On the Solutions of the b -Family of Novikov Equation

: In this paper, we study the symmetric travelling wave solutions of the b -family of the Novikov equation. We show that the b -family of the Novikov equation can provide symmetric travelling wave solutions, such as peakon, kink and smooth soliton solutions. In particular, the single peakon, two-peakon, stationary kink, anti-kink, two-kink, two-anti-kink, bell-shape soliton and hat-shape soliton solutions are presented in an explicit formula.


Introduction
The b-family of the Camassa-Holm equation where b is an arbitrary constant and u(x, t) is fluid velocity. Eq.(1) was first proposed by Holm and Stanley in studying the exchange of stability in the dynamics of solitary waves under changes in the nonlinear balance in a 1 + 1 evolutionary PDE related to shallow water waves and turbulence [1,2]. In the case of b = 0, peakon solutions of Equation (1) were discussed in [1,2]. In the case of b = 0, Xia and Qiao showed that Equation (1) provides N-kink, bell-shape and hat-shape solitary solutions [3]. For b = 2, Equation (1) becomes the well-known Camassa-Holm (CH) equation which was originally implied in Fokas and Fuchssteiner in [4], but became well-known when Camassa and Holm [5] derived it as a model for the unidirectional propagation of shallow water over a flat bottom. The CH equation was found to be completely integrable with a Lax pair and associated bi-Hamiltonian structure [4][5][6]. The famous feature of the CH equation is that it provides peaked soliton (peakon) solutions [4,5], which present an essential feature of the travelling waves of largest amplitude [7][8][9]. For b = 3, Equation (1) becomes the Degasperis-Procesi (DP) equation which can be regarded as another model for nonlinear shallow water dynamics with peakons [10,11]. The integrability of the DP equation was shown by constructing a Lax pair, and deriving two infinite sequences of conservation laws in [12].
In this paper, we are concerned with the b-family of the Novikov equation where b is an arbitrary constant. It is easy to see that the b-family of the Novikov Equation (4) has nonlinear terms that are cubic, rather than quadratic, of the b-family of CH Equation (1). The Cauchy problem of the b-family of the Novikov Equation (4) was studied in [13]. For b = 3, Equation (4) becomes the Novikov equation which was discovered by Vladimir Novikov [14] in a symmetry classification of nonlocal PDEs with quadratic or cubic nonlinearity. In [15,16], it was shown that the Novikov equation provides peakon solutions such as the CH and DP equations. Additionally, the Novikov Equation (5) has a Lax pair in matrix form and a bi-Hamiltonian structure.
Moreover, it has infinitely many conserved quantities. The purpose of this paper is to investigate the solutions of the b-family of the Novikov Equation (4) in the case of b = 0 and b = 0. We will show that Equation (4) possesses symmetric travelling wave solutions, such as peakon, kink and smooth soliton solutions. In particular, the single peakon, two-peakon, stationary kink, anti-kink, two-kink, two-antikink, bell-shape soliton and hat-shape soliton solutions are presented in an explicit formula and plotted.
The rest of this paper is organized as follows. In Section 2, we derive the N-peakon solutions in the case of b = 0. In Section 3, we discuss the N-kink and smooth soliton solutions in the case of b = 0.

Peakon Solutions
In this section, we derive the N-peakon solutions in the case of b = 0. We assume the N-peakon solution as the form where p j (t) and q j (t) are to be determined. The derivatives of (6) do not exist at x = q j (t), thus (6) can not satisfy Equation (4) in a classical sense. However, in the distribution, we have Substituting (6)-(10) into (4) and integrating against the test function with compact support, we obtain that p j (t) and q j (t) evolve according to the dynamical system: For N = 1, (11) is reduced to Thus, the single peakon solution ( See Figure 1 ) is For N = 2, (11) is reduced to Solving (13), we have In particular, for C = 1, q 2 (t) = t, b = 1, the solution (See Figure 2) becomes

Kink and Smooth Soliton Solutions
In this section, we discuss the N-kink and smooth soliton solutions in the case of b = 0, namely We suppose that the N-kink solution as the form where c j are arbitrary constants and q j (t) are to be determined. The derivatives of (17) do not exist at x = q j (t), thus (17) can not satisfy Equation (4) in a classical sense. However, in the distribution, we have Substituting (17)-(20) into (16) and integrating against the test function with compact support, we obtain that q j (t) evolves according to the dynamical system: For N = 1, we have q 1,t = 0, which yields q 1 = c, where c is an arbitrary constant. Thus the single kink solution (See Figure 3 and Figure 4) is stationary and it reads  For N = 2, (21) is reduced to If c 2 1 = c 2 2 , we obtain where C 1 is an arbitrary constant. The solution (See Figure 5 and Figure 6) becomes where q 1 and q 2 are given by (24).  If c 2 1 = c 2 2 , we obtain In particular, for q 1 (t) = 1 2 g(t) and q 2 (t) = − 1 2 g(t), the solution (See Figure 7 and Figure 8) becomes