, we have
(12) 
where . The second is the classical 4thorder RungeKutta method, where can be viewed as a function of by solving the above tridiagonal linear system. Therefore, in one time step, we have to solve tridiagonal linear system four times.
In summary, the numerical computation in one timestep only involves a ODE solver for nonuniform mesh and a tridiagonal linear system solver. Hence, the computation cost is much less than other existing numerical methods. A Matlab code is made to perform all the computations. Iterative methods, for instance, the biconjugate gradient method bicg in Matlab are used to solve the tridiagonal system.
For the sake of numerical experiments in the subsequent section, we list exact one and two soliton/cuspon and peakon solutions.
(1). One soliton/cuspon solution: The functions for the one soliton/cuspon solution are
(13) 
with , . This leads to a solution
(14) 
(15) 
where the positive case in Eq.(14) stands for the one smooth soliton solution when , while the negative case in Eq.(14) stands for the onecuspon solution when . Otherwise, the solution is singular. Thus Eq.(14) for nonsingular cases can be expressed by
(16) 
Similarly, for the semidiscrete case, we have
(17) 
with , resulting in a solution of the form
(18) 
in conjunction with a transform between an uniform mesh and a nonuniform mesh
(2). Two soliton/cuspon solutions: The functions for the two soliton/cuspon solution are
with , , , . The parametric solution can be calculated through
(19) 
whose form is complicated and is omitted here. Note that the above expression includes the twosoliton solution (, ), the twocuspon solution (, ), or the solitoncuspon solution (, ).
Similarly, for the semidiscrete case, we have
with , . The solution can be calculated through
(20) 
with a transform
(21) 
Again, the explicit form of the solution is complicated and is omitted here.
(3). Peakon solutions: In the continuous CH equation, it is possible to construct peakon solutions from soliton solutions by taking the peakon limit CHH (); LiOlver (); Schiff (); Johnson (); Parker (); PKMatsuno (); MatsunoPeakon ().
For the continuous case, we can express the 1soliton solution as
where , , . Taking the peakon limit , , , the solution , where , gives the 1peakon solution MatsunoPeakon (). In Fig.1, one can see that the 1soliton solution approaches to the 1peakon solution as approaches to 0.
We can also consider the peakon limit for the semidiscrete CH equation. For the semidiscrete case, we can express the 1soliton solution as
where , , . The peakon limit for the semidiscrete CH equation is again , , Taking the peakon limit, the solution , where , approaches to a solution which approaches to the peakon solution of the CH equation as taking the continuous limit. In Fig.2, one can see that the 1soliton solution approaches to the 1peakon like solution as approaches to 0. Taking the continuous limit, this solution approaches to the 1peakon solution of the CH equation.
4 Numerical experiments
In this section, we apply our scheme to several test problems. They include: 1) propagation and interaction of nearlypeakon solutions; 2) propagation and interaction of cuspon solutions; 3) interactions of solitoncuspon solutions; 4) nonexact initial value problems.
4.1 Propagation and interaction of nearlypeakon solutions
Example 1: One peakon propagation. It has been shown in PKMatsuno (); MatsunoPeakon () that the analytic soliton solution of the CH equation converges to the nonanalytic peakon solution when (). To show this, we choose one soliton solution with parameters , . Thus the speed of the soliton () is . Its profile is plotted and is compared with one peakon solution in Fig. 3. These two solutions are indistinguishable from the graph. The error in , where , is calculated to be , and the discrepancy for the first conserved quantity is less than . Therefore, this soliton solution can be viewed as an approximate peakon solution with amplitude .
The propagation of the above designed approximate peakon solution is solved by the selfadaptive mesh scheme with two different time advancing methods: the modified forward Euler method (MFE) and the classical RungeKutta method (RK4). The length of the interval in the domain is chosen to be and the number of grid is . For the above parameters of onepeakon solution, the length of the computation domain turns out to be about . Figures 4 (a)(d) display the numerical solutions at , together with the selfadjusted mesh. It can be seen that the nonuniform mesh is dense around the crest. The most dense part of the nonuniform mesh moves along with the peakon point with the same speed. With the same grid points , the relative errors in norm and the first conserved quantity are computed and compared in Table LABEL:t:peakon. Here, , where and represent the numerical and analytical solutions at the grid points , respectively. indicates the relative error in , where stands for the counterpart of by the numerical solution. Trapezoidal rule on the nonuniform mesh is employed for the evaluation of the integrals.
T  

MFE  0.001  
0.001  
RK4  0.01  
0.01 
T  

MFE  0.001  
0.001  
RK4  0.01  
0.01 
Example 2: Two peakon interaction. For , we initially choose two approximate peakon solutions moving with velocity , and , respectively. Their interaction is numerically solved by MFE and RK4, respectively, with a fixed grid number of . Figure 5 displays the process of collision at different times. Table LABEL:t:peakon2 presents the errors in norm and . It could be seen that, in spite of a small number of grid points and a large time step, RK4 simulates the collision of two approximate peakons with good accuracy.
In regard to the propagation and interaction of approximate peakon solutions, we summarize as follows:

Due to the integrability of the scheme and the selfadaptive feature of the nonuniform mesh, the norm is small and the first conserved quantity is preserved extremely well even for a small number of grid points.

The errors is mainly due to the time advancing methods. MFE is first order in time, so it produces relatively large and , roughly changing in proportional with time. RK4 is fourthorder in time, so up to , and are of the orders and for a grid number of and a time step .
4.2 Propagation and interaction of cuspon solutions
The classical 4thorder RungeKutta method fails whenever the cuspon solution is involved. It seems that a kind of instability occurs in this case, whose theoretical reason is still unclear. Therefore, only MFE is employed to conduct the numerical experiments hereafter.
Example 3: Onecuspon propagation. The parameters taken for the onecuspon solution are , . The number of grid is taken as in an interval of width of in the domain. Through the hodograph transformation, this corresponds to an interval of width in the domain. Figure 6(a) shows the initial profile and the initial mesh. Figures 6(b)(d) display the numerical solutions (solid line) and exact solutions (dotted line) at , together with the selfadjusted mesh. It can be seen that the nonuniform mesh is dense around the cuspon point, and moves to the left in accordance with the movement of the cuspon point. Table 3 exhibits the results of relative errors in norm and .
2.0  

4.0  
2.0  
4.0 
Example 4: Twocuspon interaction. The parameters taken for the twocuspon solutions are , , . Figures 7(a)(d) display the process of collision at several different times, along with the exact solution. Meanwhile, the selfadaptive mesh is also shown in the graph. It can be seen that two cuspon solutions undertake elastic collision, regaining their shapes after the collision is complete. As mentioned in DaiLi (), the two cuspon points are always present during the collision. The grid points are automatically adapted with the movement of the cuspons, and are always concentrated at the cuspon points. In compared with the exact solutions, we can comment that the numerical solutions are in a good agreement with exact solutions. As far as we know, what is shown here is the first numerical demonstration for the cusponcuspon interaction.
4.3 Solitoncuspon interactions
Here we show two examples for the solitoncuspon interaction with . In Fig.8, we plot the interaction process between a soliton of and a cuspon of at several different times where the soliton and the cuspon have almost the same amplitude. It can be seen that when the collision starts (), another singularity point with infinite derivative () occurs. As collision goes on (), the soliton seems ’eats up’ the cuspon, and the profile looks like a complete elevation. However, the cuspon point exists at all times, especially, at , the profile becomes one symmetrical hump with a cuspon point in the middle of the hump.
In Fig.9, we present another example of a collision between a soliton () and a cuspon () where the cuspon has a larger amplitude () than the soliton (). Again, when the collision starts, another singularity point appears. As collision goes on, the soliton is gradually absorbed by the cuspon. At , the whole profile looks like a single cuspon when the soliton is completely absorbed. Later on, the soliton reappears from the right until , the soliton and cuspon recover their original shapes except for a phase shift when the collision is complete.
4.4 Nonexact initial value problems
Here, we show that the integrable scheme can also be applied for the initial value problem starting with non exact solutions. To the end, we choose an initial condition whose mesh size is determined by
(22) 
then, the initial profile can be calculated through the second equation of the semidiscretization, which is plotted in Fig.10 (a). Figures 10 (b), (c) and (d) show the evolutions at , respectively. Note that in this computation. It can be seen that a soliton with large amplitude is firstly developed, and moving fast to the right. By , a second soliton with small amplitude is to be developed.
Next, we increase the value of to 90, which implies a very small dispersion term, corresponding to the dispersionless CH equation. The initial profile and the evolutions at are shown in Fig.11. It is seen that four nearlypeakons are developed from the initial profile at . Later on, an array of nearlypeakons of seven and eight are developed at , respectively. This result is similar to the result for the KdV type equations with a small dispersion, i.e. the peakon trains are generated. (For the KdV type equations, soliton trains are generated. For example, see Kamchatnov (); El () for numerical simulations and Karpman () for a theoretical analysis for the KdV equation.) A theoretical analysis for the dispersionless CH equation to explain the above intriguing numerical result is called for.
5 Concluding Remarks
In the present paper, we have proposed a selfadaptive moving mesh method for the CH equation, which based on an integrable semidiscretization of the CH equation. It has the properties: (1) it is integrable in the sense that the scheme itself admits the soliton solution approaching to the soliton solution of the CH equation in the limit of mesh size going to zero; (2) the mesh is nonuniform and is automatically adjusted so that it is concentrated in the region where the solution changed sharply, for example, the cuspon point; (3) once the nonuniform mesh is evolved, the solution is determined from the evolved mesh by solving a tridiagonal linear system. Therefore, either from the accuracy or from the computation cost, the proposed method is expected to be superior than other existing numerical methods of the CH equation. This is indeed true. The numerical results in this paper indicate that a very good accuracy is obtained.
Two time advancing methods, the modified forward Euler method and the classical 4thorder RungeKutta method, are used to solve the evolution of nonuniform mesh. The RungeKutta method gains much better accuracy than the modified forward Euler method. However, it fails for the computations of cuspons. Using the selfadaptive moving mesh method for the CH equation, we have obtained interesting numerical computation results starting with nonexact solutions. When is very small, the peakon train is generated from the nonexact initial condition.
As further topics, it is interesting to construct integrable discretizations, or, the selfadaptive moving mesh methods for a class of integrable nonlinear wave equations possessing soliton solutions with singularities such as peakon, cuspon or loop solutions. For example, such equations include the short pulse equation which was derived as a model for the propagation of ultrashort optical pulses in nonlinear media SP (),
(23) 
and the DegasperisProcesi (DP) equation DP ()
(24) 
It is worth pointing out that the authors have constructed semi and fulldiscretization for the short pulse equation, which is another example of the selfadaptive moving mesh method FMO09 (), in which we have succeeded in computing the one and twoloop soliton propagations and interactions.
Acknowledgments
The work of B.F. was partially supported by the U.S. Army Research Office under Contract No. W911NF0510029. The work of Y.O. was partly supported by JSPS GrantinAid for Scientific Research (B19340031, S19104002). Y.O. and K.M. are grateful for the hospitality of the Isaac Newton Institute for Mathematical Sciences (INI) in Cambridge where this article was completed during the programme Discrete Integrable Systems (DIS). The authors are grateful to the anonymous referee for valuable comments.
References
 (1) R. Camassa, D.D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett. 71 (1993) 16611664.
 (2) B. Fuchssteiner, A. Fokas, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Physica D 4 (1981) 4766.
 (3) R. Camassa, D.D. Holm, J.M. Hyman, A new integrable shallow water equation, Adv. Appl. Mech. 31 (1994) 133.
 (4) R. Beals, D. H. Sattinger and J. Szmigielski, Multipeakons and the Classical Momemt Problem, Adv. Math. 154 (2000) 229257.
 (5) R.S. Johnson, CamassaHolm, KortewegdeVries and related models for waterwaves, J. Fluid Mech. 457 (2002) 6382.
 (6) A. Constantin, D. Lannes, The Hydrodynamical relevance of the CamassaHolm and DegasperisProcesi equations, Arch. Rational Mech. Anal. 192 (2009) 165186
 (7) R.S. Johnson, The CamassaHolm equation for water waves moving over a shear flow, Fluid Dynam. Res. 33 (2003) 97111.
 (8) V. Busuioc, On second grade fluids with vanishing viscosity, C. R. Acad. Sci. Paris Ser. I 328 (1999) 12411246.
 (9) H.H. Dai, Exact travellingwave solutions of an integrable equation arising in hyperelastic rods, Wave Motion 28 (1998) 367381.
 (10) A. Constantin, On the scattering problem for the CamassaHolm equation, Proc. R. Soc. London A 457 (2001) 953970.
 (11) A. Constantin, V. S. Gerdjikov and R. I. Ivanov, Inverse Scattering Transform for the CamassaHolm equation, Inv. Prob. 22 (2006) 21972207.
 (12) J. Schiff, The CamassaHolm equation: A loop group approach, Physica D 121 (1998) 2443
 (13) M. C. Ferreira, R. A. Kraenkel, A. I. Zenchuk, Solitoncuspon interaction for the CamassaHolm equation, J. Phys. A: Math. Gen. 32 (1999) 86658670.
 (14) R.S. Johnson, On solutions of the CamassaHolm equation, Proc. R. Soc. London A459 (2003) 16871708.
 (15) Y. Li, J.E. Zhang, The multiplesoliton solution of the CamassaHolm equation, Proc. R. Soc. London A460 (2004) 26172627.
 (16) A. Parker, On the CamassaHolm equation and a direct method of solution. I. Bilinear form and solitary waves, Proc. R. Soc. London A460 (2004) 29292957.
 (17) A. Parker, On the CamassaHolm equation and a direct method of solution. II. Soliton solutions, Proc. R. Soc. London A461 (2005) 36113632.
 (18) H.H. Dai, Y. Li, The interaction of the soliton and the cuspon of the CamassaHolm equation, J. Phys. A: Math. Gen. 38 (2005) L685L694.
 (19) Y. Matsuno, Parametric Representation for the Multisoliton Solution of the CamassaHolm Equation, J. Phys. Soc. Jpn. 74 (2005) 19831987.
 (20) H. Kalisch, J. Lenells, Numerical study of travelingwave solutions for the CamassaHolm equation, Chaos, Solitons & Fractals 25 (2005) 287298.
 (21) H. Holden, X. Raynaud, Convergence of a Finite Difference Scheme for the CamassaHolm Equation, SIAM J. Numer. Anal. 44 (2006) 16551680.
 (22) G. M. Coclite, K. H. Karlsen, N. H. Risebro, A Convergent Finite Difference Scheme for the CamassaHolm Equation with General Initial Data, SIAM J. Numer. Anal. 46 (2008) 15541579.
 (23) R. Artebrant, H. J. Schroll, Numerical simulation of CamassaHolm peakons by adaptive upwinding, Appl. Numer. Math. 56 (2006) 695711.
 (24) Y. Xu, C.W. Shu, A local discontinuous Galerkin method for the CamassaHolm equation, SIAM J. Numer. Anal. 46 (2008) 19982021.
 (25) T. Matsuo, H. Yamaguchi, An energyconserving Galerkin scheme for a class of nonlinear dispersive equations, J. Comput. Phys. 228 (2009) 43364358.
 (26) T. Matsuo, A Hamiltonianconserving Galerkin scheme for the CamassaHolm, J. Comput. Appl. Math. (2009) In press.
 (27) D. Cohen, B. Owren, X. Raynaud, Multisymplectic integration of the CamassaHolm equation, J. Comp. Phys. 227 (2008) 54925512.
 (28) R. Camassa, Characteristics and the initial value problem of a completely integrable shallow water equation, Discrete Cont. Dyn.B 3 (2003) 115139.
 (29) R. Camassa, J. Huang, L. Lee, On a completely integrable numerical scheme for a nonlinear shallowwater wave equation, J. Nonlinear Math. Phys. 12 (2005) 146162.
 (30) R. Camassa, J. Huang, L. Lee, Integral and integrable algorithms for a nonlinear shallowwater wave equation, J. Comp. Phys. 216 (2006) 547572.
 (31) R. Camassa, L. Lee, Complete integrable particle methods and the recurrence of initial states for a nonlinear shallowwater wave equation, J. Comp. Phys. 227 (2008) 72067221.
 (32) H. Holden, X. Raynaud, A convergent numerical scheme for the CamassaHolm equation based on multipeakons, Discrete Contin. Dyn. Syst. 14 (2006) 505523.
 (33) Y. Ohta, K. Maruno, B.F. Feng, An integrable semidiscretization of the CamassaHolm equation and its determinant solution, J. Phys. A 41 (2008) 355205.
 (34) A. Harten, J.M. Hyman, Selfadjusting grid methods for onedimensional hyperbolic conservation laws, J. Comput. Phys. 50 (1983) 235315.
 (35) K. Miller, R. N. Miller, Moving finite elements.I, SIAM J. Numer. Anal. 18 (1981) 10191032.
 (36) E. A. Dorfi, T. J. Kaper, Simple adaptive grids for 1d initial value problems, J. Comput. Phys. 69 (1987) 175195.
 (37) J. U. Brackbill, An adaptive grid with directional control, J. Comput. Phys. 108 (1993) 3850.
 (38) W. M. Cao, W. Z. Huang, R. D. Russell, An radaptive finite element method based upon moving mesh PDEs, J. Comput. Phys. 149 (1999) 221244.
 (39) J. M. Stockie, J. A. Mackenzie, R. D. Russell, A moving mesh method for onedimensional hyperbolic conservation laws, SIAM J. Sci. Comput. 22 (2001) 17911813.
 (40) H. Z. Tang, T. Tang, Adaptive mesh methods for one and twodimensional hyperbolic conservation laws, SIAM J. Numer. Anal. 41 (2003) 487515.
 (41) Y. Li, P. Olver, Convergence of solitary wave solutions in a perturbed biHamiltonian dynamical system. I. Compactons and peakons, Discrete Contin. Dynam. Syst. A, 3 (1997) 419432.
 (42) A. Parker, Y. Matsuno, The Peakon Limits of Soliton Solutions of theCamassaHolm Equation, J. Phys. Soc. Jpn. 75 (2006) 124001.
 (43) Y. Matsuno, The Peakon Limit of the Soliton Solution of the CamassaHolm Equation, J. Phys. Soc. Jpn. 76 (2007) 034003.
 (44) A. M. Kamchatnov, R. A. Kraenkel, and B. A. Umarov, Asymptotic soliton train solution of KaupBoussinesq equations, Wave Motion 38 (2003) 355365.
 (45) G. A. El, R. H. J. Grimshaw, and N. F. Smyth, Asymptotic description of solitary wave trains in fully nonlinear shallowwater theory, Physica D 237 (2008) 24232435.
 (46) V. I. Karpman, An asymptotic solution of the Kortewegde Vries equation, Phys. Lett. 25A (1967) 708709.
 (47) T. Schäfer, C. E. Wayne, Propagation of ultrashort optical pulses in cubic nonlinear media, Physica D 196 (2004) 90105.
 (48) A. Degasperis, M. Procesi, Asymptotic integrability, in: A. Degasperis and G. Gaeta (Eds.), Symmetry and Perturbation Theory , World Scientific, Singapore (1999) 2337.
 (49) B.F. Feng, K. Maruno, Y. Ohta, Integrable discretizations of the short pulse equation, J. Phys. A 43 (2010) 085203.