Bifurcations of Travelling Wave Solutions for the B(m,n) Equation
|
|
- Gwendoline Whitehead
- 5 years ago
- Views:
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
15 References [] Song M. and Shao S.G. ( Exact Solitary Wave Solutions of the Generalized ( + Dimensional Boussinesq Equation. Applied Mathematics and Computation [] Chen H.T. and Zhang H.Q. (4 New Double Periodic and Multiple Soliton Solutions of Thegeneralized ( + -Dimensional Boussinesq Equation. Chaos Solitons & Fractals [3] Li J.B. (8 Bifurcation of Traveling Wave Solutions for Two Types Boussinesq Equations. Science China Press [4] Tang S.Q. Xiao Y.X. and Wang Z.J. (9 Travelling Wave Solutions for a Class of Nonlinear Fourth Order Variant of a Generalized Camassa-Holm equation. Applied Mathematics and Computation [5] Rong J.H. Tang S.Q. and Huang W.T. ( Bifurcation of Traveling Wave Solutions for a Class of Nonlinear Fourth Order Variant of a Generalized Camassa-Holm equation. Communications in Nonlinear Science and Numerical Simulation [6] Tang S.Q. and Huang W.T. (8 Bifurcations of Travelling Wave Solutions for the K(n-nn Equations. Applied Mathematics and Computation [7] Li J.B. and Liu Z.R. ( Travelling Wave Solutions for a Class of Nonlinear Dispersive Equations. Chinese Annals of Mathematics Series B [8] Li J.B. and Liu Z.R. ( Smooth and Non-Smooth Travelling Waves in a Nonlinearly Dispersive Equation. Applied Mathematical Modelling [9] Bibikov Y.N. (979 Local Theory of Nonlinear Analytic Ordinary Differential Equations. Lecture Notes in Mathematics Vol. 7. Springer-Verlag New York. [] Wang Z.J. and Tang. S.Q. (9 Bifurcation of Travelling Wave Solutions for the Generalizedzk Equations. Communications in Nonlinear Science and Numerical Simulation [] Takahashi M. (3 Bifurcations of Ordinary Differential Equations of Clairaut Type. Journal of Differential Equations
Bifurcations of Traveling Wave Solutions for a Generalized Camassa-Holm Equation
Computational and Applied Mathematics Journal 2017; 3(6): 52-59 http://www.aascit.org/journal/camj ISSN: 2381-1218 (Print); ISSN: 2381-1226 (Online) Bifurcations of Traveling Wave Solutions for a Generalized
More informationIntegral Bifurcation Method and Its Application for Solving the Modified Equal Width Wave Equation and Its Variants
Rostock. Math. Kolloq. 62, 87 106 (2007) Subject Classification (AMS) 35Q51, 35Q58, 37K50 Weiguo Rui, Shaolong Xie, Yao Long, Bin He Integral Bifurcation Method Its Application for Solving the Modified
More informationEXACT TRAVELLING WAVE SOLUTIONS FOR THE GENERALIZED KURAMOTO-SIVASHINSKY EQUATION
Journal of Mathematial Sienes: Advanes and Appliations Volume 3, 05, Pages -3 EXACT TRAVELLING WAVE SOLUTIONS FOR THE GENERALIZED KURAMOTO-SIVASHINSKY EQUATION JIAN YANG, XIAOJUAN LU and SHENGQIANG TANG
More informationEXACT SOLITARY WAVE AND PERIODIC WAVE SOLUTIONS OF THE KAUP-KUPERSCHMIDT EQUATION
Journal of Applied Analysis and Computation Volume 5, Number 3, August 015, 485 495 Website:http://jaac-online.com/ doi:10.11948/015039 EXACT SOLITARY WAVE AND PERIODIC WAVE SOLUTIONS OF THE KAUP-KUPERSCHMIDT
More informationResearch Article New Exact Solutions for the 2 1 -Dimensional Broer-Kaup-Kupershmidt Equations
Hindawi Publishing Corporation Abstract and Applied Analysis Volume 00, Article ID 549, 9 pages doi:0.55/00/549 Research Article New Exact Solutions for the -Dimensional Broer-Kaup-Kupershmidt Equations
More informationEXACT DARK SOLITON, PERIODIC SOLUTIONS AND CHAOTIC DYNAMICS IN A PERTURBED GENERALIZED NONLINEAR SCHRODINGER EQUATION
CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 17, Number 1, Spring 9 EXACT DARK SOLITON, PERIODIC SOLUTIONS AND CHAOTIC DYNAMICS IN A PERTURBED GENERALIZED NONLINEAR SCHRODINGER EQUATION JIBIN LI ABSTRACT.
More informationNew Exact Solutions of the Modified Benjamin-Bona-Mahony Equation Yun-jie YANG and Li YAO
06 International Conference on Artificial Intelligence and Computer Science (AICS 06) ISBN: 978--60595-4-0 New Exact Solutions of the Modified Benamin-Bona-Mahony Equation Yun-ie YANG and Li YAO Department
More informationSUB-MANIFOLD AND TRAVELING WAVE SOLUTIONS OF ITO S 5TH-ORDER MKDV EQUATION
Journal of Applied Analysis and Computation Volume 7, Number 4, November 07, 47 430 Website:http://jaac-online.com/ DOI:0.94/0706 SUB-MANIFOLD AND TRAVELING WAVE SOLUTIONS OF ITO S 5TH-ORDER MKDV EQUATION
More informationTravelling Wave Solutions for the Gilson-Pickering Equation by Using the Simplified G /G-expansion Method
ISSN 1749-3889 (print, 1749-3897 (online International Journal of Nonlinear Science Vol8(009 No3,pp368-373 Travelling Wave Solutions for the ilson-pickering Equation by Using the Simplified /-expansion
More informationNew explicit solitary wave solutions for (2 + 1)-dimensional Boussinesq equation and (3 + 1)-dimensional KP equation
Physics Letters A 07 (00) 107 11 www.elsevier.com/locate/pla New explicit solitary wave solutions for ( + 1)-dimensional Boussinesq equation and ( + 1)-dimensional KP equation Yong Chen, Zhenya Yan, Honging
More informationExact Solutions for the Nonlinear (2+1)-Dimensional Davey-Stewartson Equation Using the Generalized ( G. )-Expansion Method
Journal of Mathematics Research; Vol. 6, No. ; 04 ISSN 96-9795 E-ISSN 96-9809 Published by Canadian Center of Science and Education Exact Solutions for the Nonlinear +-Dimensional Davey-Stewartson Equation
More informationTwo Loop Soliton Solutions. to the Reduced Ostrovsky Equation 1
International Mathematical Forum, 3, 008, no. 31, 159-1536 Two Loop Soliton Solutions to the Reduced Ostrovsky Equation 1 Jionghui Cai, Shaolong Xie 3 and Chengxi Yang 4 Department of Mathematics, Yuxi
More informationFrom bell-shaped solitary wave to W/M-shaped solitary wave solutions in an integrable nonlinear wave equation
PRAMANA c Indian Academ of Sciences Vol. 74, No. journal of Januar 00 phsics pp. 9 6 From bell-shaped solitar wave to W/M-shaped solitar wave solutions in an integrable nonlinear wave equation AIYONG CHEN,,,
More informationTopological Solitons and Bifurcation Analysis of the PHI-Four Equation
BULLETIN of the MALAYSIAN MATHEMATICAL SCIENCES SOCIETY http://math.usm.my/bulletin Bull. Malays. Math. Sci. Soc. () 37(4) (4), 9 9 Topological Solitons Bifurcation Analysis of the PHI-Four Equation JUN
More informationInfinite Sequence Soliton-Like Exact Solutions of (2 + 1)-Dimensional Breaking Soliton Equation
Commun. Theor. Phys. 55 (0) 949 954 Vol. 55, No. 6, June 5, 0 Infinite Sequence Soliton-Like Exact Solutions of ( + )-Dimensional Breaking Soliton Equation Taogetusang,, Sirendaoerji, and LI Shu-Min (Ó
More informationKINK DEGENERACY AND ROGUE WAVE FOR POTENTIAL KADOMTSEV-PETVIASHVILI EQUATION
THERMAL SCIENCE, Year 05, Vol. 9, No. 4, pp. 49-435 49 KINK DEGENERACY AND ROGUE WAVE FOR POTENTIAL KADOMTSEV-PETVIASHVILI EQUATION by Hong-Ying LUO a*, Wei TAN b, Zheng-De DAI b, and Jun LIU a a College
More informationExact solutions through symmetry reductions for a new integrable equation
Exact solutions through symmetry reductions for a new integrable equation MARIA LUZ GANDARIAS University of Cádiz Department of Mathematics PO.BOX, 1151 Puerto Real, Cádiz SPAIN marialuz.gandarias@uca.es
More informationApplication of the trial equation method for solving some nonlinear evolution equations arising in mathematical physics
PRMN c Indian cademy of Sciences Vol. 77, No. 6 journal of December 011 physics pp. 103 109 pplication of the trial equation method for solving some nonlinear evolution equations arising in mathematical
More informationRational Form Solitary Wave Solutions and Doubly Periodic Wave Solutions to (1+1)-Dimensional Dispersive Long Wave Equation
Commun. Theor. Phys. (Beijing, China) 43 (005) pp. 975 98 c International Academic Publishers Vol. 43, No. 6, June 15, 005 Rational Form Solitary Wave Solutions and Doubly Periodic Wave Solutions to (1+1)-Dimensional
More informationOn Universality of Transition to Chaos Scenario in Nonlinear Systems of Ordinary Differential Equations of Shilnikov s Type
Journal of Applied Mathematics and Physics, 2016, 4, 871-880 Published Online May 2016 in SciRes. http://www.scirp.org/journal/jamp http://dx.doi.org/10.4236/jamp.2016.45095 On Universality of Transition
More informationNew Exact Travelling Wave Solutions for Regularized Long-wave, Phi-Four and Drinfeld-Sokolov Equations
ISSN 1749-3889 print), 1749-3897 online) International Journal of Nonlinear Science Vol.008) No.1,pp.4-5 New Exact Travelling Wave Solutions for Regularized Long-wave, Phi-Four and Drinfeld-Sokolov Euations
More informationA robust uniform B-spline collocation method for solving the generalized PHI-four equation
Available at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 1932-9466 Vol. 11, Issue 1 (June 2016), pp. 364-376 Applications and Applied Mathematics: An International Journal (AAM) A robust uniform B-spline
More informationResearch Article Two Different Classes of Wronskian Conditions to a (3 + 1)-Dimensional Generalized Shallow Water Equation
International Scholarly Research Network ISRN Mathematical Analysis Volume 2012 Article ID 384906 10 pages doi:10.5402/2012/384906 Research Article Two Different Classes of Wronskian Conditions to a 3
More informationThe Exact Solitary Wave Solutions for a Family of BBM Equation
ISSN 749-3889(print),749-3897(online) International Journal of Nonlinear Science Vol. (2006) No., pp. 58-64 The Exact Solitary Wave Solutions f a Family of BBM Equation Lixia Wang, Jiangbo Zhou, Lihong
More informationTWELVE LIMIT CYCLES IN A CUBIC ORDER PLANAR SYSTEM WITH Z 2 -SYMMETRY. P. Yu 1,2 and M. Han 1
COMMUNICATIONS ON Website: http://aimsciences.org PURE AND APPLIED ANALYSIS Volume 3, Number 3, September 2004 pp. 515 526 TWELVE LIMIT CYCLES IN A CUBIC ORDER PLANAR SYSTEM WITH Z 2 -SYMMETRY P. Yu 1,2
More informationA NEW VARIABLE-COEFFICIENT BERNOULLI EQUATION-BASED SUB-EQUATION METHOD FOR SOLVING NONLINEAR DIFFERENTIAL EQUATIONS
U.P.B. Sci. Bull., Series A, Vol. 76, Iss., 014 ISSN 1-707 A NEW VARIABLE-COEFFICIENT BERNOULLI EQUATION-BASED SUB-EQUATION METHOD FOR SOLVING NONLINEAR DIFFERENTIAL EQUATIONS Bin Zheng 1 In this paper,
More informationAn Improved F-Expansion Method and Its Application to Coupled Drinfel d Sokolov Wilson Equation
Commun. Theor. Phys. (Beijing, China) 50 (008) pp. 309 314 c Chinese Physical Society Vol. 50, No., August 15, 008 An Improved F-Expansion Method and Its Application to Coupled Drinfel d Sokolov Wilson
More informationResearch Article Soliton Solutions for the Wick-Type Stochastic KP Equation
Abstract and Applied Analysis Volume 212, Article ID 327682, 9 pages doi:1.1155/212/327682 Research Article Soliton Solutions for the Wick-Type Stochastic KP Equation Y. F. Guo, 1, 2 L. M. Ling, 2 and
More informationThe Solitary Wave Solutions of Zoomeron Equation
Applied Mathematical Sciences, Vol. 5, 011, no. 59, 943-949 The Solitary Wave Solutions of Zoomeron Equation Reza Abazari Deparment of Mathematics, Ardabil Branch Islamic Azad University, Ardabil, Iran
More information) -Expansion Method for Solving (2+1) Dimensional PKP Equation. The New Generalized ( G. 1 Introduction. ) -expansion method
ISSN 749-3889 (print, 749-3897 (online International Journal of Nonlinear Science Vol.4(0 No.,pp.48-5 The New eneralized ( -Expansion Method for Solving (+ Dimensional PKP Equation Rajeev Budhiraja, R.K.
More informationExact Solutions for Generalized Klein-Gordon Equation
Journal of Informatics and Mathematical Sciences Volume 4 (0), Number 3, pp. 35 358 RGN Publications http://www.rgnpublications.com Exact Solutions for Generalized Klein-Gordon Equation Libo Yang, Daoming
More informationNew Analytical Solutions For (3+1) Dimensional Kaup-Kupershmidt Equation
International Conference on Computer Technology and Science (ICCTS ) IPCSIT vol. 47 () () IACSIT Press, Singapore DOI:.776/IPCSIT..V47.59 New Analytical Solutions For () Dimensional Kaup-Kupershmidt Equation
More informationExact Solutions of Kuramoto-Sivashinsky Equation
I.J. Education and Management Engineering 01, 6, 61-66 Published Online July 01 in MECS (http://www.mecs-press.ne DOI: 10.5815/ijeme.01.06.11 Available online at http://www.mecs-press.net/ijeme Exact Solutions
More informationExact Solutions of the Generalized- Zakharov (GZ) Equation by the Infinite Series Method
Available at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 193-9466 Vol. 05, Issue (December 010), pp. 61 68 (Previously, Vol. 05, Issue 10, pp. 1718 175) Applications and Applied Mathematics: An International
More informationDepartment of Applied Mathematics, Dalian University of Technology, Dalian , China
Commun Theor Phys (Being, China 45 (006 pp 199 06 c International Academic Publishers Vol 45, No, February 15, 006 Further Extended Jacobi Elliptic Function Rational Expansion Method and New Families of
More informationNew Homoclinic and Heteroclinic Solutions for Zakharov System
Commun. Theor. Phys. 58 (2012) 749 753 Vol. 58, No. 5, November 15, 2012 New Homoclinic and Heteroclinic Solutions for Zakharov System WANG Chuan-Jian ( ), 1 DAI Zheng-De (à ), 2, and MU Gui (½ ) 3 1 Department
More informationNew approach for tanh and extended-tanh methods with applications on Hirota-Satsuma equations
Volume 28, N. 1, pp. 1 14, 2009 Copyright 2009 SBMAC ISSN 0101-8205 www.scielo.br/cam New approach for tanh and extended-tanh methods with applications on Hirota-Satsuma equations HASSAN A. ZEDAN Mathematics
More informationOptical solitary wave solutions to nonlinear Schrödinger equation with cubic-quintic nonlinearity in non-kerr media
MM Research Preprints 342 349 MMRC AMSS Academia Sinica Beijing No. 22 December 2003 Optical solitary wave solutions to nonlinear Schrödinger equation with cubic-quintic nonlinearity in non-kerr media
More informationNew Exact Solutions to NLS Equation and Coupled NLS Equations
Commun. Theor. Phys. (Beijing, China 4 (2004 pp. 89 94 c International Academic Publishers Vol. 4, No. 2, February 5, 2004 New Exact Solutions to NLS Euation Coupled NLS Euations FU Zun-Tao, LIU Shi-Da,
More informationExact Solutions for a Fifth-Order Two-Mode KdV Equation with Variable Coefficients
Contemporary Engineering Sciences, Vol. 11, 2018, no. 16, 779-784 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ces.2018.8262 Exact Solutions for a Fifth-Order Two-Mode KdV Equation with Variable
More informationDear Dr. Gaetana Gambino,
-------- Messaggio originale -------- Oggetto: Your Submission - CNSNS-D-13-93R1 Data:28 Oct 213 19:37:56 + Mittente:Stefano Ruffo A:gaetana@math.unipa.it, gaetana.gambino@unipa.it
More informationResearch Article The Extended Hyperbolic Function Method for Generalized Forms of Nonlinear Heat Conduction and Huxley Equations
Journal of Applied Mathematics Volume 0 Article ID 769843 6 pages doi:0.55/0/769843 Research Article The Extended Hyperbolic Function Method for Generalized Forms of Nonlinear Heat Conduction and Huxley
More informationExact Travelling Wave Solutions of the Coupled Klein-Gordon Equation by the Infinite Series Method
Available at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 93-9466 Vol. 6, Issue (June 0) pp. 3 3 (Previously, Vol. 6, Issue, pp. 964 97) Applications and Applied Mathematics: An International Journal (AAM)
More informationStability and Hopf Bifurcation for a Discrete Disease Spreading Model in Complex Networks
International Journal of Difference Equations ISSN 0973-5321, Volume 4, Number 1, pp. 155 163 (2009) http://campus.mst.edu/ijde Stability and Hopf Bifurcation for a Discrete Disease Spreading Model in
More informationHomotopy Perturbation Method for the Fisher s Equation and Its Generalized
ISSN 749-889 (print), 749-897 (online) International Journal of Nonlinear Science Vol.8(2009) No.4,pp.448-455 Homotopy Perturbation Method for the Fisher s Equation and Its Generalized M. Matinfar,M. Ghanbari
More informationSymmetry reductions and travelling wave solutions for a new integrable equation
Symmetry reductions and travelling wave solutions for a new integrable equation MARIA LUZ GANDARIAS University of Cádiz Department of Mathematics PO.BOX 0, 50 Puerto Real, Cádiz SPAIN marialuz.gandarias@uca.es
More informationBifurcation control and chaos in a linear impulsive system
Vol 8 No 2, December 2009 c 2009 Chin. Phys. Soc. 674-056/2009/82)/5235-07 Chinese Physics B and IOP Publishing Ltd Bifurcation control and chaos in a linear impulsive system Jiang Gui-Rong 蒋贵荣 ) a)b),
More informationA Generalized Extended F -Expansion Method and Its Application in (2+1)-Dimensional Dispersive Long Wave Equation
Commun. Theor. Phys. (Beijing, China) 6 (006) pp. 580 586 c International Academic Publishers Vol. 6, No., October 15, 006 A Generalized Extended F -Expansion Method and Its Application in (+1)-Dimensional
More informationPeriodic and Soliton Solutions for a Generalized Two-Mode KdV-Burger s Type Equation
Contemporary Engineering Sciences Vol. 11 2018 no. 16 785-791 HIKARI Ltd www.m-hikari.com https://doi.org/10.12988/ces.2018.8267 Periodic and Soliton Solutions for a Generalized Two-Mode KdV-Burger s Type
More informationThe General Form of Linearized Exact Solution for the KdV Equation by the Simplest Equation Method
Applied and Computational Mathematics 015; 4(5): 335-341 Published online August 16 015 (http://www.sciencepublishinggroup.com/j/acm) doi: 10.11648/j.acm.0150405.11 ISSN: 38-5605 (Print); ISSN: 38-5613
More informationDynamical analysis and circuit simulation of a new three-dimensional chaotic system
Dynamical analysis and circuit simulation of a new three-dimensional chaotic system Wang Ai-Yuan( 王爱元 ) a)b) and Ling Zhi-Hao( 凌志浩 ) a) a) Department of Automation, East China University of Science and
More informationExact Periodic Solitary Wave and Double Periodic Wave Solutions for the (2+1)-Dimensional Korteweg-de Vries Equation*
Exact Periodic Solitary Wave Double Periodic Wave Solutions for the (+)-Dimensional Korteweg-de Vries Equation* Changfu Liu a Zhengde Dai b a Department of Mathematics Physics Wenshan University Wenshan
More informationarxiv: v1 [nlin.ps] 3 Sep 2009
Soliton, kink and antikink solutions o a 2-component o the Degasperis-Procesi equation arxiv:0909.0659v1 [nlin.ps] 3 Sep 2009 Jiangbo Zhou, Liin Tian, Xinghua Fan Nonlinear Scientiic Research Center, Facult
More informationCompacton Solutions and Peakon Solutions for a Coupled Nonlinear Wave Equation
ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol 4(007) No1,pp31-36 Compacton Solutions Peakon Solutions for a Coupled Nonlinear Wave Equation Dianchen Lu, Guangjuan
More informationDynamical behaviour of a controlled vibro-impact system
Vol 17 No 7, July 2008 c 2008 Chin. Phys. Soc. 1674-1056/2008/17(07)/2446-05 Chinese Physics B and IOP Publishing Ltd Dynamical behaviour of a controlled vibro-impact system Wang Liang( ), Xu Wei( ), and
More informationAuto-Bäcklund transformation and exact solutions for compound KdV-type and compound KdV Burgers-type equations with nonlinear terms of any order
Physics Letters A 305 (00) 377 38 www.elsevier.com/locate/pla Auto-Bäcklund transformation and exact solutions for compound KdV-type and compound KdV Burgers-type equations with nonlinear terms of any
More informationStable Propagating Waves and Wake Fields in Relativistic Electromagnetic Plasma
Commun. Theor. Phys. (Beijing, China) 49 (2008) pp. 753 758 c Chinese Physical Society Vol. 49, No. 3, March 15, 2008 Stable Propagating Waves and Wake Fields in Relativistic Electromagnetic Plasma XIE
More informationMath 4200, Problem set 3
Math, Problem set 3 Solutions September, 13 Problem 1. ẍ = ω x. Solution. Following the general theory of conservative systems with one degree of freedom let us define the kinetic energy T and potential
More informationA New Integrable Couplings of Classical-Boussinesq Hierarchy with Self-Consistent Sources
Commun. Theor. Phys. Beijing, China 54 21 pp. 1 6 c Chinese Physical Society and IOP Publishing Ltd Vol. 54, No. 1, July 15, 21 A New Integrable Couplings of Classical-Boussinesq Hierarchy with Self-Consistent
More informationResearch Article A Note about the General Meromorphic Solutions of the Fisher Equation
Mathematical Problems in Engineering, Article ID 793834, 4 pages http://dx.doi.org/0.55/204/793834 Research Article A Note about the General Meromorphic Solutions of the Fisher Equation Jian-ming Qi, Qiu-hui
More informationYong Chen a,b,c,qiwang c,d, and Biao Li c,d
Jacobi Elliptic Function Rational Expansion Method with Symbolic Computation to Construct New Doubly-periodic Solutions of Nonlinear Evolution Equations Yong Chen abc QiWang cd and Biao Li cd a Department
More informationSolitary Wave Solution of the Plasma Equations
Applied Mathematical Sciences, Vol. 11, 017, no. 39, 1933-1941 HIKARI Ltd, www.m-hikari.com https://doi.org/10.1988/ams.017.7609 Solitary Wave Solution of the Plasma Equations F. Fonseca Universidad Nacional
More informationGeneralized bilinear differential equations
Generalized bilinear differential equations Wen-Xiu Ma Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620-5700, USA Abstract We introduce a kind of bilinear differential
More informationFibonacci tan-sec method for construction solitary wave solution to differential-difference equations
ISSN 1 746-7233, England, UK World Journal of Modelling and Simulation Vol. 7 (2011) No. 1, pp. 52-57 Fibonacci tan-sec method for construction solitary wave solution to differential-difference equations
More informationAdditive resonances of a controlled van der Pol-Duffing oscillator
Additive resonances of a controlled van der Pol-Duffing oscillator This paper has been published in Journal of Sound and Vibration vol. 5 issue - 8 pp.-. J.C. Ji N. Zhang Faculty of Engineering University
More informationThe Modified (G /G)-Expansion Method for Nonlinear Evolution Equations
The Modified ( /-Expansion Method for Nonlinear Evolution Equations Sheng Zhang, Ying-Na Sun, Jin-Mei Ba, and Ling Dong Department of Mathematics, Bohai University, Jinzhou 11000, P. R. China Reprint requests
More informationAnti-synchronization of a new hyperchaotic system via small-gain theorem
Anti-synchronization of a new hyperchaotic system via small-gain theorem Xiao Jian( ) College of Mathematics and Statistics, Chongqing University, Chongqing 400044, China (Received 8 February 2010; revised
More informationJACOBI ELLIPTIC FUNCTION EXPANSION METHOD FOR THE MODIFIED KORTEWEG-DE VRIES-ZAKHAROV-KUZNETSOV AND THE HIROTA EQUATIONS
JACOBI ELLIPTIC FUNCTION EXPANSION METHOD FOR THE MODIFIED KORTEWEG-DE VRIES-ZAKHAROV-KUZNETSOV AND THE HIROTA EQUATIONS ZAI-YUN ZHANG 1,2 1 School of Mathematics, Hunan Institute of Science Technology,
More informationThe (G'/G) - Expansion Method for Finding Traveling Wave Solutions of Some Nonlinear Pdes in Mathematical Physics
Vol.3, Issue., Jan-Feb. 3 pp-369-376 ISSN: 49-6645 The ('/) - Expansion Method for Finding Traveling Wave Solutions of Some Nonlinear Pdes in Mathematical Physics J.F.Alzaidy Mathematics Department, Faculty
More informationTraveling Wave Solutions For Three Non-linear Equations By ( G G. )-expansion method
Traveling Wave Solutions For Three Non-linear Equations By ( )-expansion method Qinghua Feng Shandong University of Technology School of Science Zhangzhou Road 1, Zibo, 55049 China fqhua@sina.com Bin Zheng
More informationSolitary Shock Waves and Periodic Shock Waves in a Compressible Mooney-Rivlin Elastic Rod
Solitary Shock Waves and Periodic Shock Waves in a Compressible Mooney-Rivlin Elastic Rod Hui-Hui Dai Department of Mathematics, City University of Hong Kong Email: mahhdai@cityu.edu.hk This is a joint
More informationTraveling Wave Solutions for a Generalized Kawahara and Hunter-Saxton Equations
Int. Journal of Math. Analysis, Vol. 7, 2013, no. 34, 1647-1666 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2013.3483 Traveling Wave Solutions for a Generalized Kawahara and Hunter-Saxton
More informationJournal of Inequalities in Pure and Applied Mathematics
Journal of Inequalities in Pure and Applied Mathematics THE METHOD OF LOWER AND UPPER SOLUTIONS FOR SOME FOURTH-ORDER EQUATIONS ZHANBING BAI, WEIGAO GE AND YIFU WANG Department of Applied Mathematics,
More informationNew solutions for a generalized Benjamin-Bona-Mahony-Burgers equation
AMERICAN CONFERENCE ON APPLIED MATHEMATICS (MATH '8) Harvard Massachusetts USA March -6 8 New solutions for a generalized Benjamin-Bona-Mahony-Burgers equation MARIA S. BRUZÓN University of Cádiz Department
More informationSolution of the Coupled Klein-Gordon Schrödinger Equation Using the Modified Decomposition Method
ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.4(2007) No.3,pp.227-234 Solution of the Coupled Klein-Gordon Schrödinger Equation Using the Modified Decomposition
More informationMath 216 First Midterm 18 October, 2018
Math 16 First Midterm 18 October, 018 This sample exam is provided to serve as one component of your studying for this exam in this course. Please note that it is not guaranteed to cover the material that
More informationCitation Acta Mechanica Sinica/Lixue Xuebao, 2009, v. 25 n. 5, p The original publication is available at
Title A hyperbolic Lindstedt-poincaré method for homoclinic motion of a kind of strongly nonlinear autonomous oscillators Author(s) Chen, YY; Chen, SH; Sze, KY Citation Acta Mechanica Sinica/Lixue Xuebao,
More informationComputational Solutions for the Korteweg devries Equation in Warm Plasma
COMPUTATIONAL METHODS IN SCIENCE AND TECHNOLOGY 16(1, 13-18 (1 Computational Solutions for the Korteweg devries Equation in Warm Plasma E.K. El-Shewy*, H.G. Abdelwahed, H.M. Abd-El-Hamid. Theoretical Physics
More informationENVELOPE SOLITONS, PERIODIC WAVES AND OTHER SOLUTIONS TO BOUSSINESQ-BURGERS EQUATION
Romanian Reports in Physics, Vol. 64, No. 4, P. 95 9, ENVELOPE SOLITONS, PERIODIC WAVES AND OTHER SOLUTIONS TO BOUSSINESQ-BURGERS EQUATION GHODRAT EBADI, NAZILA YOUSEFZADEH, HOURIA TRIKI, AHMET YILDIRIM,4,
More informationGeneralized projective synchronization of a class of chaotic (hyperchaotic) systems with uncertain parameters
Vol 16 No 5, May 2007 c 2007 Chin. Phys. Soc. 1009-1963/2007/16(05)/1246-06 Chinese Physics and IOP Publishing Ltd Generalized projective synchronization of a class of chaotic (hyperchaotic) systems with
More informationExact Solutions for a BBM(m,n) Equation with Generalized Evolution
pplied Mathematical Sciences, Vol. 6, 202, no. 27, 325-334 Exact Solutions for a BBM(m,n) Equation with Generalized Evolution Wei Li Yun-Mei Zhao Department of Mathematics, Honghe University Mengzi, Yunnan,
More information2. The generalized Benjamin- Bona-Mahony (BBM) equation with variable coefficients [30]
ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.12(2011) No.1,pp.95-99 The Modified Sine-Cosine Method and Its Applications to the Generalized K(n,n) and BBM Equations
More informationResearch Article Application of the Cole-Hopf Transformation for Finding Exact Solutions to Several Forms of the Seventh-Order KdV Equation
Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 21, Article ID 194329, 14 pages doi:1.1155/21/194329 Research Article Application of the Cole-Hopf Transformation for Finding
More informationA NEW APPROACH FOR SOLITON SOLUTIONS OF RLW EQUATION AND (1+2)-DIMENSIONAL NONLINEAR SCHRÖDINGER S EQUATION
A NEW APPROACH FOR SOLITON SOLUTIONS OF RLW EQUATION AND (+2-DIMENSIONAL NONLINEAR SCHRÖDINGER S EQUATION ALI FILIZ ABDULLAH SONMEZOGLU MEHMET EKICI and DURGUN DURAN Communicated by Horia Cornean In this
More informationNew families of non-travelling wave solutions to a new (3+1)-dimensional potential-ytsf equation
MM Research Preprints, 376 381 MMRC, AMSS, Academia Sinica, Beijing No., December 3 New families of non-travelling wave solutions to a new (3+1-dimensional potential-ytsf equation Zhenya Yan Key Laboratory
More informationAvailable online at J. Math. Comput. Sci. 2 (2012), No. 1, ISSN:
Available online at http://scik.org J. Math. Comput. Sci. 2 (2012), No. 1, 15-22 ISSN: 1927-5307 BRIGHT AND DARK SOLITON SOLUTIONS TO THE OSTROVSKY-BENJAMIN-BONA-MAHONY (OS-BBM) EQUATION MARWAN ALQURAN
More information7 Planar systems of linear ODE
7 Planar systems of linear ODE Here I restrict my attention to a very special class of autonomous ODE: linear ODE with constant coefficients This is arguably the only class of ODE for which explicit solution
More informationHandling the fractional Boussinesq-like equation by fractional variational iteration method
6 ¹ 5 Jun., COMMUN. APPL. MATH. COMPUT. Vol.5 No. Å 6-633()-46-7 Handling the fractional Boussinesq-like equation by fractional variational iteration method GU Jia-lei, XIA Tie-cheng (College of Sciences,
More informationexp Φ ξ -Expansion Method
Journal of Applied Mathematics and Physics, 6,, 6-7 Published Online February 6 in SciRes. http://www.scirp.org/journal/jamp http://dx.doi.org/.6/jamp.6. Analytical and Traveling Wave Solutions to the
More informationLecture 3. Dynamical Systems in Continuous Time
Lecture 3. Dynamical Systems in Continuous Time University of British Columbia, Vancouver Yue-Xian Li November 2, 2017 1 3.1 Exponential growth and decay A Population With Generation Overlap Consider a
More informationResearch Article Traveling Wave Solutions of the Kadomtsev-Petviashvili-Benjamin-Bona-Mahony Equation
Abstract and Applied Analsis, Article ID 943167, 9 pages http://dx.doi.org/10.1155/2014/943167 Research Article Traveling Wave Solutions of the Kadomtsev-Petviashvili-Benjamin-Bona-Mahon Equation Zhengong
More informationA Novel Three Dimension Autonomous Chaotic System with a Quadratic Exponential Nonlinear Term
ETASR - Engineering, Technology & Applied Science Research Vol., o.,, 9-5 9 A Novel Three Dimension Autonomous Chaotic System with a Quadratic Exponential Nonlinear Term Fei Yu College of Information Science
More informationNew Approach of ( Ǵ/G ) Expansion Method. Applications to KdV Equation
Journal of Mathematics Research; Vol. 6, No. ; ISSN 96-9795 E-ISSN 96-989 Published by Canadian Center of Science and Education New Approach of Ǵ/G Expansion Method. Applications to KdV Equation Mohammad
More informationSymmetry Reductions of (2+1) dimensional Equal Width. Wave Equation
Authors: Symmetry Reductions of (2+1) dimensional Equal Width 1. Dr. S. Padmasekaran Wave Equation Asst. Professor, Department of Mathematics Periyar University, Salem 2. M.G. RANI Periyar University,
More informationHomotopy perturbation method for the Wu-Zhang equation in fluid dynamics
Journal of Physics: Conference Series Homotopy perturbation method for the Wu-Zhang equation in fluid dynamics To cite this article: Z Y Ma 008 J. Phys.: Conf. Ser. 96 08 View the article online for updates
More informationExact Travelling Wave Solutions to the (3+1)-Dimensional Kadomtsev Petviashvili Equation
Vol. 108 (005) ACTA PHYSICA POLONICA A No. 3 Exact Travelling Wave Solutions to the (3+1)-Dimensional Kadomtsev Petviashvili Equation Y.-Z. Peng a, and E.V. Krishnan b a Department of Mathematics, Huazhong
More informationElements of Applied Bifurcation Theory
Yuri A. Kuznetsov Elements of Applied Bifurcation Theory Third Edition With 251 Illustrations Springer Introduction to Dynamical Systems 1 1.1 Definition of a dynamical system 1 1.1.1 State space 1 1.1.2
More informationScattering of Solitons of Modified KdV Equation with Self-consistent Sources
Commun. Theor. Phys. Beijing, China 49 8 pp. 89 84 c Chinese Physical Society Vol. 49, No. 4, April 5, 8 Scattering o Solitons o Modiied KdV Equation with Sel-consistent Sources ZHANG Da-Jun and WU Hua
More informationAsymptotic Position And Shape Of The Limit Cycle In A Cardiac Rheodynamic Model
Applied Mathematics E-Notes, 6(2006), 1-9 c ISSN 1607-2510 Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/ Asymptotic Position And Shape Of The Limit Cycle In A Cardiac Rheodynamic
More informationWeierstraß-Institut. für Angewandte Analysis und Stochastik. Leibniz-Institut im Forschungsverbund Berlin e. V. Preprint ISSN
Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut im Forschungsverbund Berlin e. V. Preprint ISSN 198-8 Construction of generalized pendulum equations with prescribed maximum
More information