Exact Traveling Wave Solutions of Explicit Type , Implicit Type , and Parametric Type for K m , n Equation

By using the integral bifurcation method, we study the nonlinearK m,n equation for all possible values of m and n. Some new exact traveling wave solutions of explicit type, implicit type, and parametric type are obtained. These exact solutions include peculiar compacton solutions, singular periodic wave solutions, compacton-like periodic wave solutions, periodic blowup solutions, smooth soliton solutions, and kink and antikink wave solutions. The great parts of them are different from the results in existing references. In order to show their dynamic profiles intuitively, the solutions of K n, n , K 2n − 1, n , K 3n − 2, n , K 4n − 3, n , and K m, 1 equations are chosen to illustrate with the concrete features.


Introduction
In this paper, we will investigate some new traveling-wave phenomena of the following nonlinear dispersive K m, n equation 1 : where m and n are integers and σ is a real parameter.This is a family of fully KdV equations.When σ 1, 1.1 as a role of nonlinear dispersion in the formation of patterns in liquid drops was studied by Rosenau and Hyman 1 .In 2-6 , the studies show that the model

The Equivalent Two-Dimensional Planar System to 1.1 and Its First Integral Equations
We make a transformation u t, x φ ξ with ξ x − vt, where the v is a nonzero constant as wave velocity.Thus, 1.1 can be reduced to the following ODE: Integrating 2.1 once and setting the integral constant as zero yields Let φ dφ/dξ y.Equation 2.2 can be reduced to a 2D planar system: where φ / 0. Obviously, the solutions of 2.2 include the solutions of 2.3 and constant solution φ 0. We notice that the second equation in 2.3 is not continuous when φ 0; that is, the function φ ξ is not defined by the singular line φ 0. Therefore, we make the following transformation: where τ is a free parameter.Under the transformation 2.4 , 2.3 , and φ 0 combine to make one 2D system as follows: Clearly, 2.5 is equivalent to 2.2 .It is easy to know that 2.3 and 2.5 have the same first integral as follows: where h is an integral constant.From 2.6 , we define a function as follows: It is easy to verify that 2.5 satisfies dφ dτ Therefore, 2.5 is a Hamiltonian system and 1/2φ n−1 is an integral factor.In fact, 2.7 can be rewritten as the form H E T , where E 1/2 My 2 1/2 M φ 2 and T 2σ/ n m φ m n − 2v/ n 1 φ n 1 with M 2nφ 2n−2 .E denotes kinetic energy, and T denotes potential energy.Especially, when n 1, M becomes a constant 2. In this case, the kinetic energy E only depends on movement velocity φ of particle; it does not depend on potential function φ.So, according to Theorem 3.2 in 37 , it is easy to know that 2.5 is a stable and nonsingular system when n 1; in this case its solutions have not singular characters.When n > 1, 2.5 becomes a singular system; in this case some solutions of 2.5 have singular characters.
For the equilibrium points of the system 2.5 , we have the following conclusion.

2.10
Obviously, if n is odd, then Then h O H 0, 0 0 whether m is odd number or even number.

Exact Solutions of Explicit Type, Implicit Type, and
Parametric Type and Their Properties

Exact Solutions and Their Properties of 1.1 under h h O
Taking h h O 0, 2.6 can be reduced to Substituting 3.2 into the first expression in 2.5 yields Noticing that equation 2nv/ n 1 φ n−1 − σ φ n−1 2 0 has two roots φ 0 and φ 2nv/ n 1 σ 1/ n−1 , we take 2nv/ n 1 σ 1/ n−1 , 0 as the initial value.Using this initial value, integrating 3.2 yields After completing the aforementioned integral, we solve this equation; thus we obtain

3.6
Thus, we respectively obtain a periodic wave solution and solitary wave solution of parametric type for the equation K n, n as follows:

3.8
On the other hand, 3.1 can be rewritten as 3.9 Using 2nv/ n 1 σ 1/ n−1 , 0 as the initial value, substituting 3.9 into the first expression in 2.3 directly, we obtain an integral equation as follows: Completing the aforementioned integral equation, then solving it, we obtain a periodic solution and a hyperbolic function solution as follows: Obviously, the solution 3.7 is equal to the solution 3.11 ; also the solution 3.8 is equal to the solution 3.12 .Similarly, taking the 0, 0 as initial value, substituting 3.9 into the first expression in 2.3 , then integrating them, we obtain another periodic solution and another hyperbolic function solution of K n, n equation as follows.
In fact, the solutions 3.11 and 3.13 have been appeared in 35 , so we do not list similar solutions anymore at here.Next, we discuss a interesting problem as follows.
When σ > 0, from 3.11 and 3.13 , we can construct two compacton solutions as follows:

3.16
The shape of compacton solutions 3.15 and 3.16 changes gradually as the value of parameter n increases.For example, when n 2, 15, 400, respectively, the shapes of compacton solution 3.15 are shown in Figure 1.
ii When n 1, m > 1, 3.1 can be directly reduced to 3.17 Equation 3.17 is a nonsingular equation.Using 2σ/ m 1 v n−1 , 0 as initial value and then substituting 3.17 into the first expression in 2.3 directly, we obtain a smooth solitary wave solution and a periodic wave solution of K m, 1 equation as follows: Also, the shape of solitary wave solution 3.18 changes gradually as the value of parameter m increases.When m 2, 20, 200, respectively, its shapes of compacton solution 3.18 are shown in Figure 2. iii When n is even number and m 2n − 1, 3.1 can be reduced to

3.20
It is easy to know that 2nv/ n 1 φ n−1 − 2nσ/ 3n − 1 φ 3 n−1 0 has three roots φ 0 and In fact, γ −α.Using these three roots as initial value, respectively, then substituting 3.20 into the first expression in 2.3 , we obtain three integral equations as follows:

3.21
Completing the previous three integral equations, then solving them, we obtain three periodic solutions of Jacobian elliptic function for K 2n − 1, n equation as follows: , n even number, 3.22 , n even number, 3.23 , n even number.

3.24
The solutions 3.22 and 3.24 show two shapes of periodic wave with blowup form, which are shown in Figures 3 a and 3 c .The solution 3.23 shows a shape of periodic cusp wave, which is shown in Figure 3 b .iv When m 3n − 2, n > 1, 3.1 can be directly reduced to

3.25
It is easy to know that the function 2nσ/ 4n Using a 1/ n−1 , 0 and 0, 0 as initial values, respectively, substituting 3.25 into the first expression in 2.3 , we obtain four elliptic integral equations as follows.

3.29
Corresponding to 3.26 , 3.27 , 3.28 , and 3.29 , respectively, we obtain four periodic solutions of elliptic function type for K 3n − 2, n equation as follows: a, and g 1/ √ AB 4 √ 27/3a with a 3 4n − 2 v / n 1 σ given previously.The solution 3.30 shows a shape of periodic wave with blowup form, which is shown in Figure 4 a .The solution 3.31 shows s shape of compacton-like periodic wave, which is shown in Figure 4 b .The profile of solution 3.32 is similar to that of solution 3.30 .Also the profile of solution 3.33 is similar to that of solution 3.31 .So we omit the graphs of their profiles here.

Journal of Applied Mathematics
1 can be directly reduced to

3.34
Suppose that φ 0 φ 0 is one of roots for equation 2nv/ n 1 φ n−1 − 2nσ/ k n − 1 2 φ k n−1 0. Clearly, the 0 is its one root.Anyone solution of K k − 1 n − k 2, n equation can be obtained theoretically from the following integral equations: The left integral of 3.35 is called hyperelliptic integral for φ n−1 when the degree k is greater than four.Let φ n−1 z.Thus, 3.35 can be reduced to

3.36
In fact, we cannot obtain exact solutions by 3.36 when the degree k is grater than five.But we can obtain exact solutions by 3.36 when k 5, v −σ n 1 /k n − 1 2, and σ < 0. Under these particular conditions, taking φ 0 z 0 1/ n−1 0 as initial value, 3.36 becomes 1/ ρ − 2 and 0 < Z ≤ 1.Thus, 3.37 can be transformed to

3.38
Completing 3.38 and refunded the variable z φ n−1 , we obtain two implicit solutions of elliptic function type for K 4n − 3, n equation as follows: where The solutions also can be rewritten as where the function F ϕ, k Elliptic F ϕ, k is the incomplete Elliptic integral of the first kind.The two solutions in 3.40 are asymptotically stable.Under The graphs of their profiles are shown in Figure 5.

Exact Solutions and Their Properties of 1.1 under h / 0
In this subsection, under the conditions h h A 0 , and h h A 1 , h h A 2 , we will investigate exact solutions of 1.1 and discuss their properties.When h / 0, 2.6 can be reduced to

3.41
Substituting 3.41 into the first expression of 2.3 yields where φ * is one of roots for equation h 2nv/ n 1 φ n 1 − 2nσ/ n m φ n m 0. However we cannot obtain any exact solutions by 3.42 when the degrees m and n are more great, because we cannot obtain coincidence relationship among different degrees n, n 1 and n m.But, we can always obtain some exact solutions when the degree m n is not greater than four.For example, by using 3.42 directly, we can also obtain many exact solutions of K 2, 1 and K 3, 1 equations; see the next computation and discussion.
i If m n 2, then 3.41 can be reduced to

3.43
Taking  Then − v 4 /6σ 3 4v/3 φ 3 − σφ 4 0 has four roots, two real roots, and two complex roots as follows: where is an elliptic integral of the third kind, and the function f 1 satisfies the following three cases, respectively: In the previous three cases, k 2 1 − k 2 .Thus, by using 3.47 and 3.48 , we obtain a parametric solution of Jacobian elliptic function for K 2, 2 equation as follows:

3.50
Solving the aforementioned integral equation yields

3.52
where is an elliptic integral of the third kind, and the function f 1 satisfies the following three cases, respectively: In the previous three cases, k 2 1 − k 2 .Thus, by using 3.51 and 3.52 , we obtain another parametric solution of Jacobian elliptic function for K 2, 2 equation as follows:

3.53
In addition, when h < −v 4 /6σ 3 , h 4v/3 φ 3 − σφ 4 0 has four complex roots; in this case, we cannot obtain any useful results for K 2, 2 equation.When h > −v 4 /6σ 3 , the case is very similar to 3.52 ; that is, the equation h 4v/3 φ 3 − σφ 4 0 has two real roots and two complex roots.So we omit the discussions for these parts of results.
In order to describe the dynamic properties of the traveling wave solutions 3.49 and 3.53 intuitively, as an example, we draw profile figure of solution 3.53 by using the software Maple, when v 4, and σ −2, see Figure 6 a .
Figure 6 a shows a shape of peculiar compacton wave; its independent variable ξ is bounded region i.e., |ξ| < α 1 1 ; see Figure 6 b .From Figure 6 a , we find that its shape is very similar to that of the solitary wave, but it is not solitary wave because when |ξ| ≥ α 1 1, u ≡ 0. So, this is a new compacton.
ii Under m 2, n 1, taking h h A 0 | m 2,n 1 −v 3 /3σ 2 as Hamiltonian quantity, 3.42 can be reduced to where φ * is one of roots for the equation − v 3 /3σ 2 vφ 2 − 2σ/3 φ 3 0. Clearly, this equation has three real roots, one single root −v/2σ and two double roots v/σ, v/σ.If σ > 0, then the In these two conditions, taking φ * − v/2σ as initial value and completing the 3.54 , we obtain a periodic solution and a solitary wave solutions for K 2, 1 as follows:

3.58
Integrating the 3.58 , then solving them, respectively, we obtain three periodic solutions of elliptic function type for K 2, 1 as follows: where as Hamiltonian quantity, 3.42 can be reduced to Clearly, v/σ − φ 2 0 has two real roots v/σ and − v/σ.Taking φ * v/σ − v/σ /2 0 as initial value, solving 3.62 , we obtain a kink wave solution and an antikink wave solution for K 3, 1 as follows: where v < 0 shows that the waves defined by 3.63 are reverse traveling waves.

3.66
where r 1 > r 2 > r 3 > r 4 .Solving the integral equations 3.66 , we obtain two periodic solutions of Jacobian elliptic function for K 3, 1 equation as follows: where The case for taking φ * r 2 , r 3 , r 4 as initial values can be similarly discussed; here we omit these discussions because these results are very similar to the solutions 3.67 and 3.68 .

3.69
Solving the aforementioned two integral equations, we obtain two periodic solutions of Jacobian elliptic function for K 3, 1 equation as follows: Among these aforementioned solutions, 3.59 shows a shape of solitary wave for given parameters v 4, and σ 1 which is shown in Figure 7 a .Equation 3.60 shows a shape of smooth periodic wave for given parameters v 2, σ 1, and h 4 which is shown in Figure 7 b .Also 3.61 shows a shape of smooth periodic wave for given parameters

Conclusion
In this work, by using the integral bifurcation method, we study the nonlinear K m, n equation for all possible values of m and n.Some travelling wave solutions such as normal compactons, peculiar compacton, smooth solitary waves, smooth periodic waves, periodic blowup waves, singular periodic waves, compacton-like periodic waves, asymptotically stable waves, and kink and antikink waves are obtained.In order to show their dynamic properties intuitively, the solutions of K n, n , K 2n − 1, n , K 3n − 2, n , K 4n − 3, n , and K m, 1 equations are chosen to illustrate with the concrete features; using software Maple, we display their profiles by graphs; see Figures 1-7.These phenomena of traveling waves are different from those in existing literatures and they are very interesting.Although we do not know how they are relevant to the real physical or engineering problem for the moment, these interesting phenomena will attract us to study them further in the future works.

Figure 1 :
Figure 1: The solution u in 3.15 shows a shape of compacton for parameters v 2, and σ 1.

Figure 2 :
Figure 2: The solution u in 3.18 shows a shape of compacton for parameters v 2, and σ 1.

n 9 , 1 Figure 4 :
Figure 4: Two different periodic waves on solutions 3.30 and 3.31 for given parameters.
− b 2 /AB, and A and B are given in case 1 .Substituting 3.51 and n 2 into 2.4 yields