Application of the Kudryashov method for finding exact solutions of the high order nonlinear evolution equations
aa r X i v : . [ n li n . S I] A ug Application of the Kudryashov method for finding exactsolutions of the high order nonlinear evolution equations
Pavel N. Ryabov ∗ , Dmitry I. Sinelshchikov, Mark B. KochanovDepartment of Applied Mathematics, National Research Nuclear University MEPHI,31 Kashirskoe Shosse, 115409 Moscow, Russian Federation Abstract
The application of the Kudryashov method for finding exact solutions of the high order non-linear evolution equations is considered. Some classes of solitary wave solutions for the families ofnonlinear evolution equations of fifth, sixth and seventh order are obtained. The efficiency of theKudryashov method for finding exact solutions of the high order nonlinear evolution equations isdemonstrated.
Keywords:
Kudryashov method; Nonlinear evolution equations; Nonlinear differential equations;Ordinary differential equations; Exact solutions.PACS 02.30.Hq - Ordinary differential equations
The powerful and effective method for finding exact solutions of nonlinear ordinary differentialequations was proposed in work [1]. In works [2–4] author applied this method to construct theexact solutions of the nonlinear nonintegrable equations. The first modification of this method waspresented in work [5]. The final and the most successful modification of this method was proposedin [6]. Thus, we refer this method as the Kudryashov method. The Kudryashov method allow us inthe straightforward manner to construct solitary wave solutions for a wide class of nonlinear ordinarydifferential equations. The main idea of the Kudryashov method is to use special form of the singularitymanifold in the truncation method [1–5]. This approach allows us to reduce the problem of constructingexact solutions to solving the overdetermined system of algebraic equations. The main advantage of theKudryashov method is that we can more effectively construct exact solutions of high order nonlinearevolution equations in comparison with other methods for finding exact solutions [7–13].Using this method in works [1–3] exact solutions of the generalized Kuramoto–Sivashinsky equa-tion, the Burgers–Korteweg–de Vries eqution, the Bretherton eqaution and the Kawahara equationwere obtained. Traveling wave solutions for class of third order nonlinear evolution equations wereconstructed in [14] with help of the Kudryashov method. The Kudryashov method was used forconstructing traveling wave solutions of several nonlinear evolution equation in works [15, 16] as well.The aim of this work is to demonstrate efficiency of the Kudryashov method for finding exactsolitary wave solutions of high order nonlinear evolution equations. For this purpose we consider three ∗ E-mail: [email protected]
The aim of this section is to present the algorithm of the Kudryashov method for finding exactsolutions of the nonlinear evolution equations. To reach this purpose we will follow the works [5,6,14].Let us consider the nonlinear partial differential equation in the form E [ u t , u x , . . . , x, t ] = 0 (1)Using the following ansatz u ( x, t ) = y ( z ) , z = kx − ωt. (2)from Eq. (1) we obtain the ordinary nonlinear differential equation E [ − ωy z , ky z , k y zz , k y zzz , . . . ] = 0 . (3)Now we show how one could obtain the exact solution of the Eq. (3) using the approach byKudryashov. This method consist of the following steps [5, 6, 14]. The first step. Determination of the dominant terms.
To find dominant terms we substitute y = z p , (4)into all terms of Eq. (3). Then we compare degrees of all terms in Eq. (3) and choose two or more withthe smallest degree. The minimum value of p define the pole of Eq. (3) solution and we denote it as N . We have to point out that method can be applied when N is integer. If the value N is nonintegerone can transform the equation studied and repeat the procedure. The second step. The solution structure .We look for exact solution of Eq. (3) in the form y = a + a Q ( z ) + a ( z ) Q ( z ) + ... + a N Q ( z ) N , (5)where a i - unknown constants, Q ( z ) is the following function Q ( z ) = 11 + e z . (6)This function satisfies to the first order ordinary differential equation Q z = Q − Q. (7)The Eq. (7) is necessary to calculate the derivatives of function y ( z ). The third step. Derivatives calculation . 2e should calculate all derivatives of function y . One can do it using the computer algebrasystems Maple or Mathematica. As an example we consider following case: The derivatives of function y ( z ) in the case of N = 2 can be written in the form y = a + a Q + a Q ,y z = − a Q + ( a − a ) Q + 2 a Q ,y zz = a Q + (4 a − a ) Q + (2 a − a ) Q + 6 a Q . (8)The relations (8) can be generalized for any value of N . Differentiating the expression (5) withrespect to z and taking into account (7) we have y z = N X i =1 a i i ( Q − Q i ,y zz = N X i =1 a i i (( i + 1) Q − (2 i + 1) Q + i ) Q i . (9)The high order derivatives of function y ( z ) can be found in works [5, 6]. The fourth step. Defining the values of unknown parameters .We substitute expressions (9) in Eq. (1). After it we have take into account (5). Thus Eq. (1)takes the form P [ Q ( z )] = 0 , (10)where P [ Q ( z )] - is a polynomial of function Q ( z ). Then we collect all items with the same powers offunction Q ( z ) and equate this expressions equal to zero. As a result we obtain algebraic system ofequations. Solving this system we get the values of unknown parameters.The Kudryashov method is a very powerful method for finding exact solutions of the nonlineardifferential equations. It has a set of advantages. They are:1. The first step is not necessary, because all redundant terms in (5) becomes equal to zero whenwe start to define unknown parameters from the system of algebraic equations;2. We construct the solution as a set of Q -blocks. The function Q does not contain any parameters.It is very comfortable, because all of them is in equation;3. Given method can be easily programmed in Maple or Mathematica, because we use the substi-tutions (5) and (9) in Eq. (3) it takes the polynomial form;4. This method is powerful and effective even if we construct the exact solutions of the high ordernonlinear evolution equations.5. It easy to show that tanh, coth, (G’/G) methods and Kudryashov method can be reduced to eachother [6]. Moreover the Kudryashov method gives the same results as an Exp-function method.However it is well known that Exp-function method can not be applied for the equations of hightorder;Let us give several examples to demonstrate it’s efficiency.3 Exact solitary wave solutions of the fifth order evolutionequation
As an example let us consider the fifth order nonlinear evolution equation in the form u t + uu x + 10 uu xxx + 20 u x u xx + 30 u u x + αu xx + βu xxx ++ γu xxxx + u xxxxx = 0 . (11)Eq. (11) is new and does not present in the periodic literature. However it is an interesting equationbecause this equation consist of the fifth order Korteweg-de Vries equation with additional dispersiveand dissipative terms. The equation (11) can arise in the physical applications when we consider thewave processes in active dispersive-dissipative media.Using the traveling waves (2) we have − wy z + kyy z + 10 k yy zzz + 20 k y z y zz + 30 ky y z + αk y zz + βk y zzz ++ γk y zzzz + k y zzzzz = 0 . (12)The pole of the Eq. (12) is equal to N = 2, thus we look for exact solution in the form y = a + a Q + a Q , (13)where a , a and a – are unknown constants.Substituting (9) in Eq. (12) and taking into account (13) we obtain the polynomial of function Q ( z ). Collecting all terms with the same power of function Q ( z ) and equate this expressions to zerowe obtain the system of algebraic equations. Solving this system we find that solution of Eq. (12)exists only in eight cases. They are k (1 , = ± γ , k (3 , = ± γ , k (5 , = ± p γ + 2646 β − ,k (7 , = ± γ p γ (972 γ + 343 α ) . (14)However in the case of k = k (5 , and k = k (7 , the parameters and solution presentation is verycumbersome and we do not give them in the present manuscript.In the case of k = k (1 , and k = k (3 , the values of parameters α, β, w, a , a , a are k = γ , α = 584 γ − γβ − γ ,a = β
20 + 16 γ − , a = 4147 γ , a = − γ ,ω = γ (cid:0) γ − γ − − β + 62496 βγ + 111132 β (cid:1) , (15) k = − γ , α = 584 γ − γβ − γ ,a = β
20 + 46 γ − , a = 0 , a = − γ ,ω = − γ (cid:0) γ − γ − − β + 62496 βγ + 111132 β (cid:1) , (16)4 = 3 γ , β = 16 − γ ,a = 7 α γ −
160 + 2221225 γ , a = 0 , a = − γ ,ω = 67228 α − αγ + 347328 γ − γ γ , (17) k = − γ , β = 16 − γ ,a = 7 α γ − − γ , a = 3649 γ , a = − γ ,ω = − α − γ α + 347328 γ − γ γ , (18)The solutions of the Eq. (12) which corresponds to the relations (15)–(18) are y ( z ) = a + 2147 γ [2 − Q ( z )] Q ( z ) , (19) y ( z ) = a − γ Q ( z ) , (20) y ( z ) = a − γ Q ( z ) , (21) y ( z ) = a + 18 γ
49 [2 − Q ( z )] Q ( z ) . (22)Graphical presentation of solutions (19), (20) is shown on Fig. 1.Figure 1: Exact solutions (19) and (20) – 1, 2 respectively at β = γ = 1 . The solutions (21), (22) have the same structure so we have decided not to picture them.
Let us consider the following equation u t + uu x + αu xx + βu xxxx + γu xxxxxx = 0 . (23)This equation was proposed in work [17] for describing the longitudinal seismic waves in a vis-coelastic medium. Moreover this equation is used for modeling the ”soft” – type of the turbulence [18].Also this equation describes the chemical reactions in reaction-diffusion systems [19]. The numerical5odeling of the wave processes describing by Eq. (23) was performed in work [20]. Exact solutions ofthis equation was obtained in work [21].Taking into account the traveling waves (2) in Eq. (23) we obtain − ωy z + kyy z + αk y zz + βk y zzzz + γk y zzzzzz = 0 . (24)Integrating the Eq. (24) we have C − ωy + k y αk y z + βk y zzz + γk y zzzzz = 0 . (25)Dominant terms of the Eq. (25) are k y zzzzz , ky /
2. Thus, the pole order of the Eq. (25) solutionis N = 5. So we look for solution in the form y ( z ) = a + a Q + a Q + a Q + a Q + a Q , (26)where a - a – constants to be determined.Using the (9) in Eq. (25) and taking into account ansatz (26) we obtain a system of algebraicequations. Solving this system we find four real families of unknown parameters. However we giveonly three because one of them is cumbersome. The following families are α = − k β, γ = − β k , C = ω k − k β ,a = ωk − k β , a = a = 0 , a = 10080 k β ,a = − k β , a = 6048 k β
11 (27) α = 21944 k β, γ = − βk , C = ω k − k β ,a = ωk + 315 k β , a = 0 , a = − k β , a = 6930 k β,a = − k β , a = 37800 k β
11 (28) α = − βk , γ = − βk , C = ω k − k β ,a = ωk − k β , a = 0 , a = 1890 k β , a = 3780 k β ,a = − k β , a = 3024 k β
11 (29)From relations (27), (28) and (29) we see that the real solution of Eq. (25) exists in the case of k = − αβ , β = 275114 αγ,k = 44219 αβ , β = − αγ,k = − αβ , β = 325912100 αγ. (30)6he solutions of Eq. (25) which corresponds to the relations (27), (28) and (29) takes the form y ( z ) = ωk + βk (cid:0) −
504 + 10080 Q ( z ) − Q ( z ) + 6048 Q ( z ) (cid:1) , (31) y ( z ) = ωk + βk (cid:18) − Q ( z ) + 76230 Q ( z ) − Q ( z ) ++37800 Q ( z ) (cid:1) , (32) y ( z ) = ωk + βk (cid:0) −
567 + 1890 Q ( z ) + 3780 Q ( z ) − Q ( z ) ++3024 Q ( z ) (cid:1) . (33)The solutions (31), (32) of the Eq. (23) is presented on Fig. 2. The third solution has the form ofthe kink as well as (31). z y y Figure 2:
Exact solutions (31) and (32) – 1, 2 respectively at β = ω = k = 1 . Let us consider the nonlinear evolution equation of seventh order u t + u n u x + αu xxx + βu xxxxx + γu xxxxxxx = 0 , (34)where n = 1 , , n = 2. We generalizes theresults of this work.Taking the traveling wave ansatz u ( x, t ) = y ( z ) , z = kx − wt into account from Eq.(34) we have − ωy z + ky n y z + αk y zzz + βk y zzzzz + γk y zzzzzzz = 0 . (35)Integrating the Eq. (35) with respect to variable z we obtain C − ωy + k y n +1 n + 1 + αk y zz + βk y zzzz + γk y zzzzzz = 0 . (36)7n the case of n = 1 we have to look exact solution of Eq. (36) in the form y ( z ) = a + a Q + a Q + a Q + a Q + a Q + a Q , (37)because the pole order of Eq. (36) solution is N = 6.Substituting (9) in Eq. (36) with n = 1 and taking (37) into account we found the followingsolutions y ( z ) = ωk + γk (cid:0) − Q ( z ) + 332640 Q ( z ) −− Q ( z ) + 1995840 Q ( z ) − Q ( z ) (cid:1) , (38) y ( z ) = ωk + γk (cid:0) − Q ( z ) − Q ( z ) +1995840 Q ( z ) − Q ( z ) (cid:1) , (39)The corresponding parameters of Eq. (36) and (37) are β = − k γ, α = 2159 k γ, C = ω k − k γ ,a = ωk − k γ, a = 0 , a = 166320 k γ, a = 332640 k γ,a = − k γ, a = 1995840 k γ, a = − k γ, (40) β = − k γ, α = 769 k γ, C = ω k − k γ ,a = ωk − k γ, a = a = 0 , a = 665280 k γ,a = − k γ, a = − a , a = − a , (41)where β/γ <
0. Thus, the solutions (38) and (39) exist when k = − β γ , β = 10002159 αγ,k = − β γ , β = 2500769 αγ. (42)The illustration of (38) is given on Fig. 3. y Figure 3:
Exact solution (38) at ω = 10 , γ = 0 . , k = 1 . The solution (39) has the same form. 8et us consider the case of n = 2. The pole order of Eq. (36) solution in that case is equal to N = 3. Thus, we look for exact solution of Eq. (36) in the form y ( z ) = a + a Q + a Q + a Q , (43)where a , a , a , a the constants to be determined.Taking into account (9) and relations (43) in Eq. (36) we have the solution in the form y ( z ) = ± p − γk (cid:0) − Q ( z ) + 4 Q ( z ) (cid:1) , (44)where γ < β, α, w, C are defined by relations β = − k γ, α = 946 k γ, w = − γk , C = 0 ,a = ± p − γk , a = 0 , a = ∓ p − γk ,a = ± p − γk , (45)So, the solution (44) exist when the wave number satisfies to equivalence k = − β γ , β = 6889946 αγ. (46)The solution (44) is a kink, see Fig. 4. z -80800 y Figure 4:
Exact solution (44) at γ = − , k = 1 . In the case of n = 3 the pole order of the Eq. (36) solution is N = 2. So, we use the followingansatz y ( z ) = a + a Q + a Q , (47)where a , a , a – unknown constants.Then we substitute (9) and (47) in Eq. (5.3) with n = 3. Collecting all terms with the same powerof function Q ( z ) and equate them to zero we obtain the system of algebraic equations on unknown9arameters. Solving this system we have α = 14 k γ + 5 β γ , w = k γ (cid:0) k γ − βγ k − β (cid:1) ,C = √ kγ / (cid:0) β − k γ + 9408 k γ β + 392 k γ β (cid:1) ,a = − √ βγ / − √ γk , a = 4 p γk , a = − a . (48)The solution of Eq. (36) with n = 3 which corresponds to (48) takes the form y ( z ) = − √ βγ / − √ γk (cid:0) − Q ( z ) + 12 Q ( z ) (cid:1) . (49)This (49) has the form of the traveling wave (Fig. 5). z -34 y Figure 5:
Exact solution (49) at β = γ = k = 1 . In this work we have demonstrated efficiency of the Kudryashov method for finding exact solutionsof high order nonlinear evolution equations. We have obtained solitary wave solutions for the threefamilies of nonlinear evolution equations of fifth, six and seven orders. Graphical representation ofthese exact solution is presented. We believe that some of exact solutions obtained in this work arenew.
This work was supported by the federal target programm ”Research and scientific-pedagogicalpersonnel of innovation in Russia” on 2009-2011.