Bifurcations of Travelling Wave Solutions for the B(m,n) Equation

Transcription

1 American Journal of Computational Mathematics Published Online March 4 in SciRes. Bifurcations of Travelling Wave Solutions for the B(mn Equation Minzhi Wei * Yujian Gan Shengqiang Tang School of Information and Statistics Guangxi University of Finance and Economics Nanning China School of Mathematics and Computing Science Guilin University of Electronic Technology Guilin China * weiminzhi@cn.com Received December 3; revised January 4; accepted 9 January 4 Copyright 4 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY. Abstract Using the bifurcation theory of dynamical systems to a class of nonlinear fourth order analogue of the B(mn equation the existence of solitary wave solutions periodic cusp wave solutions compactons solutions and uncountably infinite many smooth wave solutions are obtained. Under different parametric conditions various sufficient conditions to guarantee the existence of the above solutions are given. Some exact explicit parametric representations of the above waves are determined. Keywords Solitary Wave Solution; Periodic Cusp Wave Solution; Periodic Wave Solution; Smoothness of Wave; B(mn Equation. Introduction Recently Song and Shao [] employed bifurcation method of dynamical systems to investigate bifurcation of solitary waves of the following generalized ( + -dimensional Boussinesq equation u αu βu γ u δu = (. tt xx yy xx xxxx where αβγ and δ are arbitrary constants with γδ. Chen and Zhang [] obtained some double periodic and multiple soliton solutions of Equation (. by using the generalized Jacobi elliptic function method. Further Li [3] studied the generalized Boussinesq equation: * Corresponding author. ( ς ς + + u = au (. t t x xxxx How to cite this paper: Wei M.Z. Gan Y.J. and Tang S.Q. (4 Bifurcations of Travelling Wave Solutions for the B(mn Equation. American Journal of Computational Mathematics

2 by using bifurcation method. In this paper we shall employ bifurcation method of dynamical systems [4]-[] to investigate bifurcation of solitary waves of the following equation: m n n n ( u ( u ( u au δ ( u = (.3 tt xx yy xx xxxx Numbers of solitary waves are given for each parameter condition. Under some parameter conditions exact solitary wave solutions will be obtained. It is very important to consider the dynamical bifurcation behavior for the travelling wave solutions of (.3. In this paper we shall study all travelling wave solutions in the parameter u x y t = φ x + y ct = φ ξ where c is the wave speed. Then (.3 becomes to space of this system. Let m n n ( φ ( φ ( φ ( 4 c a bu = + + (.4 where ' is the derivative with respect to ξ. Integrating Equation (.4 twice using the constants of integration to be zero we find where m n n n qφ + pφ + φ + r n n φ φ + nφ φ = c b p= q= r =. Equation (.5 is equivalent to the two-dimensional systems as follows a a a with the first integral (.5 m n n dφ dy qφ + pφ + φ + rn n φ y = y = (.6 n dξ dξ rnφ ( n n+ q m p n H( φ y = rnφ + φ φ φ φ h = n+ m n + n. mnpqr. For different m n and a fixed r we shall investigate the bifurcations of phase portraits of System (.6 in the phase plane ( φ y as the parameters pq are changed. Here we are considering a physical model where only bounded travelling waves are meaningful. So we only pay attention to the bounded solutions of System (.6. System (.6 is a 5-parameter planar dynamical system depending on the parameter group (. Bifurcations of Phase Portraits of (.6 In this section we study all possible periodic annuluses defined by the vector fields of (.6 when the parameters pq are varied. n Let dξ = rnφ dζ Then except on the straight lines φ = the system (.6 has the same topological phase portraits as the following system (.7 dφ d n y rn p n rn n = φ = φ + φ + n φ y dζ dζ (. Now the straight lines φ = is an integral invariant straight line of (.. Denote that m n n 3 m n ( φ = + φ + φ ( φ = φ φ + ( f q p f q M p n + For m n l ( l Z m m n n When = = = p n φ = φ = q m f ( ± φ =. m n p( n p( n We have f ( ± φ = + q + p q( m q( m and which imply respectively the relations in the ( pq -parameter plane (. 5

3 + For m n l ( l Z m m n n m n n n La : q= p p> q< m m m n n n m n Lb : q= ( p p< q> m m = = = when m n p n p n f ( φ = + q + p q m q m parameter plane + For m n l ( l Z m m n n p n φ = φ = q m f ( φ =. We have and which imply respectively the relations in the ( pq - m n n n m n Lb : q= ( p p< q> m m m n n n m n Lc : q= ( p p< q< m m p n = = = whenφ = φ = q m m n p n p n f ( φ = + q + p q m q m parameter plane + For m n l ( l Z m m n n f ( φ =. We have which imply respectively the relations in the ( pq - m n n n m n+ 4 Ld : q= p. m m = = = when m n p( n p( n m n f ( φ = + q + p q( m q( m which imply respectively the relations in the ( p n φ = φ = q m f ( ± φ =. We have m n p n p n and f ( φ = q p q m q m pq -parameter plane m n n n Le : q = p pq <. m m Let M( φ y be the coefficient matrix of the linearized system of (. at an equilibrium point ( y Then we have e e = ( = + J φ M φ rnφ q m φ p n φ n 3 m 3 n 3 i det e i i i. e e φ. By the theory of planar dynamical systems we know that for an equilibrium point of a planar integrable system if J < then the equilibrium point is a saddle point; if Trace M φ y = then it is a center point; if ( Trace M e ye J ( φe ye J > and ( ( φ J > and ( ( e e 4 >.then it is a node; if J = and the index of the equilibrium point is then it is a cusp otherwise it is a high order equilibrium point. For the function defined by (.7 we denote that 6

4 ( n m p m h H i ( n ( m+ n n m+ n n+ n i = φi = φi + φi = - 4. pq parameter plane the curves partition it into 4 regions for m n= l m n= l shown in Figure (- (- (-3 and (-4 respectively. The case q We use Figure Figure 3 Figure 4 and Figure 5 to show the bifurcations of the phase portraits of (.. The case q =. We consider the system We next use the above statements to consider the bifurcations of the phase portraits of (.. In the ( with the first integral dφ d n y rn p n rn n = φ = φ + φ + n φ y dζ dζ (.3 H ( φ y rnφ y h. = n + n ( n n+ p n = φ φ + Figure 6 and Figure 7 show respectively the phase portraits of (.3 for n= n and n= n Exact Explicit Parametric Representations of Traveling Wave Solutions of (.6 In this section we give some exact explicit parametric representations of periodic cusp wave solutions. n= 4 m= 6 r< pq A In this case we have the phase portrait of (. shown in Figure. Suppose that 4 (-5. Corresponding to the orbit defined by H( y arch curve has the algebraic equation φ = to the equilibrium point S ± ± (.4 p+ p 4 q the q 5p p 4q 5p p 4q y = φ φ (3. 4( r q q Thus by using the first Equation of (.6 and (3. we obtain the parametric representation of this arch as follows: 5p p 4q φ( ξ =± cn ( Ω ξ k (3. q where p q p p q Ω= k =. 4qr p 4q 6 5 We will show in Section 4 that (3. gives rise to two periodic cusp wave solutions of peak type and valley type of (.3.. Suppose that n= m= 4 r> ( pq A3 In this case we have the phase portrait of (. shown in Figure (-4. corresponding to the orbit defined by H( φ y = to the equilibrium point A the arch curve has the algebraic equation q p y r φ 6 φ + = + (3.3 4 Thus by using the first equation of (.6 and (3.3 we obtain the parametric representation of this arch as follows: 7

5 (- (- (-3 (-4 Figure. (- m n = l n = n ; (- m n = l n = n ; (-3 m n = l n = n + ; (-4 m n = l- n = n +. (- (- (-3 (-4 (-5 (-6 Figure. The phase portraits of (.6 for m n = l n = n l n Z +. (- r < n = (p q (A 3 ; (- r < n (p q (A 3 ; (-3 r > n (p q (A 3 ; (-4 r > n = (p q (A 3 ; (-5 r < n = (p q (A 4 ; (-6 r < n 3 (p q (A 3. 8

6 (3- (3- (3-3 (3-4 (3-5 (3-6 (3-7 (3-8 Figure 3. The phase portraits of (.6 for m n = l n = n l n Z +. (3- r < n = (p q (B ; (3- r < n = (p q (B B ; (3-3 r > n (p q (B (B ; (3-4 r > n (p q (B 3 ; (3-5 r < n = (p q (B 3 ; (3-6 r < n (p q (B B ; (3-7 r < n = (p q (B 4 (B ; (3-8 r > n (p q (B 4. q q φ( ξ = ξ 3( p+ 6r sin We will show in Section 4 that (3. gives rise to a solitary wave solutions of peak type and valley type of ( Suppose that n= 3 m= 5 r< ( pq C In this case we have the phase portrait of (. shown in Figure 4 (4-5. corresponding to the orbit defined by H( φ y = to the equilibrium point A the arch curve has the algebraic equation y = 3 ( φ φ( φ φ( φ3 φ( φ4 φ ( r q 3 p where φ < φ < φ3 < φ4 φi φi + φi + = i = Thus by using the first equation of (.6 and (3.5 we obtain the parametric representation of this arch as follows: φ ξ = ( φ4 φ φ φ( φ4 φ sn ( Ωξ; k ( φ φ ( φ φ sn ( Ωξ; k 4 4 where sn( x; k is the Jacobin elliptic functions with the modulo k Ω = ( φ φ ( φ φ 3 3 6r k = ( φ3 φ( φ4 φ ( φ φ ( φ φ 4 3 We will show in Section 4 that (3.6 gives rise to a smooth compacton solution of ( Suppose that n= m= 3 r< ( pq B. In this case we have the phase portrait of (. shown in Figure 3 (3- corresponding to the orbit defined by H( φ y = to the equilibrium point A the arch curve has the algebraic equation (3.4 (3.5 (3.6 9

7 (4- (4- (4-3 (4-4 (4-5 (4-6 (4-7 (4-8 (4-9 (4- (4- (4- Figure 4. (4- r > n = (p q (C ; (4- r > n (p q (C ; (4-3 r > n = (p q (C ; (4-4 r > n (p q (C ; (4-5 r < n = (p q (C ; (4-6 r < n (p q (C ; (4-7 r > n (p q (C 3 ; (4-8 r < n = (p q (C 3 ; (4-9 r > n = (p q (C 4 ; (4- r < n = (p q (C 4 ; (4- r > n (p q (C 4 ; (4- r < n (p q (C 4.

8 (5- (5- (5-3 (5-4 (5-5 (5-6 (5-7 (5-8 Figure 5. The phase portraits of (.6 for m n = l n =n + l n Z + (5- r > n (p q (D ; (5- r > n = (p q (D ; (5-3 r > n (p q (D 3 (D 4 ; (5-4 r > n = (p q (D 3 (D 4 ; (5-5 r < n (p q (D 3 (D 4 ; (5-6 r < n = (p q (D 3 (D 4 ; (5-7 r < n (p q (D ; (5-8 r < n = (p q (D. ( p + q 5 y = φ φ + 5r 4q Thus by using the first equation of (.6 and (3.7 we obtain the parametric representation of this arch as follows: ( p + 5 4q φ( ξ =. 4q tanh ξ 5r (3.7 (3.8

9 (6- (6- (6-3 Figure 6. The phase portraits of (.6 for n = n n Z +. (6- r < n = m n p < ; (6- r < n m > n p < ; (6-3 r > m > n p <. (7- (7- (7-3 (7-4 Figure 7. The phase portraits of (.6 for n = n + n Z +. (7- r > n = m n p > ; (7- r < n = m > n p < ; (7-3 r < n m > n p > (7-4 r < n m > n p <. We will show in Section 4 that (3.8 gives rise to a solitary wave solution of peak type or valley type of (.3. n= 3 m= 4 r> pq D D in this case we have the phase portrait of (. shown 5. Suppose that 3 4 in Figure 5 (5-4. corresponding to the orbit defined by H( y S ± ± p+ p 4 q the arch curve has the algebraic equation q y φ = to the equilibrium point p p q p p q = ( φ φ + φ. 3r q q Thus by using the first equation of (.6 and (3.9 we obtain the parametric representation of this arch as follows: 7 p 7 p 4 q 7 p 7 p 4 q φ ξ = + + sn ( Ω3ξ; k 3 (3. q q q q where sn( x; k is the Jacobin elliptic functions with the modulo k and p 4q Ω 3 = k3 = + 6qr p 4q (3.9 We will show in Section 4 that (3. gives rise to a smooth compacton solution of (.3.

10 n= 3 m= 4 r< pq D D In this case we have the phase portrait of (. shown in φ = to the equilibrium point 6. Suppose that 3 4 Figure 5 (5-6 corresponding to the orbit defined by H( y S ± ± p± p 4 q the arch curve has the algebraic equation q y p p q p p q = ( φ φ + φ. 3r q q (3. Thus by using the first equation of (.6 and (3. we obtain the parametric representation of this arch as follows: 7 p 4q 7p 7 p 4q p p q q q q φ( ξ = q q p 7 p 4q + sn Ω4 k 4 q q ( ξ; (3. where p 4q Ω 4 = k4 = + 6qr p 4q we will show in Section 4 that (3. gives rise to a smooth compacton solution of (.3 7. Suppose n= 4 m= 5 r< ( pq B B B4 that. In this case we have the phase portrait of (. shown in Figure 3 (3- and (3-7 corresponding to the orbit defined by H( φ y = to the equilibrium point A the arch curve has the algebraic equation y q p = φ + φ +. r (3.3 Thus by using the first equation of (.6 and (3.3 we obtain the parametric representation of this arch as follows: where 3 7 p 7 p 6 g 3 3p q 8q 7r = + ξ g g φ ξ (3.4 g =. 6q = 8q q We will show in Section 4 that (3.4 gives rise to a smooth compacton solution of ( Suppose n= 4 m= 6 r< ( pq A3. In this case we have the phase portrait of (. shown in Figure (- corresponding to the orbit defined by H( φ y = to the equilibrium point A the arch curve has the algebraic equation 5p 5 p 4q 5p 5 p 4q y = φ + + φ +. (3.5 4r 8q q 6 5 8q q 6 5 Thus by using the first equation of (.6 and (3.5 we obtain the parametric representation of this arch as follows: φ ξ = ± p p q + 8q q 6 5 cn ( Ω ξ; k 5 5 (3.6 3

11 where Ω = 5 p 4q qr k 4 p = + p 4q We will show in Section 4 that (3.6 gives rise to two periodic cusp wave solutions of peak type and valley type of ( Suppose n= 3 m= 5 r> ( pq C4. In this case we have the phase portrait of (. shown in Figure 4 (4-5 corresponding to the orbit defined by H( φ y = to the equilibrium point A the arch curve has the algebraic equation y = ( 3( 4 3r φ φ φ φ φ φ φ φ (3.7 q 3 p where φ < φ < φ3 < φ4 φi φi φi = i = 4. Thus by using the first equation of (.6 and ( we obtain the parametric representation of this arch as follows: φ ξ = ( φ4 φ3 φ φ3( φ4 φ sn ( Ω6ξ; k6 ( φ φ ( φ φ sn ( Ωξ; k where sn( x; k is the Jacobin elliptic functions with the modulo k and Ω = ( φ φ ( φ φ r k 6 = ( φ4 φ3( φ φ ( φ φ ( φ φ 3 3 We will show in Section 4 that (3.6 gives rise to a smooth compacton solution of ( The Existence of Smooth and Non-Smooth Travelling Wave Solutions of (.6 (3.8 In this section we use the results of Section to discuss the existence of smooth and non-smooth solitary wave and periodic wave solutions. We first consider the existence of smooth solitary wave solution and periodic wave solutions. Theorem 4. m n= n= n + 5 ln Z + r> pq C : Then corresponding to a branch of the. Suppose that curves H( y h( h3 corresponding to a branch of the curves H( y hh ( h φ = defined by (.7 Equation (.3 has a smooth solitary wave solution of peak type φ = defined by (.7 Equation (.3 has a smooth family of periodic wave solutions (see Figure 4 (4-4. m n= n= 3 m Z + r> pq C: Then corresponding to a branch of the curves. Suppose that H( y h to a branch of the curves H( y hh ( h φ = defined by (.7 equation (.3 has a smooth solitary wave solution of peak type corresponding 3 3 φ = defined by (.7 Equation (.3 has a smooth family of pe- riodic wave solutions (see Figure 4 (4-3. m n= n= n + 5 ln Z r> pq A Then corresponding to a branch of the φ = defined by (.7 equation (.3 has a smooth solitary wave solution of peak type corres- 3. Suppose that curves H( y ponding to a branch of the curves H( y hh ( h φ = defined by (.7 equation (.3 has a smooth family of periodic wave solutions (see Figure 4 (4-. m n= ln = 3 m Z r> pq C Then corresponding to a branch of the curves 4. Suppose that H( y to a branch of the curves H( y hh ( h φ = defined by (.7 equation (.3 has a smooth solitary wave solution of valley type corresponding odic wave solutions (see Figure 4 (4-3. φ = defined by (.7 equation (.3 has a smooth family of peri- 3 4

12 5. Suppose that m n ln 4 m Z + r ( pq A H( y h to a branch of the curves H( y hh ( h = > Then corresponding to a branch of the curves φ = defined by (.7 equation (.3 has a smooth solitary wave solution of peak type corresponding φ = defined by (.7 Equation (.3 has a smooth family of pe- 3 4 riodic wave solutions (see Figure (-3. m n= ln 4 m Z r> pq A Then corresponding to a branch of the curves 6. Suppose that H( y h3 to a branch of the curves H( y hh ( h φ = defined by (.7 equation (.3 has a smooth solitary wave solution of peak type corresponding φ = defined by (.7 Equation (.3 has a smooth family of peri- odic wave solutions (see Figure (-3. m n= ln = 3 m Z r> pq C Then corresponding to a branch of the curves 7. Suppose that 3 H( y to a branch of the curves H( y hh ( h φ = defined by (.7 Equation (.3 has a smooth solitary wave solution of valley type corresponding φ = defined by (.7 Equation (.3 has a smooth family of peri- odic wave solutions (see Figure (-3. m n= l n= 4 l Z r> pq B Then corresponding to a branch of the curves 8. Suppose that 3 ( ( H φ y = hh h h defined by (.7 Equation (.3 has a smooth family of periodic wave solutions (see Figure 3 ( Suppose that: m n l n 4 l Z + r ( pq B3 H( y h to a branch of the curves H( y hh ( h = > Then corresponding to a branch of the curves φ = defined by (.7 equation (.3 has a smooth solitary wave solution of valley type corresponding φ = defined by (.7 Equation (.3 has a smooth family of peri- odic wave solutions (see Figure 3 (3-4. m n= ln = l Z r< pq B Then corresponding to a branch of the curves. Suppose that H( y to a branch of the curves H( y hh ( φ = defined by (.7 equation (.3 has a smooth solitary wave solution of valley type corresponding φ = defined by (.7 Equation (.3 has a smooth family of peri- odic wave solutions (see Figure (-4. m n= l n= l Z r< pq B: Then corresponding to a branch of the curves. Suppose that H( y to a branch of the curves H( y hh ( φ = defined by (.7 Equation (.3 has a smooth solitary wave solution of valley type corresponding φ = defined by (.7 Equation (.3 has a smooth family of peri- odic wave solutions (see Figure 3 (3-. We shall describe what types of non-smooth solitary wave and periodic wave solutions can appear for our system (.6 which correspond to some orbits of (. near the straight line φ =. To discuss the existence of cusp waves we need to use the following lemmarelating to the singular straight line. Lemma 4. The boundary curves of a periodic annulus are the limit curves of closed orbits inside the annulus; If these boundary curves contain a segment of the singular straight line φ = of (.4 then along this segment and near this segment in very short time interval y = φ jumps rapidly. ξ Base on Lemma 4. Figure and Figure 3 we have the following result. Theorem 4.3. Suppose that m n= ln = 4 l Z +. a. For r< ( pq A4 corresponding to the arch curve H( y two periodic cusp wave solutions; corresponding to two branches of the curves H( y hh ( φ = defined by (.7 Equation (.3 has φ = defined by (.7 Equation (.3 has two families of periodic wave solutions. When h varies from h to these periodic travelling waves will gradually lose their smoothness and evolve from smooth periodic travelling waves to periodic cusp travelling waves finally approach a periodic cusp wave of valley type and a periodic cusp wave of H φ y = of (.7 (see Figure (-5. peak type defined by 5

13 φ = defined by (.7 Equation b. For r> ( pq A3 corresponding to the arch curve H( y hh ( h (.3 has two periodic cusp wave solutions; corresponding to two branches of the curves H( φ y = hh ( defined by (.7 Equation (.3 has two families of periodic wave solutions. When h varies from h to these periodic travelling waves will gradually lose their smoothness and evolve from smooth periodic travelling waves to periodic cusp travelling waves finally approach a periodic cusp wave of valley type and a periodic H φ y = of (.7 (see Figure (-. cusp wave of peak type defined by. Suppose that m n= l n= 4 l Z +. c. For r< ( pq B4 corresponding to the arch curve H( y two periodic cusp wave solutions; corresponding to two branches of the curves H( y hh ( φ = defined by (.7 Equation (.3 has φ = defined by (.7 Equation (.3 has two families of periodic wave solutions. When h varies from to h these periodic travelling waves will gradually lose their smoothness and evolve from smooth periodic travelling waves to periodic cusp travelling waves finally approach a periodic cusp wave of valley type and a periodic cusp wave of H φ y = of (.7 (see Figure 3 (3-7. peak type defined by d. For r< ( pq B3 corresponding to the arch curve H( y two periodic cusp wave solutions; corresponding to two branches of the curves H( y hh ( φ = defined by (.7 Equation (.3 has φ = defined by (.7 Equation (.3 has two families of periodic wave solutions. When h varies from to h 3 these periodic travelling waves will gradually lose their smoothness and evolve from smooth periodic travelling waves to periodic cusp travelling waves finally approach a periodic cusp wave of valley type and a periodic cusp wave of H φ y = of (.7 (see Figure 3 (3-5. peak type defined by d. For r< ( pq B B corresponding to the arch curve H( y has two periodic cusp wave solutions; corresponding to two branches of the curves H( y hh ( φ = defined by (.7 Equation (.3 φ = de- fined by (.7 Equation (.3 has two families of periodic wave solutions. When h varies from to h these periodic travelling waves will gradually lose their smoothness and evolve from smooth periodic travelling waves to periodic cusp travelling waves finally approach a periodic cusp wave of valley type and a periodic H φ y = of (.7 (see Figure 3 (3-3. cusp wave of peak type defined by We can easily see that there exist two families of closed orbits of (.3 in Figure (-6 Figure 3 (3-6 and in Figure 6 (6-. There is one family of closed orbits in Figure 3 (3-3 (3-8 Figure 4 (4- (4-5 - (4-7 (4-9 (4- and in Figure 5 (5-3 (5-5 - (5-7 and in Figure 7 (7-3 (7-4. In all the above cases there exists at least one family of closed orbits (.3 for which as whichh from e H φ to where φ e is the abscissa of the center the closed orbit will expand outwards to approach the straight line φ = and y = φ will ap- proach to. As a result we have the following conclusions. Theorem 4.4. Suppose that m n= ln = n + ln Z +. r n pq C h h in (.7 Equation (.3 has a family of uncountably in- a. If < ; then when finite many periodic traveling wave solutions; where h varies from h to these periodic traveling wave solutions will gradually lose their smoothness and evolve from smooth periodic traveling waves to periodic cusp traveling waves (see Figure 4 (4-. r n pq C h h in (.7 Equation (.3 has a family of uncountably in- b. If < ; then when 3 finite many periodic traveling wave solutions; when h varies from to h these periodic traveling wave solutions will gradually lose their smoothness and evolve from smooth periodic traveling waves to periodic cusp traveling waves (see Figure 4 (4-7. r n pq C h h in (.7 Equation (.3 has a family of uncountably in- c. If < ; then when finite many periodic traveling wave solutions; when h varies from to h these periodic traveling wave solutions will gradually lose their smoothness and evolve from smooth periodic traveling waves to periodic cusp traveling waves (see Figure 4 (

14 d. If r> n ( pq C ; then when h ( h 4 in (.7 Equation (.3 has a family of uncountably in- finite many periodic traveling wave solutions; when h varies from h to these periodic traveling wave solutions will gradually lose their smoothness and evolve from smooth periodic traveling waves to periodic cusp traveling waves (see Figure 4 (4-. r n pq C h h in (.7 Equation (.3 has a family of uncountably in- e. If < = ; then when finite many periodic traveling wave solutions; when h varies from to h these periodic traveling wave solutions will gradually lose their smoothness and evolve from smooth periodic traveling waves to periodic cusp traveling waves (see Figure 4 (4-5. r n pq C h h in (.7 Equation (.3 has a family of uncountably in- f. If > = ; then when 4 finite many periodic traveling wave solutions; when h varies from to h these periodic traveling wave solutions will gradually lose their smoothness and evolve from smooth periodic traveling waves to periodic cusp traveling waves (see Figure 4 (4-9. h h in (.7 Equation (.3 has two family of uncountably infinite many. Suppose that then when periodic traveling wave solutions; when h varies from to h these periodic traveling wave solutions will gradually lose their smoothness and evolve from smooth periodic traveling waves to periodic cusp traveling waves (see Figure (-6.Parallelling to Figure (-6 we can see the periodic travelling wave solutions implied in Figure 3 (3-6 and Figure 6 (6- have the same characters. 3. Suppose that m n= l n= n + ln Z +. r n pq D D h h in (.7 Equation (.3 has a family of uncounta- g. If < = ; then when 4 bly infinite many periodic traveling wave solutions; when h varies from to h these periodic traveling wave solutions will gradually lose their smoothness and evolve from smooth periodic traveling waves to periodic cusp traveling waves (see Figure 5 (5-6. r n pq D h h in (.7 Equation (.3 has a family of uncountably in-. If < ; then when finite many periodic traveling wave solutions; when h varies from to h these periodic traveling wave solutions will gradually lose their smoothness and evolve from smooth periodic traveling waves to periodic cusp traveling waves (see Figure 5 (5-7. r n pq D D h h in (.7 Equation (.3 has a family of uncounta- i. If < then when 3 4 bly infinite many periodic traveling wave solutions; when h varies from h to these periodic traveling wave solutions will gradually lose their smoothness and evolve from smooth periodic traveling waves to periodic cusp traveling waves (see Figure 5 (5-3. r n pq D D h h in (.7 Equation (.3 has a family of uncounta- j. If < then when 3 4 bly infinite many periodic traveling wave solutions; when h varies from to h these periodic traveling wave solutions will gradually lose their smoothness and evolve from smooth periodic traveling waves to periodic cusp traveling waves (see Figure 5 ( Suppose that n= n + ln Z q=. k. If r < n p> then when h ( h in (.7 Equation (.3 has a family of uncountably infinite many periodic traveling wave solutions; when h varies from h to these periodic traveling wave solutions will gradually lose their smoothness and evolve from smooth periodic traveling waves to periodic cusp traveling waves (see Figure 5 (5-5. Equation (.3 has one family of uncountably infinite many periodic traveling wave solutions; when h varies from to h these periodic travelling wave solutions will gradually lose their smoothness and evolve from smooth periodic travelling waves to periodic cusp travelling waves (see Figure 7 (7-4. Acknowledgements This work is supported by t NNSF of China (6 63. The authors are grateful for this financial support. 7

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