Abstract
The generalized HD type equation is studied by using the bifurcation method of dynamical systems. From a dynamic point of view, the existence of different kinds of traveling waves which include periodic loop soliton, periodic cusp wave, smooth periodic wave, loop soliton, cuspon, smooth solitary wave, and kink-like wave is proved and the sufficient conditions to guarantee the existence of the above solutions in different regions of the parametric space are given. Also, all possible exact parametric representations of the bounded waves are presented and their relations are stated.
1. Introduction
The search for the exact solutions of nonlinear partial differential equations has been one of the most important concerns of mathematicians throughout the world for a long time. Because for understanding the nonlinear phenomena, which are usually described by partial differential equations, the study of the exact solutions is essential. Many powerful methods have been presented for finding the traveling wave solutions of nonlinear partial differential equations, such as the Bäcklund transformation [1], Darboux transformation [2], inverse scattering method [3], Hirota bilinear method [4], Lie group analysis method [5], transformed rational function method [6], extended transformed rational function method [7], exp-function method [8], multiple exp-function algorithm [9], and symbolic computation method [10].
The Harry Dym (HD) equation [11] is one of the most interesting and exotic nonlinear integrable equations, which was first discovered in an unpublished paper by Harry Dym in 1973. Many fields related to the HD equation are studied. Leo et al. [12] considered the Lie-Bäcklund symmetries for the HD equation. Fuchssteiner et al. [13] show that the HD equation is completely integrable. Direct links were found between either the KdV and the HD equations [14, 15] or the mKdV and the HD equations [16]. Different kinds of solutions to the HD equation have been derived by some authors [17–19]. Moreover, the -dimensional HD equation [20, 21], coupled HD hierarchy [22, 23], and generalized HD equation [24, 25] were studied.
In 2015, Geng et al. [26] first propose a hierarchy of a generalized HD type (gHD type) equation: where is a parameter. Obviously, (2) is reduced to the HD equation (1) with . Furthermore, under the transformation , (2) can be written as Geng et al. [26] also stated that (2) and (3) are almost equivalent except for that an unbounded solution might be bounded for the other, and for , they obtained some exact solutions. To review conveniently, we list them as follows.
Implicit soliton for : Peakon for : (Anti-)kink soliton for : Cuspon for : Periodic solution for : where and are the elliptic integrals of the second and first kinds, respectively, with the modulus To be specific, Periodic cuspon for : Explicit soliton for : Rational solution for : Periodic solution for :
In this paper, by employing the bifurcation method of dynamical systems [27–34], we will obtain some new bounded traveling wave solutions for (3). These solutions contain periodic loop soliton solutions, periodic cusp wave solutions, smooth periodic wave solutions, loop soliton solutions, cuspon solutions, smooth solitary wave solutions, and kink-like wave solutions. Under some parameter conditions, we also show the convergence of certain solutions. For example, when the parameter value varies, the periodic loop soliton solutions converge to the loop soliton solutions.
2. Preliminaries
Using a transformationwhere is a constant parameter, (3) can be rewritten as where “” is the derivative with respect to Multiplying both sides of (15) by and integrating with respect to and setting the integral constant as , we obtain
Letting , we can obtain from (16) that
From (17), we get the following planar dynamical system:
Using , it carries (18) into the Hamiltonian system with the following first integral:
For a fixed , the level curve defined by (20) determines a set of invariant curves of system (19) which contains different branches of curves. As is varied, it defines different families of orbits of system (19) with different dynamical behaviors.
Write that Clearly, when and , system (19) has three equilibrium points at , , and in -axis, where When and , system (19) has two equilibrium points at and in -axis. When or , system (19) has only one equilibrium point at in -axis. There is not any equilibrium point of system (19) in line .
Let be the coefficient matrix of the linearized system of (19) at equilibrium point , and then we have , , , and .
For an equilibrium point of system (19), we know that is a saddle point if , a center point if , and a cusp if and the Poincaré index of is zero.
Notice that for , we have and , respectively. And implies that , , and Therefore, we can obtain four bifurcation curves of (19) as follows: which divide the -parameter plane into 18 subregions.
Since both (18) and (19) have the same first integral (20), then two systems above have the same topological phase portraits. Therefore we can obtain the phase portraits of (18) from that of (19). We show that bifurcation sets and phase portraits of (18) are drawn in Figure 1.
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The remainder of this paper is organized as follows. In Section 3, we state our main results for (3). In Section 4, we give the derivations for our main results. Some remarks and a short conclusion will be given in Section 5.
3. Main Results
Let and then our main results are stated in the following seven propositions.
Proposition 1. When , (3) has a periodic loop soliton solution as follows: where is the Jacobian elliptic function with the modulus reads amplitude [35], are three real roots of , and is a new parametric variable.
When , (3) has a periodic loop soliton solution as follows: where
When , (3) has a periodic loop soliton solution as follows: where are three real roots of .
Example 2. Taking , we get the approximations of in formula (24), where , , and . The profile of (24) is shown in Figure 2.
Proposition 3. When , (3) has a periodic cusp wave solution as follows: where , , and
When , (3) has a periodic cusp wave solution as follows: where , , and .
Example 4. Taking , , we get the approximations of in formula (26), where , , , and The profile of (26) is shown in Figure 3.
Proposition 5. When , , , or , , , (3) has a smooth periodic wave solution as follows: where and are given in (23). are three real roots of .
When , , , , or , , , (3) has a smooth periodic wave solution as follows: where , , are three real roots of
Example 6. Taking , we get the approximations of in formula (29), where , , , , and The profile of (29) is shown in Figure 4.
Proposition 7. When , (3) has a loop soliton solution as follows: where , ,
When , (3) has a loop soliton solution as follows:where
Example 8. Taking , we get the approximations of in formula (31), where , The profile of (31) is shown in Figure 5.
Proposition 9. When , (3) has a cuspon solution as follows: where
When , (3) has a cuspon solution as follows: where
Example 10. Taking , we get the approximations of in formula (33), where The profile of (33) is shown in Figure 6.
Proposition 11. When , or , or , , (3) has a smooth solitary wave solution as follows: where and and are given in (30).
When , (3) has a smooth solitary wave solution as follows: where , ,
When , , (3) has a smooth solitary wave solution as follows: where and are given in (31).
Example 12. Taking , , we get the approximations of in formula (35), where , , , The profile of (35) is shown in Figure 7.
Proposition 13. When , (3) has two kink-like wave solutions of implicit expression as follows: where , and are given in (35), and .
When , (3) has two kink-like wave solutions of implicit expression as follows: where , ,
When , , (3) has two kink-like wave solutions of implicit expression as follows: where , , and is given in (31).
When , (3) has two kink-like wave solutions of implicit expression as follows: where , ,
When , (3) has two kink-like wave solutions of implicit expression as follows: where , , and are given in (30), and
When , (3) has two kink-like wave solutions of implicit expression as follows: where , ,
Example 14. Taking , we get the approximations of in formula (37), where , , , , The profiles of (37) are shown in Figures 8(a) and 8(b), respectively.
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4. The Derivations to Main Results
4.1. The Derivations for Proposition 1
For given in Figures 1(e), 1(f), and 1(g), respectively, the level curves are shown in Figures 9(a), 9(b), and 9(c), respectively. From Figures 9(a), 9(b), and 9(c), we see that there are two open curves of system (18) defined by passing points and , respectively, and approaching the line when , and , where are three real roots of Their expressions are Substituting (43) into and integrating it along the open curves yield equation Completing (44) and using transformations and , we can get a periodic loop soliton solution of (3) same as (23).
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For given in Figure 1(h), the level curve is shown in Figure 9(d). From Figure 9(d), we see that there are two open curves of system (18) defined by passing points and , respectively, and approaching the line when , and , where Their expressions are Substituting (45) into and integrating it along the open curves yield equation Completing (46) and using transformations and , we can get a periodic loop soliton solution of (3) same as (24).
For given in Figures 1(i), 1(p), 1(q), and 1(r), respectively, the level curves are shown in Figures 9(e), 9(f), 9(g), and 9(h), respectively. From Figures 9(e), 9(f), 9(g), and 9(h), we see that there are two open curves of system (18) defined by passing points and , respectively, and approaching the line when , and , where are three real roots of Their expressions are Substituting (47) into and integrating it along the open curves yield equation Completing (48) and using transformations and , we can get a periodic loop soliton solution of (3) same as (25).
4.2. The Derivations for Proposition 3
For given in Figure 1(d), the level curve is shown in Figure 9(i). From Figure 9(i), we see that there is one open curve of system (18) defined by passing point and approaching the line when , and , where Its expression is Substituting (49) into and integrating it along the open curve yield equation Completing (50) and using transformations and , we can get a periodic cusp wave solution of (3) same as (26).
For given in Figure 1(o), the level curve is shown in Figure 9(j). From Figure 9(j), we see that there is one open curve of system (18) defined by passing point and approaching the line when , and , where Its expression isSubstituting (51) into and integrating it along the open curve yield equation Completing (52) and using transformations and , we can get a periodic cusp wave solution of (3) same as (27).
4.3. The Derivations for Proposition 5
For given in Figure 1(c), in Figure 1(e), and in Figure 1(g), respectively, the level curves are shown in Figures 9(k), 9(l), and 9(m), respectively. From Figures 9(k), 9(l), and 9(m), we see that there is one periodic orbit of system (18) defined by passing points and when , , , , or , , , where are three real roots of Its expression is Substituting (53) into and integrating it along the periodic orbit yield equation Completing (54) and using transformations and , we can get a smooth periodic wave solution of (3) same as (28).
For given in Figure 1(l), in Figure 1(n), and in Figure 1(p), respectively, the level curves are shown in Figures 9(n), 9(o), and 9(p), respectively. From Figures 9(n), 9(o), and 9(p), we see that there is one periodic orbit of system (18) defined by passing points and when , , , , or , , , where are three real roots of Its expression is Substituting (55) into and integrating it along the periodic orbit yield equation