DOI 10.1007/s11182-017-0966-1 Russian Physics Journal, Vol. 59, No. 10, February, 2017 (Russian Original No. 10, October, 2016)

BRIEF COMMUNICATIONS KINK-TYPE SOLUTIONS OF DISTURBED BURGERS’ EQUATION M. A. Knyazev

UDC 530.182

Keywords: kink, Burgers' equation, Hirota’s method.

Recently, several authors [1, 2] have  considered generalization of the non-linear Burgers' equation. The approach used in these works accounts not only for the first-order but also the second-order members when deriving a generalized equation with respect to a small parameter. The equation obtained in this way is interesting in terms of the integrability of non-linear problems. In dimensionless coordinates x and t the generalized Burgers' equation takes the form [3, 4]

 ut  uu x  u xx    22      uu xx   21      u x2  –

  1  2  2  u u x     2 u xxx  , 2 





(1)

where , , ,  are dimensionless physical parameters; 1 ,  2 are arbitrary parameters; ut  u t and so on. For a special case of 1  2        , the following transformations are introduced [2]:

x  x 

  2   . u  t , t     2 t , u     3      6     





The use of new variables gives Eq. (1) in the form

ut  3  uux  x  3u 2ux  uxxx  0 , where  



   2



2     

(2)

. In Eq. (2) primes are omitted. This equation is called the disturbed Burgers' equation. The

kink solution of Eq. (2) is known for propagating waves [4]:

Belarusian National Technical University, Minsk, the Republic of Belarus, e-mail: [email protected]. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 172–173, October, 2016. Original article submitted March 25, 2016. 1064-8887/17/5910-1715 2017 Springer Science+Business Media New York

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u  x 

3  9  8 B tanh  B  z  z0   , 2

where z  x  C0t ; C0 is the kink velocity; z0 is the initial position of the kink; B 

(3)

2C0  9  3 9  8  4

.

The method of simplest equation is used to obtain a solution (3) [5]. In order to obtain a new solution to Eq. (2), the Hirota’s method is used [6]. Let us introduce the new dependent variable

u

Fx , F

(4)

where F  F  x, t  is the new unknown function;  is the constant which will be determined below. Substituting Eq. (4) into Eq. (2) and requiring the fulfillment of the condition

 2  3  2  0 ,

(5)

Fxt F 2  Fx Ft F  AFxx2  DFx Fxxx F  Fxxxx F 2  CFx2 Fxx  0 ,

(6)

Eq. (2) can be written as

where A  3    1 ; D  3  4 ; C  15  3 2  12 . Let us represent F as formal series

F  1   f1  2 f 2  ,

(7)

where f i  f i  x, t  , i  1,2,3, , are new unknown functions;  is a non-small parameter. Substituting Eq. (7) into Eq. (6) and equating the coefficients to zero at equal powers of  , the infinite system of equations is obtained. The first three equations of this system have a form:

f1, xt  f1, xxxx  0 ,

(8)

f 2, xt  f 2, xxxx  f1, x f1,t  Af1,2xx  Df1, x f1, xxx  0 ,

(9)

f 3, xt  f 3, xxxx  f1, x f 2,t  f1,t f 2, x  f1 f 2, xxxx  f1 f 2, xt  2 Af1, xx f 2, xx 2

+ Df1, x f 2, xxx  Df1, xxx f 2, x  Cf1, x f1, xx .

(10)

For the solution corresponding to the single kink we use the function f1 in the following form:

f1  exp  kx  t  0  ,

(11)

where k , , 0 are the certain parameters. Substituting Eq. (11) in Eq. (8), we get the dispersion relation of the form:

  k3.

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(12)

In accordance with the Hirota’s method, the right-hand side of Eq. (9) should go to zero. Consequently,

   A  Dk3 .

(13)

For matching the conditions of (12) and (13), the condition of A  D  1 should be satisfied, whereof it follows that

  1 . Using this value, we obtain   1 from (5). The value of  is obtained at the same  value as the solution (3). The final solution of (2) can be written as

k  kx   t  0   u  x, t    1  tanh   . 2 2  

(14)

since k is an arbitrary parameter, we see that the function

u  x, t  

k  kx   t  0   1  tanh    2 2  

(15)

is also the solution of Eq. (2). These solutions describe the kink-type states as along with the kink they contain a constant summand relevant to the vacuum contribution. Such solutions arise when damping is taken into account in the theory 4 [7] and in the model describing the deformation of construction composites [8]. Unlike these two models for which the bound states of the two kink solutions can be constructed, such solutions cannot be obtained for the disturbed Burgers' equation at   1 and Sharma–asso–Olver equation [9] using the Hirota’s method.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9.

M. B. Kochanov and N. A. Kudryashov, Rep. Math. Phys., 74, 399 (2014). N. A. Kudryashov and D. I. Sinelshchikov, J. Math. Phys. 55, 103504 (2014). N. A. Kudryashov and D. I. Sinelshchikov, Wave Mot., 50, 351 (2013). N. A. Kudryashov and D. I. Sinelshchikov, Int. J. Nonlin. Mech., 63, 31 (2014). N. A. Kudryashov, Chaos Soliton Fract., 24, 1217 (2005). M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform [Russian translation], Mir, Мoscow (1982). M. A. Knyazev, Kinks in Scalar Model with Damping [in Russian], Tekhnalogiya, Minsk (2003). M. A. Knyazev, Solitons in Non-linear Elastoplastic Model [in Russian], Belarusian National Technical University, Minsk (2013). M. A. Knyazev, Russ. Phys. J., 54, No. 3, 391 (2011).

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