The Camassa-Holm (CH) equation
was derived by Camassa and Holm  as a model for the unidirectional propagation of the shallow water waves over a flat bottom   , where . It was found earlier by Fuchssteiner and Fokas  using the recursion operator of the KdV equation. More interestingly, the CH equation could also be obtained by the tri-Hamiltonian duality approach from the bi-Hamiltonian structure of the KdV equation   . So the CH equation could be regarded as a dual system of the KdV equation. The CH equation had attracted much attention because of its nice features: existence of peaked solitons   (which has a discontinuous first derivative at the spike), complete integrability   , nice geometric formulations    and wave breaking phenomena     etc. It was noticed that the peaked solitons were not allowed by the well-known integrable equations such as the KdV equation, the mKdV equation, and the Schrodinger equation etc. The stability of peakons of the CH Equation (1.1) was proved in   . A similar integrable equation with quadratic nonlinearities was the Degasperis-Procesi (DP) equation  , which took the form
It was regarded as a model for nonlinear shallow water dynamics and could be derived from the governing equation for water waves  . Analogous to the CH equation, the DP equation also possessed an infinite number of conservation laws, bi-Hamiltonian structure, peaked solitons etc. Moreover, it admitted shocked peaked solitons  . Its integrability, existence of peaked solitons, stability of peaked solitons, and wave breaking phenomena were studied extensively in     .
Notice that both the CH and DP equations had the quadratic nonlinearities and conservation laws. So it was of great interest to search for such type of equations with higher-order nonlinearities. To the best of our knowledge, two integrable equations with cubic nonlinearities had been proposed. One was a modified CH equation
which was obtained by the tri-Hamiltonian duality approach from the bi-Hamiltonian structure of the mKdV equation   . Its well-posedness, blow-up, wave breaking, peaked solitons, and their stability were studied in recent works     . The mCH Equation (1.3) exhibits new features of peaked solitons, wave-breaking mechanism, and blow-up criteria. The mCH equation could also be obtained from an invariant non-stretching planar curve flow in Euclidean geometry  . So it was regarded as an Euclidean version of the CH equation in this sense. The other one was the so-called Novikov equation
which was obtained by Novikov  in the symmetry classification of such type equations. The integrability, existence of peaked solitons and their stability, global well-posedness and wave breaking phenomena of Equation (1.4) were discussed in     . Applying the tri-Hamiltonian duality approach to the Gardner equation
they deduced the following generalized CH equation with the cubic and quadratic nonlinearities 
This equation admitted Lax-pair and peaked solitons  .
There are few studies on numerical solutions of the generalized Novikov equation. Therefore in this paper, we will construct a finite difference scheme for the initial boundary problem of equations as follows,
where is a function of time and a single spatial variable , and , , are constants. It is clear that Equation (1.6) is reduced respectively to the Novikov Equation (1.4), the CH Equation (1.1) and the DP Equation (1.2), when , , ; , , ; , , . Clearly, Equation (1.6) is a linear combination of the Novikov equation, the CH equation, and the DP equation.
We shall give an energy conservative finite scheme for Equations (1.6)-(1.8) and obtain the module estimation. Moreover, we prove the convergence and stability of the finite scheme by use of discrete energy method.
For convenience, we denote in the following section. Let , be any positive integers and , ,
for . Denote , , for and .
For simplicity, we introduce some notations as follows:
, , ,
, , ,
For , we define a discrete inner product and the discrete -norm as
In order to obtain the module estimation, investigate the convergence and stability of the finite difference scheme, we need introduce two lemmas as follows,
Lemma 2.1. (Discrete Sobolev inequality  ) There exist constants , satisfying
Lemma 2.2. (Discrete Gronwall inequality  ) Suppose that there exist negative functions , , where is decreasing. For any and , if
3. A Energy Conservative Finite Scheme
The Equation (1.6) can be rewritten in the following form:
which can be expressed as
Firstly, we construct an energy conservative finite scheme for the problem (1.6)-(1.8) as follows:
Lemma 3.1. The difference scheme satisfies discrete conservative law as follows:
Proof. Computing the inner product of the difference scheme with , we have
From the above formulas, we can simplify formula as
So the difference scheme satisfies the discrete conservative law as follows
This completes the proof of Lemma 3.1.
Theorem 3.2. The solution of the difference scheme (3.1)-(3.3) satisfies:
Proof. From Lemma 3.1, we know and . Then by Lemma 2.1, we obtain . This completes the proof of Theorem 3.2.
Theorem 3.3. There exists satisfying the difference scheme (3.1)-(3.3).
Proof. We shall use Mathematical Induction to prove Theorem 3.3.
When , it is known that there exists satisfying the difference scheme from initial condition. Next, we need to prove the case of .
Assume that there exists satisfies the difference scheme when , then we need to prove that there exists satisfies the difference scheme.
Define a operator in as follows:
It is clear that is continuous.
Computing the inner product of the operator , we obtain
By using Cauchy-Schwartz inequality, we obtain
From above discussions, we get
For , then
So we get . By Brouwer fixed point theorem, there exists such that and .
Let , it is easy to verify that satisfies the difference scheme for the problem.
So we complete the proof of Theorem 3.3.
4. Convergence and Stability of the Difference Scheme
In order to investigate the convergence and stability of the difference scheme, we need to obtain the truncation error of the scheme.
Theorem 4.1. If the solution of equation is sufficiently regular, then the truncation error of the difference scheme is .
Proof. Firstly, we can use the Taylor expansion of , , , , , at the point . Secondly, from the above Taylor expansions we reorganize difference equation at point , then
From the above expansions, we obtain the linear part of Equation (3.1) at the point satisfying
On the other hand, the nonlinear parts at the point satisfying
Then we can easily obtain that truncation error of scheme (3.1) is .
Theorem 4.2. The solution of the difference scheme (3.1)-(3.3) approaches to the solution of the original differential Equations (1.6)-(1.8) in , and the corresponding truncation error is .
Proof. Assume that is the solution of the difference scheme (3.1)-(3.3), is the solution of the original differential Equations (1.6)-(1.8). We can easily have .
For the difference scheme (3.1),
then (4,1) minus (3.1), yields
Taking the inner product of (4.2) with , we obtain
By computing , , , , , and respectively, and from Lemma 2.3 and Cauchy-Schwartz inequality, we have
Taking Schwartz inequality, it follows that
From the above discussion, we have
Let , then (4.3) becomes to
Hence we have
From the above inequalities, one has
Since , then . From discrete Gronwall inequality, , that is
Then we have
From Theorem 3.2, we assert
We complete the proof of Theorem 4.2.
Theorem 4.3. The solution of the difference scheme (3.1)-(3.3) is stable in .
Proof. Assume that is the solution of the difference scheme (3.1)-(3.3), is the solution of original differential Equations (1.6)-(1.8). We can easily obtain that .
Then from Theorem 4.2, the following inequality holds true
Thus, we complete the proof of Theorem 4.3.
In this paper, we give a difference scheme for the generalized Novikov equation. In Section 2, we give some preparation knowledge. In Section 3, we propose a conservative finite difference scheme for the generalized Novikov equation and use Brouwer fixed point theorem to obtain the existence of the solution for the corresponding difference equation. In Section 4, we prove the convergence and stability of the solution by using the discrete energy method.
This research is supported by the National Nature Science Foundation of China (Grant Nos. 11501253, 11571141 and 11571140), and the Nature Science Foundation of Jiangsu Province (Grant No. BK20140525), and the Advanced talent of Jiangsu University (Grant Nos. 14JDG070, 15JDG079).
 Constantin, A. and Lannes, D. (2009) The Hydrodynamical Relevance of the Camassa-Holmand Degasperis-Procesi Equations. Archive for Rational Mechanics and Analysis, 192, 165-186.
 Olver, P.J. and Rosenau, P. (1996) Tri-Hamiltonian Duality between Solitons and Solitary-Wave Solutions Having Compact Support. Physical Review E, 53, 1900-1906.
 Fuchssteiner, B. (1996) Some Tricks from the Symmetry-Toolbox for Nonlinear Equations: Generalizations of the Camassa-Holm Equation. Physica D, 95, 229-243.
 Cao, C.S., Holm, D.D. and Titi, E.S. (2004) Traveling Wave Solutions for a Class of One-Dimensional Nonlinear Shallow Water Wave Models. Journal of Dynamics and Differential Equations, 16, 167-178.
 Li, Y.A. and Olver, P. (2000) Well-Posedness and Blow-Up Solutions for an Integrable Nonlinearly Dispersive Model Wave Equation. Journal of Differential Equations, 162, 27-63.
 Constantin, A. and Strauss, W. (2000) Stability of Peakons. Communications on Pure and Applied Mathematics, 53, 603-610.
 Liu, Y. and Yin, Z. (2006) Global Existence and Blow-Up Phenomena for the Degasperis-Procesi Equation. Communications in Mathematical Physics, 267, 801-820.
 Escher, J., Liu, Y. and Yin, Z. (2007) Shock Waves and Blow-Up Phenomena for the Periodic Degasperis-Procesi Equation. In-diana University Mathematics Journal, 56, 87-117.
 Gui, G.L., Liu, Y., Olver, P. and Qu, C.Z. (2013) Wave-Breaking and Peakons for a Modified Camassa-Holm Equation. Communications in Mathematical Physics, 319, 731-759.
 Qu, C.Z., Liu, X.C. and Liu, Y. (2013) Stability of Peakons for an Integrable Modified Camassa-Holm Equation. Communications in Mathematical Physics, 322, 967-997.
 Qu, C.Z., Zhang, Y., Liu, X.C. and Liu, Y. (2014) Orbital Stability of Periodic Peakons to a Generalized Camassa-Holm Equation. Archive for Rational Mechanics and Analysis, 211, 593-617.
 Hone, A.N., Lundmark, H. and Szmigielski, J. (2009) Explicit Multipeakon Solutions of Novikov’s Cubically Nonlinear Integrable Camassa-Holm Type Equation. Dynamics of Partial Differential Equations, 6, 253-289.