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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 518415, 10 pages
http://dx.doi.org/10.1155/2013/518415
Research Article

Peaked and Smooth Solitons for Equation

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

Received 6 August 2013; Revised 10 October 2013; Accepted 8 November 2013

Academic Editor: Alberto Cabada

Copyright © 2013 Yongan Xie et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper is contributed to explore all possible single peak solutions for the equation . Our procedure shows that the equation either has peakon, cuspon, and smooth soliton solutions when sitting on a nonzero constant pedestal or possesses compacton solutions only when . We present a new smooth soliton solution in an explicit form. Mathematical analysis and numeric graphs are provided for those soliton solutions of the equation.

1. Introduction

It is well known that the study of nonlinear wave equations and their solutions is of great importance in many areas of physics.

In 1993, Cooper et al. [1] considered the following generalized KdV equation (GKdV): where . These equations are derived from Lagrangian where is defined by . These equations have the same terms as the following equations, considered by Rosenau and Hyman [2]: but relative weights of the terms are quite different leading to the possibility that the integrability properties might be different. Cooper et al. [1] investigated Hamiltonian structure and integrability properties for this class of KdV equations. By using bifurcation theory of dynamical systems, when , Tang and Li [3] investigated the bifurcation behavior for traveling wave solutions of (1). In [4], by using sine-cosine method and extended tanh method, some new solitary patterns solutions and compactons solutions are formally derived. In [5], by using analytic methods from the dynamical systems theory, some new exact explicit parametric representations of breaking loop-solutions under some fixed parameter conditions are formally derived.

In the development of soliton theory, there exist many different approaches to searching for exact solutions of nonlinear partial differential equations, such as mapping method [6], fan-expansion method [7], and -expansion method [8]. In particular, it is very interesting to investigate the traveling wave solutions on a constant pedestal. Qiao and Zhang [9] discussed the traveling wave solutions for the Camassa-Holm equation on the nonzero constant pedestal and found new soliton solutions, which are smooth and cusped. Later, Zhang and Qiao [10] investigated the Degasperis-Procesi equation under the boundary condition and obtained all possible single peak soliton solutions of the Degasperis-Procesi equation. Recently, Chen and Li [11] studied osmosis equation with a nonzero constant pedestal and obtained smooth, peaked, and cusped soliton solutions of the osmosis equation. More recently, Zhang et al. [12] studied the equation under an inhomogeneous boundary condition and obtained compacton solutions, loop soliton solutions, cusped soliton solutions, and smooth soliton solutions.

In the literature [1, 35], the authors did not investigate the existence of peakon soliton for the equation. We are thus interested in an important question that should be investigated: does the equation have peakon soliton? We hope to answer this problem in this paper. Assume that , . Then, (1) becomes (simply called equation) We will study single peak solitary solutions of equation under inhomogeneous boundary condition Peakon, compacton, cuspon, and smooth soliton solutions are obtained. Our method is based on phase portrait analysis technique under an inhomogeneous boundary condition.

2. Asymptotic Behavior of Solutions

In this section, we first introduce some notations. Let denote the set of all times continuously differential functions on the open set . refers to the set of all functions whose restriction on any compact subset is integrable. stands for . Assume that , , are the Jacobian elliptic functions with the modulus is the first kind of complete elliptic integral, is the normal elliptic integral of the 2nd kind, and is the normal elliptic integral of the 3rd kind [13].

Let us consider the traveling wave solution of equation (4) through a generic setting , , where is wave speed. Substituting it into (4) yields where “′” is the derivative with respect to . Taking the integration twice on both sides leads to where are two integration constants. Let us solve (7) with the boundary condition (5). From (7) we obtain that If , then (8) reduces to where Obviously, . We assume that throughout the paper since there are similar results for the case .

Definition 1. A function is said to be a single peak soliton solution for equation (4) if satisfies the following conditions:(A1) is continuous on and has a unique peak point , where attains its global maximum or minimum value;(A2) satisfies (8) on ;(A3).

Definition 2. A wave function is called a peakon if is smooth locally on either side of and , , .

Definition 3. A wave function is called a cuspon if is smooth locally on either side of and .
Without loss of generality, one assumes that .

Lemma 4. Equation (4) has trivial solution , if one of the following three conditions holds:(i), ;(ii), ;(iii), .

Proof. (i) If , , then , , , and The fact that implies that and .
If , , then , , and . The fact that implies that and .
(ii) If , , then , , , and The fact that implies that and .
If , , then , , , and The fact that implies that and .
(iii) If , , then , , , and The fact that implies and .

Theorem 5. Suppose that is a single peak solitary wave solution for (4) at the peak point . If , or , or , , then or or .

Proof. If , then for any since . Differentiating both sides of (8) yields .(i)If , or , or , , from Lemma 4, we know that (4) has trivial solution .(ii)For , , we have , if , then, according to the definition of peak point, we have ; thus, or .(iii)For , or , or , , if , then . According to the definition of peak point, we have . Thus, or since contradicts the fact that 0 is the unique peak point.

Theorem 6. Suppose that is a single peak solitary wave solution for (4) at the peak point . When , or , or , , then we have the following classification and asymptotic behavior of solutions.(i)If and or and , then is a smooth solitary wave solution.(ii)If ,, , then is a compacton solution [2].(iii)If and , then gives the peakon solution (iv)If and , and , or or , then is a cuspon soliton solution and or

Proof. (i) From the process of proving Theorem 5, we know that if or , then ; thus, is a smooth solitary wave solution.
(ii) If , then , , (9) becomes Integrating both sides of (18) on the interval leads to a compacton solution with compact support The profile of compacton is shown in Figure 2(b).
Remark 1. To the best of our knowledge, the solution (19) of (4) has not been reported in the literature.
(iii) If and , then (9) becomes Integrating both sides of (20) on the interval leads to a peakon solution with the following properties: The profile of peakon solution is shown in Figure 2(e).
Remark 2. To the best of our knowledge, the solution (21) of (4) has not been reported in the literature.
(iv) If , , and , then does not contain the factor . From (9) we obtain Let , then and Inserting into (24) and using the initial condition , we obtain Thus, which implies that . Therefore, we have Thus, .
Let ; then and Inserting into (29) and using the initial condition , we obtain Thus, which implies that . Therefore, we have Thus, .

3. Smooth, Peaked, and Cusped Single Peak Solitary Wave Solutions

Theorem 6 gives a classification for all single peak solitary wave solutions for (4). In this section, we will present all possible soliton solutions for (4). We shall discuss the four cases: Case 1: , ; Case 2: , ; Case 3: , or , .

Case 1 (, , ). In this case, according to Theorem 5 and standard phase portrait analytical technique (see Figure 1(a)), we have and Integrating both sides of (33) on the interval leads to
Thus we obtain the implicit solution defined by where  , , , , , .
In view of , we know that is strictly decreasing on with , which gives a unique cuspon soliton solution satisfying , , , . The profile of cuspon soliton solution is shown in Figure 2(a).

fig1
Figure 1: The phase portraits of (8) on the plane.
fig2
Figure 2: The profiles of waves for . (a) , , (b) , , (c) , , , (d) , , , (e) , , , (f) , , , (g) , , , (h) , , .

Case 2 (, , ). In this case, by the standard phase portrait analysis (see Figure 1(b)), we have and then obtain a compacton solution (see (19)). The profile of compacton is shown in Figure 2(b).

Case 3 (, or , ). By virtue of Theorem 5, any single peak soliton solution for (4) must satisfy the following initial and boundary values problem: Equation (36) implies (i)When , from (37) we obtain (ii)When , from (37) we obtain Since , , introducing the constant yields which implies that for and for
From the standard phase portrait analysis (see Figures 1(c)1(h)), we know that if is a single peak soliton solution of (4), then In the following part, let us separate seven cases to discuss.
(1) . In this case, by the standard phase portrait analysis (see Figure 1(e)), we have (i). If , then . In this case, by the standard phase portrait analysis (see Figure 1(e)), we obtain a peakon solution (see (21)). The profile of peakon solution is shown in Figure 2(e).(ii). In this case there is no single peak solitary wave solution for the boundary condition .
(2) . In this case, by the standard phase portrait analysis (see Figure 1(f)), we have (i). If , then . Integrating both sides of (47) on the interval leads to where .From , we know that is strictly decreasing on the interval , has the inverse denoted by Corresponding to the homoclinic orbit to the saddle point ) shown in Figure 1(f), gives a smooth soliton solution satisfying The profile of smooth soliton solution is shown in Figure 2(f).
Remark 3.  To the best of our knowledge, the solution (48) of (4) has not been reported in the literature.(ii) or . In this case there is no single peak solitary wave solution for the boundary condition .
(3) . In this case, by the standard phase portrait analysis (see Figure 1(d)), we have (i). If , then . Integrating both sides of (44) on the interval leads to where , .From , we know that is strictly decreasing on , has the inverse denoted by , and gives a cuspon soliton solution satisfying The profile of cuspon soliton solution is shown in Figure 2(d).(ii) or . In this case there is no single peak solitary wave solution for the boundary condition .
(4) . In this case, we have , by virtue of Lemma 4(iii), (4) has trivial solution .
(5) . In this case, by the standard phase portrait analysis (see Figure 1(c)), we have (i). If , then . Integrating both sides of (44) on the interval leads to From , we know that is strictly decreasing on , has the inverse denoted by , and gives a cuspon soliton solution satisfying The profile of cuspon soliton solution is shown in Figure 2(c).(ii). In this case, there is no single peak solitary wave solution for the boundary condition .
(6) . In this case, by the standard phase portrait analysis (see Figure 1(g)), we have (i). If , then . Integrating both sides of (44) on the interval leads to In view of , we know that is strictly decreasing on   and has the inverse which is denoted by , where gives a cuspon soliton solution satisfying The profile of cuspon soliton solution is shown in Figure 2(g).(ii). In this case there is no single peak solitary wave solution for the boundary condition .
(7) . In this case, by the standard phase portrait analysis (see Figure 1(h)), we have (i). If , then . This case is completely similar to the case of .(ii). If , then . Integrating both sides of (44) on the interval leads to where , .In view of , we know that is strictly increasing on and has the inverse which is denoted by , where gives a smooth soliton solution satisfying The profile of smooth soliton solution is shown in Figure 2(h).(iii). In this case there is no single peak solitary wave solution for the boundary condition .

Let us summarize our results in the following theorem.

Theorem 7. Suppose that is a single peak soliton for (4) at the peak point , which satisfies the boundary condition (5). Then we have the following conclusions.
(i)   , . If , (4) has cuspon soliton solution which can be expressed as where . It satisfies
(ii) , . If , the only possible quasi single peak soliton is compacton: with the following properties:
(iii)   , , or .(1). If , then , and (4) has peakon soliton solution which can be expressed as with the following properties: (2). If , (4) has smooth soliton solution which can be expressed as with the following properties: (3). If , (4) has cuspon soliton solution which can be expressed as with the following properties: (4). If , (4) has cuspon soliton solution which can be expressed as with the following properties: (5). If , (4) has cuspon soliton solution which can be expressed as with the following properties: (6). (i). If , then . This case is completely similar to the case of .(ii). If , (4) has cuspon soliton solution which can be expressed as with the following properties:

4. Conclusion

In this paper, we study the single peak solitary wave solutions of equation under inhomogeneous boundary condition. The conditions of the existence of peakon, compacton, cuspon, and smooth soliton solutions are given by using phase portrait analytical technique. We have obtained all peakon, compacton, cuspon, and smooth soliton solutions of equation and analyzed their analytic and dynamical behavior. We have gotten a new type of smooth soliton, which is expressed in terms of trigonometric functions (see (48)), for equation. New peaked solitons and new type of smooth soliton solutions are expected to apply in nonlinear shallow-water wave theory and Newton motion theory because they have a very close relation. Actually, the ODE (8) has a physical meaning and coincides with the Newton equation of a particle in the potential We solve the Newton equation , for all possible single peak solitary wave solutions, where .

Conflict of Interests

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

Acknowledgment

This work is supported by the National Natural Science Foundation of China (11361017, 11161013), the Natural Science Foundation of Guangxi (2012GXNSFAA053003, 2013GXNSFAA019010), and the Innovation Project of GUET Graduate Education (XJYC2012021, XJYC2012022).

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