Research

# Local well-posedness and persistence properties for a model containing both Camassa-Holm and Novikov equation

Shouming Zhou1*, Chun Wu1 and Baoshuai Zhang2

Author Affiliations

1 College of Mathematics Science, Chongqing Normal University, Chongqing, 401331, P.R. China

2 School of Economics Management, Chongqing Normal University, Chongqing, 401331, P.R. China

For all author emails, please log on.

Boundary Value Problems 2014, 2014:9  doi:10.1186/1687-2770-2014-9

 Received: 25 June 2013 Accepted: 10 December 2013 Published: 8 January 2014

This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

This paper deals with the Cauchy problem for a generalized Camassa-Holm equation with high-order nonlinearities,

where and . This equation is a generalization of the famous equation of Camassa-Holm and the Novikov equation. The local well-posedness of strong solutions for this equation in Sobolev space with is obtained, and persistence properties of the strong solutions are studied. Furthermore, under appropriate hypotheses, the existence of its weak solutions in low order Sobolev space with is established.

##### Keywords:
persistence properties; local well-posedness; weak solution

### 1 Introduction

This work is concerned with the following one-dimensional nonlinear dispersive PDE:

(1.1)

where and .

Obviously, if , , , equation (1.1) becomes the Camassa-Holm equation,

(1.2)

where the variable represents the fluid velocity at time t and in the spatial direction x, and k is a nonnegative parameter related to the critical shallow water speed [1]. The Camassa-Holm equation (1.2) is also a model for the propagation of axially symmetric waves in hyperelastic rods (cf.[2]). It is well known that equation (1.2) has also a bi-Hamiltonian structure [3,4] and is completely integrable (see [5,6] and the in-depth discussion in [7,8]). In [9], Qiao has shown that the Camassa-Holm spectral problem yields two different integrable hierarchies of nonlinear evolution equations, one is of negative order CH hierachy while the other one is of positive order CH hierarchy. Its solitary waves are smooth if and peaked in the limiting case (cf.[1]). The orbital stability of the peaked solitons is proved in [10], and the stability of the smooth solitons is considered in [11]. It is worth pointing out that solutions of this type are not mere abstractions: the peakons replicate a feature that is characteristic for the waves of great height - waves of largest amplitude that are exact solutions of the governing equations for irrotational water waves (cf.[12-14]). The explicit interaction of the peaked solitons is given in [15] and all possible explicit single soliton solutions are shown in [16]. The Cauchy problem for the Camassa-Holm equation (1.2) has been studied extensively. It has been shown that this problem is locally well-posed for initial data with [17-19]. Moreover, it has global strong solutions and also admits finite time blow-up solutions [17,18,20,21]. On the other hand, it also has global weak solutions in [22-25]. The advantage of the Camassa-Holm equation in comparison with the KdV equation (1.2) lies in the fact that the Camassa-Holm equation has peaked solitons and models the peculiar wave breaking phenomena [1,21].

For , , , equation (1.1) becomes a generalized Camassa-Holm equation,

(1.3)

Wazwaz [26,27] studied the solitary wave solutions for the generalized Camassa-Holm equation (1.3) with , , and the peakon wave solutions for this equation were studied in [28-30], and the periodic blow-up solutions and limit forms for (1.3) were obtained in [31]. In [30,32], the authors have given the traveling waves solution, peaked solitary wave solutions for (1.3).

On the other hand, taking , , in (1.1) we found the Novikov equation [33]:

(1.4)

The Novikov equation (1.4) possesses a matrix Lax pair, many conserved densities, a bi-Hamiltonian structure as well as peakon solutions [34]. These apparently exotic waves replicate a feature that is characteristic of the waves of great height-waves of largest amplitude that are exact solutions of the governing equations for water waves, as far as the details are concerned [13,35,36]. The Novikov equation possesses the explicit formulas for multipeakon solutions [37]. It has been shown that the Cauchy problem for the Novikov equation is locally well-posed in the Besov spaces and in Sobolev spaces and possesses the persistence properties [38,39]. In [40,41], the authors showed that the data-to-solution map for equation (1.4) is not uniformly continuous on bounded subsets of for . Analogous to the Camassa-Holm equation, the Novikov equation shows the blow-up phenomenon [42] and has global weak solutions [43]. Recently, Zhao and Zhou [44] discussed the symbolic analysis and exact traveling wave solutions of a modified Novikov equation, which is new in that it has a nonlinear term instead of .

Other integrable CH-type equations with cubic nonlinearity have been discovered:

(1.5)

where γ is a constant. equation (1.5) was independently proposed by Fokas [45], by Fuchssteiner [46], and Olver and Rosenau [47] as a new generalization of integrable system by using the general method of tri-Hamiltonian duality to the bi-Hamiltonian representation of the modified Korteweg-de Vries equation. Later, it was obtained by Qiao [48,49] from the two-dimensional Euler equations, where the variables and represent, respectively, the velocity of the fluid and its potential density. Ivanov and Lyons [50] obtain a class of soliton solutions of the integrable hierarchy which has been put forward in a series of woks by Qiao [48,49]. It was shown that equation (1.5) admits the Lax-pair and the Cauchy problem (1.5) may be solved by the inverse scattering transform method. The formation of singularities and the existence of peaked traveling-wave solutions for equation (1.5) was investigated in [51]. The well-posedness, blow-up mechanism, and persistence properties are given in [52]. It was also found that equation (1.5) is related to the short-pulse equation derived by Schäfer and Wayne [53].

Applying the method of pseudoparabolic regularization, Lai and Wu [54] investigated the local well-posedness and existence of weak solutions for the following generalized Camassa-Holm equation with dissipative term:

(1.6)

where , and a, k, β are constants. Hakkaev and Kirchev [55] studied the local well-posedness and orbital stability of solitary wave solution for equation (1.6) with , and .

Motivated by the results mentioned above, the goal of this paper is to establish the well-posedness of strong solutions and weak solutions for problem (1.1). First, we use Kato’s theorem to obtain the existence and uniqueness of strong solutions for equation (1.1).

Theorem 1.1Letwith. Then there exists a maximal, and a unique solutionto the problem (1.1) such that

Moreover, the solution depends continuously on the initial data, i.e. the mapping

is continuous.

In [38,56,57], the spatial decay rates for the strong solution to the Camassa-Holm Novikov equation were established provided that the corresponding initial datum decays at infinity. This kind of property is so-called the persistence property. Similarly, for equation (1.1), we also have the following persistence properties for the strong solution.

Theorem 1.2Assume thatwithsatisfies

for some (respectively, ), then the corresponding strong solutionto equation (1.1) satisfies for some

uniformly in the time interval.

Theorem 1.3Assume that, , andwithsatisfies

for some (respectively, , ), then the corresponding strong solutionto equation (1.1) satisfies for some

uniformly in the time interval.

Remark 1.1 The notations mean that

Finally, we have the following theorem for the existence of a weak solution for equation (1.1).

Theorem 1.4Suppose thatwithand. Then there exists a life spansuch that problem (1.1) has a weak solutionin the sense of a distribution and.

The plan of this paper is as follows. In the next section, the local well-posedness and persistence properties of strong solutions for the problem (1.1) are established, and Theorems 1.1-1.3 are proved. The existence of weak solutions for the problem (1.1) is proved in Section 3, and this proves Theorem 1.4.

### 2 Well-posedness and persistence properties of strong solutions

Notation The space of all infinitely differentiable functions with compact support in is denoted by . Let p be any constant with and denote to be the space of all measurable functions f such that . The space with the standard norm . For any real number s, let denote the Sobolev space with the norm defined by

where . Let denote the class of continuous functions from to and .

Proof of Theorem 1.1 To prove well-posedness we apply Kato’s semigroup approach [58]. For this, we rewrite the Cauchy problem of equation (1.1) as follows for the transport equation:

(2.1)

where . and . Let , , and . Following closely the considerations made in [17,54,59], we obtain the statement of Theorem 1.1. □

Proof of Theorem 1.2 We introduce the notation . The first step we will give estimates on . Integrating the both sides with respect to x variable by multiplying the first equation of (2.1) by with , we get

(2.2)

Note that the estimates

and

are true. Moreover, using Hölder’s inequality

From equation (2.2) we can obtain

Since as for any . From the above inequality we deduce that

where we use

Because of Gronwall’s inequality, we get

Next, we will give estimates on . Differentiating (2.1) with respect to the x-variable produces the equation

(2.3)

Multiplying this equation by with , integrating the result in the x-variable, and using integration by parts:

From the above inequalities, we also can get the following inequality:

where we use . Then passing to the limit in this inequality and using Gronwall’s inequality one can obtain

We shall now repeat the arguments using the weight

where . Observe that for all N we have

(2.4)

Using the notation , from (2.1) we get

and from (2.3), we also obtain

We need to eliminate the second derivatives in the second term in the above equality. Thus, combining integration by parts and equation (2.4) we find

Hence, as in the weightless case, we have

A simple calculation shows that there exists , depending only on such that for any ,

Thus, we have

and

Using the same method, we can estimate the other terms:

and

Thus, it follows that there exists a constant which depends only on M, m, n, k, a, and T, such that

Hence, for any and any we have

Finally, taking the limit as N goes to infinity we find that for any ,

which completes the proof of Theorem 1.2. □

Next, we give a simple proof for Theorem 1.3.

Proof of Theorem 1.3 We should use Theorem 1.3 to prove this theorem.

For any , integrating equation (2.1) over the time interval we get

(2.5)

From Theorem 1.2, it follows that

and so

We shall show that the last term in equation (2.5) is ; thus we have

From the given condition and Theorem 1.2. we know as . Since

we have

Thus

From equation (2.5) and as , we know

By the arbitrariness of , we get

uniformly in the time interval . This completes the proof of Theorem 1.3. □

### 3 Existence of solution of the regularized equation

In order to prove Theorem 1.4, we consider the regularized problem for equation (1.1) in the following form:

(3.1)

where , , and a, k are constants. One can easily check that when , equation (3.1) is equivalent to the IVP (1.1).

Before giving the proof of Theorem 1.4, we give several lemmas.

Lemma 3.1 (See [54])

Letpandqbe real numbers such that. Then

Lemma 3.2Letwith. Then the Cauchy problem (3.1) has a unique solutionwheredepends on. If, the solutionexists for all time. In particular, when, the corresponding solution is a classical globally defined solution of problem (3.1).

Proof First, we note that, for any and any s, the integral operator

defines a bounded linear operator on the indicated Sobolev spaces.

To prove the existence of a solution to the problem (3.1), we apply the operator to both sides of equation (3.1) and then integrate the resulting equations with regard to t. This leads to the following equations:

(3.2)

Suppose that is the operator in the right-hand side of equation (3.2). For fixed , we get

Since is an algebra for , we have the inequalities

Since , by Lemma 3.1, we get

and

where , only depend on n. Suppose that both u and v are in the closed ball of radius R about the zero function in ; by the above inequalities, we obtain

where and C only depend on a, k, m, n. Choosing T sufficiently small such that , we know that is a contraction. Applying the above inequality yields

Taking T sufficiently small so that , we deduce that maps to itself. It follows from the contraction-mapping principle that the mapping has a unique fixed point u in .

For , multiplying the first equation of the system (3.1) by 2u, integrating with respect to x, one derives

from which we have the conservation law

(3.3)

The global existence result follows from the integral from equation (3.2) and equation (3.3). □

Now we study the norms of solutions of equation (3.1) using energy estimates. First, recall the following two lemmas.

Lemma 3.3 (See [58])

If, thenis an algebra, and

herecis a constant depending only onr.

Lemma 3.4 (See [58])

If, then

wheredenotes the commutator of the linear operatorsAandB, andcis a constant depending only onr.

Theorem 3.1Suppose that, for some, the functionsare a solution of equation (3.1) corresponding to the initial data. Then the following inequality holds:

(3.4)

For any real number, there exists a constantcdepending only onqsuch that

(3.5)

For, there is a constantcindependent ofϵsuch that

(3.6)

Proof Using and (3.3) derives (3.4).

Since and the Parseval equality gives rise to

For any , applying to both sides of the first equation of (3.1), respectively, and integrating with regard to x again,using integration by parts, one obtains

(3.7)

We will estimate the terms on the right-hand side of (3.7) separately. For the first term, by using the Cauchy-Schwartz inequality and Lemmas 3.3 and 3.4, we have

(3.8)

Using the above estimate to the second term on the right-hand side of equation (3.7) yields

(3.9)

For the fourth term on the right-hand side of equation (3.7), using the Cauchy-Schwartz inequality and Lemma 3.3, we obtain

(3.10)

For the last term on the right-hand side of equation (3.7), using Lemma 3.3 repeatedly results in

(3.11)

It follows from equations (3.7)-(3.11) that there exists a constant c depending only on a, m, n, s such that

Integrating both sides of the above inequality with respect to t results in inequality (3.5).

To estimate the norm of , we apply the operator to both sides of the first equation of the system (3.1) to obtain the equation

(3.12)

Applying to both sides of equation (3.12) for gives rise to

(3.13)

For the right-hand of equation (3.13), we have

(3.14)

and

(3.15)

Since

(3.16)

Using Lemma 3.3, and , we have

(3.17)

and

(3.18)

By the Cauchy-Schwartz inequality and Lemma 3.3, we get

(3.19)

Substituting equations (3.14)-(3.19) into equation (3.13) yields the inequality

(3.20)

with a constant . This completes the proof of Theorem 3.1. □

For a real number s with , suppose that the function is in , and let be the convolution of the function and be such that the Fourier transform of ϕ satisfies , and for any . Thus we have . It follows from Theorem 3.1 that for each ϵ satisfying , the Cauchy problem

(3.21)

has a unique solution , in which may depend on ϵ.

For an arbitrary positive Sobolev exponent , we give the following lemma.

Lemma 3.5Forwithand, the following estimates hold for anyϵwith:

(3.22)

(3.23)

(3.24)

(3.25)

wherecis a constant independent ofϵ.

Proof This proof is similar to that of Lemma 5 in [60] and Lemma 4.5 in [61], we omit it here. □

Remark 3.1 For , using , , equations (3.4), (3.22), and (3.23), we obtain

(3.26)

where is independent of ϵ.

Theorem 3.2Ifwithsuch that. Letbe defined as in the system (3.21). Then there exist two constants, cand, which are independent ofϵ, such thatof problem (3.21) satisfiesfor any.

Proof Using the notation and differentiating equation (3.21) or equation (3.12) with respect to x give rise to

Letting be an integer and multiplying the above equality by , then integrating the resulting equation with respect to x, and using

we find the equality

(3.27)

Applying Hölder’s inequality, we get

where . Furthermore

Since as for any , integrating the above inequality with respect to t and taking the limit as result in the estimate

(3.28)

Using the algebraic property of with and the inequality (3.26) leads to

(3.29)

and

where c is a constant independent of ϵ. Using (3.6), (3.29), and the above inequality, we get

(3.30)

where c is independent of ϵ. Furthermore, for any fixed , there exists a constant such that . By (3.6) and (3.26), one has

(3.31)

Making use of the Gronwall inequality with equation (3.5), with , , and equation (3.26), yields

(3.32)

From equations (3.22)-(3.23) and (3.31)-(3.32), we have

(3.33)

For , applying equations (3.28), (3.30), and (3.33), we obtain

It follows from the contraction-mapping principle that there is a such that the equation

has a unique solution . From the above inequality, we know that the variable T only depends on c and . Using the theorem present on p.51 in [28] or Theorem II in Section 1.1 in [62] one derives that there are constants and independent of ϵ such that for arbitrary , which leads to the conclusion of Theorem 3.2. □

Using equations (3.5)-(3.6) in Theorem 3.1 and Theorem 3.2, with the notation and with Gronwall’s inequality, results in the inequalities

and

where , and . It follows from Aubin’s compactness theorem that there is a subsequence of , denoted by , such that and their temporal derivatives are weakly convergent to a function and its derivative in and , respectively. Moreover, for any real number , is convergent to the function u strongly in the space for and converges to strongly in the space for . Thus, we can prove the existence of a weak solution to equation (1.1).

Proof of Theorem 1.4 From Theorem 3.2, we know that is bounded in the space . Thus, the sequences , are weakly convergent to u, in the space for any , separately. Hence, u satisfies the equation

with and . Since is a separable Banach space and is a bounded sequence in the dual space of X, there exists a subsequence of , still denoted by , weakly star convergent to a function v in . As weakly converges to in , as a result almost everywhere. Thus, we obtain . □

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

This paper is the result of joint work of all authors who contributed equally to the final version of this paper. All authors read and approved the final manuscript.

### Acknowledgements

The authors are very grateful to the anonymous reviewers for their careful reading and useful suggestions, which greatly improved the presentation of the paper. This work is supported in part by NSFC grant 11301573 and in part by the funds of Chongqing Normal University (13XLB006 and 13XWB008) and in part by the Program of Chongqing Innovation Team Project in University under Grant No. KJTD201308.

### References

1. Camassa, R, Holm, D, Hyman, J: A new integrable shallow water equation. Adv. Appl. Mech.. 31, 1–33 (1994)

2. Constantin, A, Strauss, WA: Stability of a class of solitary waves in compressible elastic rods. Phys. Lett. A. 270, 140–148 (2000). Publisher Full Text

3. Fokas, AS, Fuchssteiner, B: Symplectic structures, their Bäcklund transformation and hereditary symmetries. Physica, D. 4, 47–66 (1981). Publisher Full Text

4. Lenells, J: Conservation laws of the Camassa-Holm equation. J. Phys. A. 38, 869–880 (2005). Publisher Full Text

5. Camassa, R, Holm, D: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett.. 71, 1661–1664 (1993). PubMed Abstract | Publisher Full Text

6. Constantin, A: On the scattering problem for the Camassa-Holm equation. Proc. R. Soc. Lond.. 457, 953–970 (2001). Publisher Full Text

7. Boutet, A, Monvel, D, Shepelsky, D: Riemann-Hilbert approach for the Camassa-Holm equation on the line. C. R. Math. Acad. Sci. Paris. 343, 627–632 (2006). Publisher Full Text

8. Constantin, A, Gerdjikov, V, Ivanov, R: Inverse scattering transform for the Camassa-Holm equation. Inverse Probl.. 22, 2197–2207 (2006). Publisher Full Text

9. Qiao, ZJ: The Camassa-Holm hierarchy, N-dimensional integrable systems, and algebro-geometric solution on a symplectic submanifold. Commun. Math. Phys.. 239, 309–341 (2003). Publisher Full Text

10. Constantin, A, Strauss, WA: Stability of peakons. Commun. Pure Appl. Math.. 53, 603–610 (2000). Publisher Full Text

11. Constantin, A, Strauss, WA: Stability of the Camassa-Holm solitons. J. Nonlinear Sci.. 12, 415–422 (2002). Publisher Full Text

12. Constantin, A: The trajectories of particles in Stokes waves. Invent. Math.. 166, 523–535 (2006). Publisher Full Text

13. Constantin, A, Escher, J: Analyticity of periodic traveling free surface water waves with vorticity. Ann. Math.. 173, 559–568 (2011). Publisher Full Text

14. Toland, JF: Stokes waves. Topol. Methods Nonlinear Anal.. 7, 1–48 (1996)

15. Beals, R, Sattinger, D, Szmigielski, J: Acoustic scattering and the extended Korteweg-de Vries hierarchy. Adv. Math.. 140, 190–206 (1998). Publisher Full Text

16. Qiao, ZJ, Zhang, GP: On peaked and smooth solitons for the Camassa-Holm equation. Europhys. Lett.. 73, 657–663 (2006). Publisher Full Text

17. Constantin, A, Escher, J: Global existence and blow-up for a shallow water equation. Ann. Sc. Norm. Super. Pisa, Cl. Sci.. 26, 303–328 (1998)

18. Li, YA, Olver, PJ: Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation. J. Differ. Equ.. 162, 27–63 (2000). Publisher Full Text

19. Rodriguez-Blanco, G: On the Cauchy problem for the Camassa-Holm equation. Nonlinear Anal.. 46, 309–327 (2001). Publisher Full Text

20. Constantin, A: Global existence of solutions and breaking waves for a shallow water equation: a geometric approach. Ann. Inst. Fourier. 50, 321–362 (2000). Publisher Full Text

21. Constantin, A, Escher, J: Wave breaking for nonlinear nonlocal shallow water equations. Acta Math.. 181, 229–243 (1998). Publisher Full Text

22. Bressan, A, Constantin, A: Global conservative solutions of the Camassa-Holm equation. Arch. Ration. Mech. Anal.. 183, 215–239 (2007). Publisher Full Text

23. Constantin, A, Escher, J: Global weak solutions for a shallow water equation. Indiana Univ. Math. J.. 47, 1527–1545 (1998)

24. Constantin, A, Molinet, L: Global weak solutions for a shallow water equation. Commun. Math. Phys.. 211, 45–61 (2000). Publisher Full Text

25. Xin, ZP, Zhang, P: On the weak solutions to a shallow water equation. Commun. Pure Appl. Math.. 53, 1411–1433 (2000). Publisher Full Text

26. Wazwaz, A: Solitary wave solutions for modified forms of Degasperis-Procesi and Camassa-Holm equations. Phys. Lett. A. 352, 500–504 (2006). Publisher Full Text

27. Wazwaz, A: New solitary wave solutions to the modified forms of Degasperis-Procesi and Camassa-Holm equations. Appl. Math. Comput.. 186, 130–141 (2007). Publisher Full Text

28. Liu, ZR, Ouyang, ZY: A note on solitary waves for modified forms of Camassa-Holm and Degasperis-Procesi equations. Phys. Lett. A. 366, 377–381 (2007). Publisher Full Text

29. Liu, ZR, Qian, ZF: Peakons and their bifurcation in a generalized Camassa-Holm equation. Int. J. Bifurc. Chaos. 11, 781–792 (2001). Publisher Full Text

30. Tian, LX, Song, XY: New peaked solitary wave solutions of the generalized Camassa-Holm equation. Chaos Solitons Fractals. 21, 621–637 (2004)

31. Liu, ZR, Guo, BL: Periodic blow-up solutions and their limit forms for the generalized Camassa-Holm equation. Prog. Nat. Sci.. 18, 259–266 (2008). Publisher Full Text

32. Shen, JW, Xu, W: Bifurcations of smooth and non-smooth travelling wave solutions in the generalized Camassa-Holm equation. Chaos Solitons Fractals. 26, 1149–1162 (2005). Publisher Full Text

33. Novikov, YS: Generalizations of the Camassa-Holm equation. J. Phys. A. 42, Article ID 342002 (2009)

34. Hone, ANW, Wang, JP: Integrable peakon equations with cubic nonlinearity. J. Phys. A, Math. Theor.. 41, Article ID 372002 (2008)

35. Constantin, A: The trajectories of particles in Stokes waves. Invent. Math.. 166, 523–535 (2006). Publisher Full Text

36. Constantin, A, Escher, J: Particle trajectories in solitary water waves. Bull. Am. Math. Soc.. 44, 423–431 (2007). Publisher Full Text

37. Hone, ANW, Lundmark, H, Szmigielski, J: Explicit multipeakon solutions of Novikov’s cubically nonlinear integrable Camassa-Holm equation. Dyn. Partial Differ. Equ.. 6, 253–289 (2009). Publisher Full Text

38. Ni, LD, Zhou, Y: Well-posedness and persistence properties for the Novikov equation. J. Differ. Equ.. 250, 3002–3201 (2011). Publisher Full Text

39. Yan, W, Li, Y, Zhang, Y: The Cauchy problem for the integrable Novikov equation. J. Differ. Equ.. 253, 298–318 (2012). Publisher Full Text

40. Grayshan, K: Peakon solutions of the Novikov equation and properties of the data-to-solution map. J. Math. Anal. Appl.. 397, 515–521 (2013). Publisher Full Text

41. Himonas, A, Holliman, C: The Cauchy problem for the Novikov equation. Nonlinearity. 25, 449–479 (2012). Publisher Full Text

42. Jiang, ZH, Ni, LD: Blow-up phenomena for the integrable Novikov equation. J. Math. Anal. Appl.. 385, 551–558 (2012). Publisher Full Text

43. Wu, SY, Yin, ZY: Global weak solutions for the Novikov equation. J. Phys. A, Math. Theor.. 44, Article ID 055202 (2011)

44. Zhao, L, Zhou, S: Symbolic analysis and exact travelling wave solutions to a new modified Novikov equation. Appl. Math. Comput.. 217, 590–598 (2010). Publisher Full Text

45. Fokas, AS: The Korteweg-de Vries equation and beyond. Acta Appl. Math.. 39, 295–305 (1995). Publisher Full Text

46. Fuchssteiner, B: Some tricks from the symmetry-toolbox for nonlinear equations: generalizations of the Camassa-Holm equation. Physica, D. 95, 229–243 (1996). Publisher Full Text

47. Olver, PJ, Rosenau, P: Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support. Phys. Rev. E. 53, 1900–1906 (1996). Publisher Full Text

48. Qiao, ZJ: A new integrable equation with cuspons and -shape-peaks solitons. J. Math. Phys.. 47, Article ID 112701 (2006)

49. Qiao, ZJ: New integrable hierarchy, its parametric solutions, cuspons, one-peak solitons, and -shape peak solitons. J. Math. Phys.. 48, Article ID 082701 (2007)

50. Ivanov, RI, Lyons, T: Dark solitons of the Qiao’s hierarchy. J. Math. Phys.. 53, Article ID 123701 (2012)

51. Gui, GL, Liu, Y, Olver, PL, Qu, CZ: Wave-breaking and peakons for a modified Camassa-Holm equation. Commun. Math. Phys.. 319, 731–759 (2012)

52. Fu, Y, Gui, GL, Liu, Y, Qu, CZ: On the Cauchy problem for the integrable modified Camassa-Holm equation with cubic nonlinearity. J. Differ. Equ.. 255, 1905–1938 (2013). Publisher Full Text

53. Schäfer-Wayne, T: Propagation of ultra-short optical pulses in cubic nonlinear media. Physica, D. 196, 90–105 (2004). Publisher Full Text

54. Lai, SY, Wu, YH: The local well-posedness and existence of weak solutions for a generalized Camassa-Holm equation. J. Differ. Equ.. 248, 2038–2063 (2010). Publisher Full Text

55. Hakkaev, S, Kirchev, K: Local well-posedness and orbital stability of solitary wave solutions for the generalized Camassa-Holm equation. Commun. Partial Differ. Equ.. 30, 761–781 (2005). Publisher Full Text

56. Himonas, AA, Misiolek, G, Ponce, G, Zhou, Y: Persistence properties and unique continuation of solutions of the Camassa-Holm equation. Commun. Math. Phys.. 271, 511–522 (2007). Publisher Full Text

57. Ni, LD, Zhou, Y: A new asymptotic behavior of solutions to the Camassa-Holm equation. Proc. Am. Math. Soc.. 140, 607–614 (2012). Publisher Full Text

58. Kato, T: Quasi-linear equations of evolution with applications to partial differential equations. Spectral Theory and Differential Equations, pp. 25–70. Springer, Berlin (1975)

59. Mu, CL, Zhou, SM, Zeng, R: Well-posedness and blow-up phenomena for a higher order shallow water equation. J. Differ. Equ.. 251, 3488–3499 (2011). Publisher Full Text

60. Bona, J, Smith, R: The initial value problem for the Korteweg-de Vries equation. Philos. Trans. R. Soc. Lond. Ser. A. 278, 555–601 (1975). Publisher Full Text

61. Lai, SY, Wu, YH: A model containing both the Camassa-Holm and Degasperis-Procesi equations. J. Math. Anal. Appl.. 374, 458–469 (2011). Publisher Full Text

62. Walter, W: Differential and Integral Inequalities, Springer, New York (1970)