EXACT PEAKON SOLUTIONS GIVEN BY THE GENERALIZED HYPERBOLIC FUNCTIONS FOR SOME NONLINEAR WAVE EQUATIONS∗

In 1993, Camassa and Holm drived a shallow water equation and found that this equation has a peakon solution with the form φ(ξ) = ce−|ξ|. In this paper, we show that three nonlinear wave systems have peakon solutions which needs to be represented as generalized hyperbolic functions. For the existence of these solutions, some constraint parameter conditions are derived.


Introduction
In 1993, Camassa and Holm used Hamiltonian methods to derive a new completely integrable dispersive shallow water equation (see [3,4]): where u is the fluid velocity in the x−direction (or equivalently the height of the water's free surface above a flat bottom), κ is a constant related to the critical shallow water wave speed, and subscripts denote partial derivatives. Considering traveling wave solutions with the form u = ϕ(x − ct) = ϕ(ξ) of equation (1.1), we have the corresponding traveling wave system [8]: (1.2) where g is an integral constant. System (1.7) has the first integral We notice that when ϕ = c, the right hand of the second equation of system (1.2) is not well-definition. In the (ϕ, y)−phase plan, ϕ = c is call a singular straight line. Taking g = 0 in system (1.2), for a fixed c > 0, near the parameter value κ = 0, we have the bifurcations of phase portraits of system (1.2) as shown in Fig.1.
It is well known that the classical Camassa-Holm equation (1.1) has been studied extensively in the last twenty years because of its many remarkable properties: infinity of conservation laws and complete integrability, existence of peakon and multipeakons and compacton solutions (i.e., breaking waves, and meaning solutions that remain bounded, while its slope becomes unbounded in finite time).
(1) For a homoclinic orbit (see Fig.1 (c)), if there exists a segment which completely lies in a left (or right) small strip neighborhood of a singular straight line, then this homoclinic orbit defines a pseudo-peakon solution of the system.
(2) For a family of periodic orbits (see Fig.1 (b)), if there exists a segment of every orbit which completely lies in a left (or right) small strip neighborhood of a singular straight line, then these periodic orbits determine a family of periodic peakon solutions of the system. Periodic peakons are two-time-scale smooth classical solutions. Cusp wave parts are locally smooth. Periodic peakons are not weak solutions in any reasonable sense.
(3) If there exists a curve triangle (see Fig.1 (b)) connecting saddle points and surrounding a periodic annulus of a center of the corresponding traveling wave system in the neighborhood of a singular straight line, for which a segment is an edge of the triangle, then as a limiting curve of a family of periodic orbits this curve triangle gives rise to a peakon solution of the system.
In fact, peakon is a limiting solution in the following sense: (i) Under fixed parameter conditions, peakon (or solitary cusp wave solution) is a limiting solution of a family of periodic peakon solutions; (ii) with changeable parameters, peakon is a limiting solution of a family of pseudo-peakons. It should be emphasized that a peakon solution is a C 0 -function, i.e., it should be a continuous solution. It is not a weak solution in the sense of distribution.
In [11], we shown that under different parameter conditions, one nonlinear wave equation can have different exact one-peakon solutions and different nonlinear wave equations can have different explicit exact one-peakon solutions. Namely, there are various exact explicit one-peakon solutions, which are different from the one-peakon solution ϕ(x, t) = ce −|x−ct| .
In this paper, for three nonlinear wave equations, we derive some new exact peakon solutions which need to be given by generalized hyperbolic functions, i.e., so called the Arai q-deformed function (see [1,2] Notice that now we have In next three sections, we consider the following three nonlinear wave equations, respectively. (i) The multicomponent Korteweg-de Vries equation with dispersion posed by Kupershmidt in 1985 [7]: System (1.5) is a bi-Hamiltonian system with an infinite number of conservation laws.
(ii) The nonlinear Schrödinger equation with fourth-order dispersion and cubicquintic nonlinearity as follows: (1.6) This equation governs wave dynamics of optical fiber system (see [14]).
, i.e., the solutions of the differential The main result of this paper is the following theorem. (ii) Considering the traveling wave solutions defined by (3.1), the nonlinear Schrödinger equation (1.6) has the peakon and anti-peakon solutions given by (3.8) and (3.9).
The proof of the conclusions of this theorem can be seen in sections 2,3 and 4.

The exact peakon solutions of the multicomponent Korteweg-de Vries equation (1.5)
To investigate the traveling wave solutions of system (1.4), let

Substituting (2.1) into the second equation of system (1.4) and integrating the obtained equation once, we have
where we take the integral constant as zero. Substituting (2.2) into the first equation of system (1.5) and integrating the obtained result onece, we obtain the planar dynamical system which has the first integral where h is a constant. Consider the associated regular system of system (2.3) as follows: 3) and (2.5) have the same first integral, but in the phase plane, two systems define different vector fields in the two sides of the singular straight lines.  Obviously, system (2.5) has the equilibrium points is a double equilibrium point of system (2.5). For the first integral given by (2.4), we write that and For a fixed pair (a 2 , c) with c > 0, when g = g s = 1 4 ca 2 − 3 16 a 4 , we have h 2 = h s . Taking g = g s , we obtain the phase portrait of system (2.3) shown in Fig.2 (a). The level curves defined by H 1 (ϕ, y) = h s are shown in Fig.2 . By using the first equation of (2.3), we have .
When h = h s , for the curve triangle, this integral becomes Thus, we obtan the following peakon solution (see Fig.2 (c)) with the parametric representation: , and tanh q0 (ξ), ctnh q0 (ξ) are the Arai q-deformed functions.

The exact peakon solution of the nonlinear Schrödinger equation (1.6)
Let Substituting (3.1) into equation (1.6) and separating the real and imaginary parts, reducing the fourth order derivative term ϕ (4) , we obtain from [3] that
To calculate the exact parametric representations of the orbits defined by H 2 (ϕ, y) = h of system (3.3), we see from (3.4) that By using the first equation of (3.3), we have (3.6) Making the transformation ψ = ϕ 2 , (3.6) becomes .

The exact peakon solutions of the rotation-twocomponent Camassa-Holm system (1.7)
To investigate the traveling wave solutions of system (1.7), by letting where c is the wave speed, we see from second equation of system (1.7) that v(ξ) = β ϕ−c , where β is an integration constant and β ̸ = 0. By the first equation of system (1.7), we have where 1 2 g is the second integration constant. Write α = 1 − 2AΩ + 2cΩ. Then, the above equation is equivalent to the following two-dimensional system: which has the following first integral: Assume that A > 0. Imposing the transformation dξ = (ϕ − c) 2 (σϕ − c − µ)dζ for ϕ ̸ = c, c+µ σ on system (4.1) leads to the following regular system: (4.3) Apparently, two singular lines ϕ = c and ϕ = c+µ σ are two invariant constant solutions of system (4.3). Near these two straight lines, the variable "ζ" is a fast variable while the variable "ξ" is a slow variable in the sense of the geometric singular perturbation theory.
To see the equilibrium points of system (4.
where H 3 is given by (4.2). In the case of α > 0, suppose that σ ̸ = 0. For a fixed c + µ, by increasing σ from σ < 1 to σ ≥ 1, i.e., letting the singular line ϕ = c+µ σ move from right to left in the (ϕ, y)-phase plane, one can obtain different topological phase portraits of system (4.3). Especially, we have two phase portraits to appear curve triangle shown in Fig.4 (a) and (b) where the corresponding values of H 3 (ϕ j , 0) and parameter conditions are given.
We discuss the exact parametric representations of peakon and anti-peakon solutions of system (4.1) corresponding to two curve triangles in Fig.4 (a) and (b). As can be seen from (4.2), for a fixed integral constant h, one has By using the first equation of system (4.1) and taking integration along a branch of the level curve defined by H 3 (ϕ, y) = h with initial value ϕ(ξ 0 ) = ϕ 0 , one obtains (i) Corresponding to the orbit triangle (see Fig.4 (c)) connecting the equilibrium points E 3 (ϕ 3 , 0) and S ∓ of system (4.3) and enclosing the center E 4 (ϕ 4 , 0) in Fig.4 (c), which is the level curve defined by Hence, taking integrals along the curves E 3 S + and S − E 3 , it yields from (4.4) that . , χ ∈ (−∞, −χ 03 ), and (χ 03 , ∞), ] , (4.6) where )) , ) .