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Cusped and smooth solitons for the generalized Camassa-Holm equation on the nonzero constant pedestal.

1. Introduction

In 1993, Camassa and Holm [1] derived a nonlinear wave equation (Camassa-Holm equation)

[u.sub.t] + 2[ku.sub.x] - [u.sub.xxt] + 3[uu.sub.x] = 2[u.sub.x][u.sub.xx] + [uu.sub.xxx] (1)

and obtained the peakon wave solution of the form u = [ce.sup.-[absolute value of x-ct]]. Where after, (1) has been researched by many authors [2-7]. Because (1) possesses rich dynamics and complex properties, recently, many authors are interested in its generalized forms. In particular, Liu and Qian [8] suggested a generalized Camassa-Holm equation,

[u.sub.t] + 2[ku.sub.x] - [u.sub.xxt] + 3[u.sup.2][u.sub.x] = 2[u.sub.x][u.sub.xx] + [uu.sub.xxx], (2)

and obtained the explicit expressions of the peakon solution of (2). Afterwards, Tian and Song [9] gave some physical significance of this equation and obtained some peakon solutions with special wave speeds. Kalisch [10] studied the stability of solitary wave solution of (2). He et al. [11] constructed some exact traveling wave solutions by using the integral bifurcation method. Liu and Liang [12] studied the explicit nonlinear wave solutions and their bifurcations of (2). When k = 0, 2) transforms into the following equation:

[u.sub.t] - [u.sub.xxt] + 3[u.sup.2][u.sub.x] = 2[u.sub.x][u.sub.xx] + [uu.sub.xxx]. (3)

For (3), there are some related works. Shen and Xu [13] discussed the existence of smooth and nonsmooth traveling waves. Khuri [14] obtained a singular wave solution composed of triangle functions. Wazwaz [15,16] acquired eleven exact traveling wave solutions composed of triangle functions or hyperbolic functions. Liu and Ouyang [17] obtained a peakon solution composed of hyperbolic functions. Liu and Guo [18] investigated the periodic blow-up solutions and their limit forms. Wang and Tang [19] obtained two exact solutions. Yomba [20, 21] gave two methods, the sub-ODE method and the generalized auxiliary equation method, to obtain the exact solution of (3). Liu and Pan [22] studied the coexistence of multifarious solutions.

In this paper, we use the Qiao and Zhang method [23] to investigate the traveling solitary wave solutions of (3) on the nonzero constant pedestal

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4)

Since Qiao and Zhang presented this method, many authors applied it to different nonlinear models and obtained a variety of new type soliton solutions. Zhang and Qiao [24] discussed the traveling wave solutions for the Degasperis-Procesi equation

[m.sub.t] + [m.sub.x]u + 3m[u.sub.x] = 0, m = u - [u.sub.xx] (5)

on the nonzero constant pedestal and found new cusped and peaked soliton solutions. Qiao [25] proposed a new completely integrable wave equation:

[m.sub.t] + [m.sub.x] ([u.sup.2] - [u.sup.2.sub.x]) + 2[m.sup.2][u.sub.x] = 0, m = u - [u.sub.xx], (6)

and obtained new cusped, one-peak, W/M-shape-peaks soliton solutions. Later, Chen et al. [26, 27] studied the osmosis K(2,2) equation

[u.sub.t] [+ or -] [([u.sup.2]).sub.x] [+ or -] [([u.sup.2]).sub.xxx] = 0 (7)

under the inhomogeneous boundary condition and obtained smooth, peaked, cusped soliton solutions of the osmosis K(2,2) equation by using the phase portrait analytical technique. Wei et al. [28] investigated the generalized KPMEW(2,2) equation

[([u.sub.t] + [([u.sup.2]).sub.x] + [([u.sup.2]).sub.xxt]).sub.x] + [u.sub.yy] = 0 (8)

on the nonzero constant pedestal and acquired smooth, peaked, cusped, and loop soliton solutions. More works on single peak soliton are reported [29-32].

2. Some Preliminary Results

Substituting u(x, t) = u([zeta]) and [zeta] = x - ct into (3), we have

-cu' + cu"' + 3[u.sup.2]u' = 2u'u" + uu"', (9)

where "'" is the derivative with respect to [zeta]. Integrating (9) once, we yield

-cu + cu" + [u.sup.3] = 1/2[(u').sup.2] + uu" + [g.sub.1], (10)

where [g.sub.1] [member of] R is an integration constant.

Further, we get

[(u').sup.2] = [u.sup.4] - 2c[u.sup.2] - 4[g.sub.1]u - 4[g.sub.2]/2 (u - c), (11)

where [g.sub.2] [member of] R is an integration constant.

Let us solve (11) with the following boundary condition:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (12)

where A is a constant. Equation (11) can be cast into the following ordinary differential equation:

[(u').sup.2] = [(u - A).sup.2] ([u.sup.2] + 2Au + 3[A.sup.2] - 2c)/2 (u -c). (13)

When c - [A.sup.2] [greater than or equal to] 0, then 13) reduces to

[(u').sup.2] = [(u - A).sup.2](u - [B.sub.1])(u - [B.sub.2])/2 (u - c), (14)

where

[B.sub.1] = -A + ([square root of 2 (c - [A.sup.2])]), [B.sub.2] = -A - ([square root of 2(c - [A.sup.2])]). (15)

Obviously, [B.sub.1] [greater than or equal to] [B.sub.2].

Remark 1. In the existing research on this method, the cases on [(u - A).sup.2] (u - [v.sub.1])/(u - [v.sub.2]) and [(u - A).sup.2] (u - [v.sub.3])(u - [v.sub.4])/ [(u - [v.sub.5]).sup.2] have been discussed, but the case on [(u - A).sup.2] (u - [v.sub.6]) (u [v.sub.7])/(u - [v.sub.8]) ([v.sub.i] (i = 1, ..., 8) [??] constant) has not been discussed. So we consider it is very meaningful researching this new case on this method, and we can obtain some new soliton solutions from this case.

Definition 2. A wave function u([zeta]) is called smooth soliton solution, if u([zeta]) is smooth and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Definition 3. A wave function u([zeta]) is called cuspon solution, if u([zeta]) is smooth locally on either side of [[zeta].sub.0] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Without loss of generality, we assume [[zeta].sub.0] = 0.

3. The Parametric Conditions and Phase Portraits of Existence of Soliton Solutions of the Generalized Camassa-Holm Equation (3)

By virtue of the above analysis, we know that soliton solitons for the generalized Camassa-Holm Equation (3) must satisfy the following initial and boundary values problem:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (16)

Lemma 4. Suppose that one of the following five conditions holds:

(i) c < [A.sup.2], A [less than or equal to] c;

(ii) c = [A.sup.2], A [less than or equal to] c;

(iii) [A.sup.2] < c < 3[A.sup.2], A [less than or equal to] c;

(iv) 3[A.sup.2] = c, c < A;

(v) 3[A.sup.2] < c, c < A.

Then (3) has trivial solution u([zeta]) = A.

Proof, (i) If c < [A.sup.2] and A [less than or equal to] c, then we have [u.sup.2] + 2Au + 3[A.sup.2] - 2c > 0. When A < c, (13) leads to [(u').sup.2] = [(u - A).sup.2]([u.sup.2] + 2Au + 3[A.sup.2] - 2c)/2(u - c) [less than or equal to] 0. For A = c, 13) can be cast into [(u').sup.2] = (1/2)(u - A)([u.sup.2] + 2Au + 3[A.sup.2] - 2c) [less than or equal to] 0.

(ii) When c = [A.sup.2] and A [less than or equal to] c, then we have [u.sup.2] + 2Au + 3[A.sup.2] - 2c = [(u + A).sup.2] [greater than or equal to] 0. If A < c, 14) changes into [(u').sup.2] = [(u - A).sup.2][(u + A).sup.2]/2(u - c) [less than or equal to] 0. If A = c, 14) transforms into [(u').sup.2] = (1/2)(u - A)[(u + A).sup.2] [less than or equal to] 0.

(iii) For [A.sup.2] < c < 3[A.sup.2] and A [less than or equal to] c, then we obtain [u.sup.2] + 2Au + 3[A.sup.2] - 2c = (u - [B.sub.1])(u - [B.sub.2]) > 0. If A < c, 14) leads to [(u').sup.2] = [(u - A).sup.2](u - [B.sub.1])(u - [B.sub.2])/2(u - c) [less than or equal to] 0. If A = c, 14) changes into [(u').sup.2] = (1/2)(u - A)(u - [B.sub.1])(u - [B.sub.2]) [less than or equal to] 0.

(iv) If 3[A.sup.2] = c and c < A, then we get [u.sup.2] + 2Au + 3[A.sup.2] 2c = (u - A)(u + 3A) < 0 and (14) can be cast into [(u').sup.2] = [(u - A).sup.3](u + 3A)/2(u - 3[A.sup.2]) [less than or equal to] 0.

(v) When 3[A.sup.2] < c and c < A, then we have [u.sup.2] + 2Au + 3[A.sup.2] - 2c = (u - [B.sub.1])(u - [B.sub.2]) < 0 and (14) transforms into [(u').sup.2] = [(u - A).sup.2] (u - [B.sub.1])(u - [B.sub.2])/2(u - c) [less than or equal to] 0.

The fact that [(u').sup.2] [greater than or equal to] 0 implies u' = 0 and u([zeta]) = A.

Obviously, we get that the generalized Camassa-Holm Equation (3) with nonzero boundary condition has soliton solutions when A and c do not belong to the above five cases. Then we obtain the generalized Camassa-Holm Equation 3) with nonzero boundary condition having soliton solutions, when c < [A.sup.2], c < A; c = [A.sup.2], c < A; [A.sup.2] < c < 3[A.sup.2], c < A; and 3[A.sup.2] = c, A [less than or equal to] c; 3[A.sup.2] < c, A [less than or equal to] c.

For the cases on [A.sup.2] < c < 3[A.sup.2], c < A, [B.sub.1] = c; 3[A.sup.2] = c, A = c, and 3[A.sup.2] < c, A [less than or equal to] c, [B.sub.1] = c, Liu and Qian [8] and Tian and Song [9] researched that the generalized CamassaHolm Equation (3) has smooth soliton and peakon solutions as similar as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (17)

where m, n, and p are constants, and s > 0 is an integration constant.

In fact, when c < [A.sup.2], c < A; c = [A.sup.2], c < A; [A.sup.2] < c < 3[A.sup.2], c < A; and 3[A.sup.2] = c, A < c; 3[A.sup.2] < c, A [less than or equal to] c, the generalized Camassa-Holm Equation (3) has also other forms of the smooth soliton and cuspon. Because (11) is equivalent to the two-dimensional system

u' = y, y' = [u.sup.3] - cu - (1/2)[y.sup.2] + [g.sub.1]/u - c. (18)

From (18), we can obtain the phase portraits of existence of soliton solutions of the generalized Camassa-Holm Equation (3) under the inhomogeneous boundary condition, when A and c belong to the above five cases (see Figure 1).

The phase portraits of (3) are shown in Figure 1 under different parametric conditions.

(1-1) c < [A.sup.2], c < A; (1-2) c = [A.sup.2], c < A; (1-3) [A.sup.2] < c < 3[A.sup.2], c < A, c < [B.sub.1]; (1-4) [A.sup.2] < c < 3[A.sup.2], c < A, [B.sub.1] < c; (1-5) c = 3[A.sup.2], 0 < A<c; (1-6) c = 3[A.sup.2], A < -1 < c; (1-7) c = 3[A.sup.2], -1 < A < 0 < c; (1-8) 3[A.sup.2] < c, A = c; (1-9) 3[A.sup.2] < c, A < c, c < [B.sub.1]; (1-10) 3[A.sup.2] < c, A < c, [B.sub.1] < c.

4. Cusped and Smooth Solitons for the Generalized Camassa-Holm Equation (3)

In this section, by using the phase portrait analytical technique, which has been developed by Li and Dai [33], we get cusped and smooth soliton solutions of the generalized Camassa-Holm Equation (3) under the inhomogeneous boundary condition.

Case 1 (c < [A.sup.2], c < A). By the standard phase portrait analysis (see Figure 1(1-1)), we have u(0) = c < A. From 13), we yield

u' = - (u - A) ([square root of [u.sup.2] + 2Au + 3[A.sup.2] - 2c])/ ([square root of 2(u - x)])sign ([zeta]). (19)

Taking the integration of both sides of (19), we can obtain the implicit cuspon solution u([zeta]) defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (20)

where [K.sub.1] = 0 is an integration constant,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (21)

Remark 5. F([phi], k) is the elliptic integral of first kind, and [PI]([phi], [tau], k) is the elliptic integral of third kind [34].

The profile of cusped soliton solution is shown in Figure 2(2-1).

Case 2 (c = [A.sup.2], c < A). Equation (14) can be cast into

[(u').sup.2] = [(u - A).sup.2] [(u + A).sup.2]/2 (u - [A.sup.2]). (22)

By the standard phase portrait analysis (see Figure 1(1-2)), we have u(0) = c < A. From 22), we get

u' = -(u - A)(u + A)/ ([square root of 2(u -[A.sup.2])]) sign([zeta]). (23)

Let h(u) = -1/(u- A)(u + A); then h(c) = -1/(c - A)(c + A), and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (24)

Inserting h(u) = h(c) + O(u) into (24) and using the initial condition u(0) = c, we obtain

1/3[[2(u - [A.sup.2])].sup.3/2] h(c)(1 + O(1)) = [absolute value of [zeta]]. (25)

Thus,

u = 1/2[[absolute value of [zeta]].sup.2/3] [(3/h(c)).sup.2/3] [(1 + O(1)).sup.-2/3] + [A.sup.2], [zeta] [right arrow] 0, (26)

which implies u = 0([[absolute value of [zeta]].sup.2/3]). Therefore, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (27)

So we can get the implicit cuspon solution u([zeta]) defined by

A - 1/([square root of 2(A - [A.sup.2])]) [I.sub.1](u) - ([square root of 2])(A + 1)/([square root of A + [A.sup.2]])[I.sub.2](u) = [absolute value of [zeta]] + [K.sub.2], (28)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (29)

Remark 6. The proof of other cuspons is similar to the above proof.

Because u(0) = c, the constant [K.sub.2] is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (30)

The profile of cusped soliton solution is shown in Figure 2 (2-2).

Case 3 ([A.sup.2] < c < 3[A.sup.2], c < A). In this case, we discuss two conditions: (1) [B.sub.1] > c; (2) [B.sub.1] < c.

(1) When [B.sub.1] > c, by the standard phase portrait analysis (see Figure 1(1-3)), we have c < u(0) [B.sub.1] = < A. From 14), we have

u' = -(u - A)([square root of (u - [B.sub.1])(u - [B.sub.2])])/([square root of 2(u - c)]) sign([zeta]). (31)

As same as the above, we can obtain the implicit smooth soliton solution u([zeta]) defined by

- 2([square root of 2])([B.sub.1] - c)/([B.sub.1] - A)([square root of [B.sub.1] - [B.sub.2]])V(u) = [absolute value of [zeta]] + [K.sub.3], (32)

where

V(u) = [PI](arcsin(([square root of u - [B.sub.1]/u - c]), A - c/A - [B.sub.1], ([square root of c - [B.sub.2]/[B.sub.1] - [B.sub.2]])). (33)

For u(0) = [B.sub.1], the constant [K.sub.3] is defined by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. For this smooth soliton solution, we get an exact explicit form [35]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (34)

The profile of smooth soliton solution is shown in Figure 2(23).

(2) When [B.sub.1] < c, by the standard phase portrait analysis (see Figure 1(1-4)), we have [B.sub.1] < u(0) = c < A. Taking the integration of both sides of (31), we can yield the implicit cuspon solution u([zeta]) defined by

2([square root of 2]) (c - [B.sub.1])/(A - [B.sub.1])([square root of c - [B.sub.2]])([O.sub.1](u) - [O.sub.2](u)) = [absolute value of [zeta]] + [K.sub.4], (35)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (36)

By view of u(0) = c, the constant [K.sub.4] is defined by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The profile of cusped soliton solution is shown in Figure 2(2-4).

Case 4 (3 [A.sup.2] = c, A < c). (1) When 0 < A < c, by the standard phase portrait analysis (see Figure 1(1-5)), we have u(0) = [B.sub.2] < 0 < [B.sub.1] = A < c. Equation 14) transforms into

[(u').sup.2] = [(u - A).sup.2](u - A)(u + 3A)/2(u - 3[A.sup.2]). (37)

From (37), we have

u' = - (u -A)([square root of (u - A)(u + 3A)/2(u - 3[A.sup.2])])sign([zeta]). (38)

Taking the integration of both sides of (38), we can obtain the implicit smooth soliton solution u([zeta]) defined by

([square root of 6])/2([square root of A(A + 1)/A]) (3A - 1/3A + 3 [P.sub.1](u) + 4/3A + 3 [P.sub.2](u)) = [absolute value of [zeta]] + [K.sub.5], (39)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (40)

For u(0) = [B.sub.2], we obtain [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The profile of smooth soliton solution is shown in Figure 2(2-5).

(2) When A < 0, by virtue of (37), we have

u' = (u - A)([square root of (u - A)(u + 3A)/2(u - 3[A.sup.2])])sign([zeta]). (41)

In this case, we discuss two conditions: (i) A < -1; (ii) -1 < A < 0.

(i) When A < -1, by the standard phase portrait analysis (see Figure 1(1-6)), we have A = [B.sub.2] [less than or equal to] u [less than or equal to] [B.sub.1] < c. Taking the integration of (41) on the interval [A, [B.sub.1]], thus, we obtain the implicit smooth soliton solution u(%) defined by

-3([square root of 2])(1 + A)/2([square root of 3[A.sup.2] - A])H(u) = [absolute value of [zeta]] + [Q.sub.1], (42)

where

H(u) = [PI](arcsin(([square root of (3A - 1)(3A + u)/4(3[A.sup.2] - u)])), 1, ([square root of -4/3A - 1])). (43)

Because u(0) = [B.sub.1], we obtain [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. For this smooth soliton solution, we get an exact explicit form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (44)

The profile of smooth soliton solution is shown in Figure 2(26).

(ii) When -1 < A < 0, by the standard phase portrait analysis (see Figure 1(1-7)), we have A = [B.sub.2] [less than or equal to] u < c < [B.sub.1]. Integrating (41) on the interval [A, c), we obtain the implicit cuspon solution u([zeta]) defined by

3([square root of 2])(A + 1)4([square root of -A])[[R.sub.1](u) - [R.sub.2](u)] = [absolute value of [zeta]] + [Q.sub.2], (45)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (46)

From u(0) = c, we obtain [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The profile of cusped soliton solution is shown in Figure 2(2-7).

Case 5 (3[A.sup.2] < c, A [less than or equal to] c). (1) When A = c, (14) can be cast into

[(u').sup.2] = 1/2(u - A)(u - [B.sub.1])(u - [B.sub.2]). (47)

Because 3[A.sup.2] < c, we have [B.sub.2] < A < [B.sub.1]. By the standard phase portrait analysis (see Figure 1(1-8)), we get [B.sub.2] [less than or equal to] u [less than or equal to] A = c < [B.sub.1]. By view of (47), we obtain

u' = ([square root of 2])/2 ([square root of (u - A)(u - [B.sub.1])(u - [B.sub.2])])sign([zeta]). (48)

As same as the above, we can get the implicit smooth soliton solution u([zeta]) defined by

2([square root of 2]) /([square root of [B.sub.1] - [B.sub.2]]) F(arcsin(([square root of u - [B.sub.2]/A - [B.sub.2]]), ([square root of A - [B.sub.2]/[B.sub.1] - [B.sub.2]])) = [absolute value of [zeta]] + [K.sup.6]. (49)

For u(0) = [B.sub.2], the constant [K.sub.6] is defined by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. For this smooth soliton solution, we can give an exact explicit form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (50)

The profile of smooth soliton solution is shown in Figure 2(2-8).

(2) When A < c, we discuss three cases: (i) [B.sub.1] > c; (ii) [B.sub.1] < c; (iii) [B.sub.1] = c.

(i) When [B.sub.1] > c, by the standard phase portrait analysis (see Figure 1(1-9)), we have u(0) = [B.sub.2] [less than or equal to] u < A or A [less than or equal to] u < u(0) = c.

For u(0) = [B.sub.2], from 14), we have

u' = (A - u)([square root of (u - [B.sub.1])(u - [B.sub.2])/2(u - c)])sign([zeta]). (51)

Taking the integration of (51) on the interval [[B.sub.2], A], thus, we obtain the implicit smooth soliton solution u([zeta]) defined by

2([square root of 2])/([square root of [B.sub.1] - [B.sub.2]])(c - A/A - [B.sub.2][G.sub.1](u) + [G.sub.2](u)) = [absolute value of [zeta]] + [W.sub.1], (52)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (53)

The constant [W.sub.1] is defined by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The profile of smooth soliton solution is shown in Figure 2(2-9).

For u(0) = c, by view of 14), we obtain

u' = (u - A)([square root of (u - [B.sub.1])(u - [B.sub.2])/2(u - c)])sign([zeta]). (54)

Taking the integration of both sides of (54), thus, we can yield the implicit solution u([zeta]) defined by

2([square root of 2])([B.sub.1] - c)/([B.sub.1] - A) ([square root of [B.sub.1] - [B.sub.2]])[[E.sub.1](u) - [E.sub.2](u)] = [absolute value of [zeta] + [W.sub.2], (55)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (56)

For u(0) = c, [W.sub.2] is defined by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The profile of cusped soliton solution is shown in Figure 2(2-10).

(ii) When [B.sub.1] < c, by the standard phase portrait analysis (see Figure 1(1-10)), we have [B.sub.2] < A < [B.sub.1] < c.

For u(0) = [B.sub.2], from 14), we get

u' = (A - u)([square root of (u - [B.sub.1])(u - [B.sub.2])/2(u - c)])sign([zeta]); (57)

we yield

[[THETA].sub.1](u) = [integral] [[theta].sub.1](u)du = [absolute value of [zeta]] + [M.sub.1], (58)

where

[[theta].sub.1](u) = -1/(u - A)([square root of 2(u - c)/(u - [B.sub.1])(u - [B.sub.2])]), (59)

and [M.sub.1] is an integration constant. Taking the integration of [[theta].sub.1](u) on the interval [[B.sub.2], A], thus, we obtain the implicit solution u([zeta]) defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (60)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (61)

The constant [M.sub.1] is defined by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Because [[theta].sub.1](u) > 0, we know that the [[THETA].sub.1](u) is strictly increasing on the interval [[B.sub.2], A];

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (62)

has the inverse denoted by u([zeta]) = [[THETA].sup.-1.sub.1]([absolute value of [zeta]]). The profile of smooth soliton solution is shown in Figure 2(2-11).

For u(0) = [B.sub.1], by view of 14), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (63)

where

[[theta].sub.2](u) = 1/(u - A) ([square root of 2(u - c)/(u - [B.sub.1])(u - [B.sub.2])]), (64)

and [M.sub.2] is an integration constant. Taking the integration of [[theta].sub.2] (u) on the interval [A, [B.sub.1]], thus, we obtain the implicit solution u([zeta]) defined by

[[THETA].sub.2](u)= -2([square root of 2])(c - [B.sub.1])/(A - [B.sub.1])([square root of c - [B.sub.2]]) N(u) = [absolute value of [zeta]] + [M.sub.2], (65)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (66)

When u(0) = [B.sub.1], the constant [M.sub.2] is defined by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. From [[theta].sub.2](u) > 0, we know that the [[THETA].sub.2](u) is strictly increasing on the interval [A, [B.sub.1]];

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (67)

has the inverse denoted by u([xi]) = [[THETA].sup.-1.sub.2] ([absolute value of [xi]]).

For this smooth soliton solution, we give an exact explicit form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (68)

The profile of smooth soliton solution is shown in Figure 2 (2-12).

The profile of soliton solution of (3) is shown in Figure 2 under special values of c and A.

(2-1) c = 1, A = 2; (2-2) c = 1/4, A = 1/2; (2-3) c = 1/10, A = 1/5; (2-4) c = 1/3, A = 1/2; (2-5) c = 3, A = 1; (2-6) c = 12, A = -2; (2-7) c = 3/16, A = -1/4; (2-8) c = A = 1/4; (2-9,10) c = 2/3, A = -1/3; (2-11,12) c = 4, A = -3/4.

5. Conclusion

In this paper, we research the soliton solutions of the generalized Camassa-Holm Equation (3) under inhomogeneous boundary condition. The parametric conditions and phase

portraits of existence of the cuspon and smooth soliton solutions are given. We obtain cuspon and smooth soliton solutions of the generalized Camassa-Holm Equation (3). Some exact explicit solutions are obtained. We show some graphs to explain our these solutions.

http://dx.doi.org/10.1155/2014/423063

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (11361017 and 11161013), the Natural Science Foundation of Guangxi (2012GXNSFAA053003), and the Innovation Project of GUET Graduate Education (XJYC2012021 and XJYC2012022).

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Dong Li, Yongan Xie, and Shengqiang Tang

School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin, Guangxi 541004, China

Correspondence should be addressed to Yongan Xie; xyan@guet.edu.cn

Received 21 November 2013; Revised 23 December 2013; Accepted 25 December 2013; Published 21 January 2014

Academic Editor: Weiguo Rui
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Author:Li, Dong; Xie, Yongan; Tang, Shengqiang
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Date:Jan 1, 2014
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