Limiting Behavior of Travelling Waves for the Modified Degasperis-Procesi Equation

Using an improved qualitative method which combines characteristics of several methods, we classify all travelling wave solutions of the modified Degasperis-Procesi equation in specified regions of the parametric space. Besides some popular exotic solutions including peaked waves, and looped and cusped waves, this equation also admits some very particular waves, such as fractal-like waves, double stumpons, double kinked waves, and butterfly-like waves. The last three types of solutions have not been reported in the literature. Furthermore, we give the limiting behavior of all periodic solutions as the parameters trend to some special values.

The DP equation is of interests because of the following two aspects.On one side, (1) is integrable [1].On another side, the DP equation presents abundant nonlinear phenomena due to the coexistence of nonlinear convections and nonlinear dispersions.Equation (1) admits wave-breaking phenomena and existence of exotic solution including peakons and cuspons [11][12][13][14][15].
To further complement the study of the DP equation, Wazwaz gave and studied the modified Degasperis-Procesi equation (mDP) [16]: It is clear that the nonlinear convection term   has been changed to  2   in (2).Wazwaz employed these modified forms as a vehicle to explore the change in the physical structure of the solution.Many researchers have obtained abundant travelling wave solutions by different methods.Ma et al. [17] applied the auxiliary equation method to obtain some new solitary and traveling wave solutions.Rui et al. [18] obtained abundant traveling wave solutions by the integral bifurcation method.In [19], a new characteristic of solitary wave solutions, bell-shaped solitary wave, and peakon coexisting for the same wave speed in mDP equation was found by Liu and Ouyang.
It is noted that the nonsmooth wave solutions of the mDP equation obtained in the previous studies have not been checked in a weak solution way.And new solutions of the mDP equation have been founded.Motivated by the above two aspects, we try to answer them.It is noticeable that our method to search solutions of the mDP equation is combined with some characteristics of several methods [10][11][12][13][14][15].Three characters lie in our method.(i) Definition of weak solutions of the mDP equation is given.(ii) More new exotic solutions are obtained.The parameter space is divided in further detail.(iii) The limiting behavior of traveling wave solutions is given.
This paper is organized as follows.In Section 2, we give the definition of weak solutions.In Section 3, we give 2 Advances in Mathematical Physics theorems of the classification of travelling waves in the mDP equation.In Section 4, we give the proof.Section 5 is the conclusion.

Definitions and Notations
In this section, we will give the classification of travelling wave solutions of (2), which is stated in Theorem 3.
For a travelling wave (, ) = (−), (2) takes the form where  is the wave speed.By integrating with respect to  and letting the integral constant be zero, (3) becomes Equation ( 4) makes sense for all  ∈  1 loc (R).The following definition is therefore natural.
Like the proof of the proceeding of Lemmas 4 and 5 in [15], we can give the following definition of weak traveling wave solutions. where and  → , at any finite endpoint of   .(ii) If  has strictly positive Lebesgue measure () > 0, we have  = −(1/3) 2 −  0 .

Main Results
Let ℎ  , ℎ  , and ℎ  be defined as in (11).All travelling wave solutions ( − ) of (1) are smooth except at points where  = .We state our main result as follows.
(2) For ℎ ∈ (ℎ  , ℎ  ], there exists a periodic wave solution. (3) For ℎ ∈ (ℎ  , 0), there exist a periodic wave solution and a looped periodic wave solution.Moreover, as ℎ → 0, the periodic wave solution converges to a solitary wave solution pointing downward and the looped periodic wave solution converges to a looped wave solution.
(2) For ℎ ∈ (ℎ  , ℎ  ], there exists a periodic wave solution. (3) For ℎ ∈ (ℎ  , 0), there exist periodic and looped periodic wave solutions.Moreover, as ℎ → 0, the periodic wave solution converges to a solitary wave solution pointing downward and the looped periodic wave solution converges to a butterfly-like wave solution.
any travelling wave solution of (1) falls into one of the following categories.
(4) If ℎ ∈ (ℎ  , 0), there exist a periodic wave solution and a cusped periodic wave solution.Moreover, as ℎ → 0, the periodic wave solution converges to a solitary wave solution pointing downward and the cusped periodic wave solution converges to a cusped wave solution.
, any travelling wave solution of (1) falls into one of the following categories.
(2) For ℎ ∈ (ℎ  , 0), there exists a periodic wave solution.Moreover, as ℎ → 0, there exists a cusped wave solution and the periodic wave solution converges to solitary wave solution pointing downward.
, any travelling wave solution of (1) falls into one of the following categories.
(4) For ℎ ∈ (ℎ  , 0), there exist a periodic wave solution and a cusped periodic wave solution.Moreover, as ℎ → 0, the periodic wave solution converges a solitary wave solution pointing downward and the cusped periodic wave solution converges to a cusped wave solution.
, any travelling wave solution of (1) falls into one of the following categories.
(1) If ℎ ≤ ℎ  , there are no bounded traveling solitary wave solutions. ( there exists a periodic wave solution. (3) If ℎ ∈ (ℎ  , 0), there are two types of periodic wave solution.Moreover, as ℎ → 0, the two periodic wave solutions converge to solitary wave solution pointing downward and peaked wave solution.
Theorem 10.If  0 ≥ , any travelling wave solution of (1) falls into one of the following categories.
Theorem 11 (composite waves).A countable number of cusped, peaked, and looped waves in the above cases corresponding to the same value of  can be joined at points where  =  to form composite waves.If ( −1 ()) = 0, one can get travelling wave solution with very strange profiles, such as the travelling waves with a fractal appearance (see Figure 1(k)).For  = −(1/3) 2 −  0 ., the composite waves are solutions of (2) even if ( −1 ()) > 0. Hence we can obtain double stumpons which contain intervals where  =  (see Figure 1(l)).

Proof of Theorem 3
In this section, we will show that the functions satisfying (a) and (b) in Definition 2 consist exactly of the waves stated in Theorem 3.
Let  be a function satisfying (a) and (b) and each wave segment solves the equation for some interval  and constants  0 , ℎ.
For determining the solutions of (3), we should give the following facts.
Lemma 12.The qualitative behavior of solutions of  2  = () near points where  has a zero or a pole is as follows.
(2) If () has a double zero at  = , the solution  of (6) satisfies () ∼  +  exp(−||) as  → ∞.It is easy to find that smooth solitary wave solutions exist if () has a simple zero.
(4) Peaked waves occur when the evolution of  according to (6) suddenly changes direction at  =  where () ̸ = 0 and ( (5) Double kinked waves occur when the right-hand side of (6) has two double zeros which are not opposite numbers, and  =  is not between the two zeros.When the pole  =  falls in the interval of the two zeros, butterfly-like waves occur.One reason for the occurrence of these two new solutions is that the solutions to (2) must be symmetry because this equation is invariable under the transformation  → −.For convenience, we define (), (), and () where

Now we apply the above analysis to
If  >  0 , we define Let Remark 13.It is not difficult to find that () has the same zero points as () except  = .It is easy to see that a change ℎ in equation ( 6) will shift the graph vertically up or down.Now we consider the existence of solutions and their limiting behavior, which have different analytical forms depending on the values of  0 and ℎ.
(1) If ℎ < ℎ  , we can get that () has only a simple zero  1 and () < 0 (see Figure 2(a)).Hence, there are no bounded traveling wave solutions.As ℎ → ℎ  , it is easy to find that, as (  ) → 0 (see Figure 2(b)), () has a double zero and a simple zero, so there are no bounded traveling wave solutions either.
(4) If ℎ > 0, it is easy to observe that () has only one simple zero and () > 0; hence there exists a cusped periodic wave solution.
Case D ( 0 = (1/3)(3 − 4 2 )).In this case, the geometric analysis of () is shown in Figure 5.The result given in Theorem 6 can be proved in a way similar to that in Case B.
Case F ( 0 = (1/5)(5 − 2 2 )).In this case, the geometric analysis of () is shown in Figure 7.The result given in Theorem 8 can be proved in a way similar to that in Case B.
Case G ((1/5)(5 − 2 2 ) <  0 < ).In this case, the geometric analysis of () is shown in Figure 8.The result given in Theorem 9 can be proved in a way similar to that in Case B.
Case H ( 0 ≥ ).In this case, the geometric analysis of () is shown in Figure 9.The result given in Theorem 3 can be proved in a way similar to that in Case B.
Then, we will study the existence of composite waves.By Theorems 3-10, any countable number of travelling waves in the above cases corresponding to the same value of  can be joined at points where  =  to form composite waves.If  = ( −1 ()) = 0, then the composite wave is a solution of (2).For  = −(1/3) 2 −  0 , the composite waves are solutions of (2) even if ( −1 ()) > 0. Consequently, we can obtain  double stumpons which contain intervals where  =  (see Figure 1(l)).Since any countable number of wave segments can be joined together, one can get travelling waves with very strange profiles, such as the travelling waves with a fractal appearance where ( −1 ()) = 0 (see Figure 1(k)).Then the proof of Theorem 11 is completed.

Conclusions
By an improved method combining some characteristics of several methods, we have obtained abundant traveling waves in the mDP equation.Those solutions include looped wave solutions, cusped wave solutions, peaked wave solutions, fractal-like waves, double stumpons, double kinked waves, and butterfly-like waves.Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given.Our method can also be applied to other models where the location of extreme value points is determined.The limiting behavior of traveling wave solutions can also be given.Based on the study, it might be concluded that the improved method is useful and efficient.
It can be widely applied to other nonlinear wave equations.
Our study may be useful to further understand the role that the nonlinearly dispersive terms pay on the optical wave solutions.

Definition 2 .
Any bounded function  belongs to  1 loc (R) and is a travelling wave solution of (2) with speed  if it satisfies the following two statements.(a) There are disjoint open intervals   ,  ≥ 1, and a closed set  such that R \  = ⋃ ∞ =1   ,  ∈  ∞ (  ) for  ≥ 1, () ̸ =  for  ∈ ⋃ ∞ =1   , and () =  for  ∈ .(b) There is an  ∈ R such that (i) For each  ∈ R, there exists  ∈ R such that