About the supersymmetric Burgers equation and its traveling wave solutions Alvaro H. Salas Department of Mathematics Universidad de Caldas, Manizales, Colombia. Universidad Nacional de Colombia, Manizales. email :
[email protected] FIZMAKO Research Group Abstract.
We find exact soliton solutions to a new integrable coupled system which corresponds to fermionic extension of the Burgers equation. This extension is related with the Burgers flows on associative algebras. One, two, three and tour soliton solutions are formally derived.
Keywords and phrases. Nonlinear equation, Burgers equation, fermionic extension, coupled system, supersymmetric equation, soliton solution, superderivative, two-soliton solution, three-soliton solution, four–soliton solution, Mathematica 7, Maple 13. 2000 Mathematics Subject Classification. 35C05.
1. Introduction As it was remarked in [3], in 1915, Harry Bateman considered a nonlinear equation whose steady solutions were thought to describe certain viscous flows [4]. This equation, modeling a diffusive nonlinear wave, is now widely known as the Burgers equation, and is given by ut uu x u xx where is a constant measuring the viscosity of the fluid. It is a nonlinear parabolic equation, simply describing a temporal evolution where nonlinear convection and linear diffusion are combined, and it can be derived as a weakly nonlinear approximation to the equations of gas dynamics. Although nonlinear, Equation (1) is very simple, and interest in it was revived in the 1940s, when Dutch physicist Jan Burgers proposed it to describe a mathematical model of turbulence in gas [5]. As a model for gas dynamics, it was then studied extensively by Burgers [6], Eberhard Hopf [7], Julian Cole [8], and others, in particular; after the discovery of a coordinate transformation that maps it to the heat equation. While as a model for gas turbulence the equation was soon rivaled by more complicated models, the linearizing transformation just mentioned added importance to the equation as a mathematical model, which has since been extensively studied. The limit 0 is an hyperbolic equation, called the inviscid Burgers equation: ut uu x 0 This limiting equation is important because it provides a simple example of a conservation law, capturing the crucial phenomenon of shock formation. Indeed, it was originally introduced as a model to describe the formation of shock waves in gas dynamics. A first-order partial differential equation for u ( x t ) is called a conservation law if it can be written in the form ut ( f (u )) x 0 . For Equation (2), f (u ) u 2 2 . Such conservation laws may exhibit the formation of shocks, which are discontinuities
(1)
(2)
appearing in the solution after a finite time and then propagating in a regular manner. When this phenomenon arises, an initially smooth wave becomes steeper and steeper as time progresses, until it forms a jump discontinuity–the shock. Nowadays, the Burgers equation is used as a simplified model of a kind of hydrodynamic turbulence [9], called Burgers turbulence. Burgers himself wrote a treatise on the equation now known by his name [10], where several variants are proposed to describe this particular kind of turbulence. Equation (1) was originally derived to describe the propagation of nonlinear waves in dissipative media, where ( 0 ) is the kinematic viscosity, and u ( x t ) represents the fluid velocity field. It plays an active role in explaining two fundamental effects characteristic of any turbulence: the nonlinear redistribution of energy over the spectrum and the action of viscosity in small scales. Over the decades, the Burgers equation has been widely used to model a large class of physical systems in which the nonlinearity is fairly weak (quadratic) and the dispersion is negligible compared to the linear damping. Hopf [7] and Cole [8] independently discovered a transformation that reduces the Burgers equation (1) to a linear diffusion equation. First, we write (1) in a form similar to a conservation law 1 ut u 2 u x 0 x 2 This may be regarded as the compatibility condition for a function to exist, such that u x
1 2
ux u 2 t We substitute the value of u from (4) in (5) to obtain 1 xx 2x t 2 Next, we introduce 2 log so that u x 2
x
(3)
(4)
(5)
(6)
(7)
This is called the Cole-Hopf transformation which, by differentiating, gives 2
2 xx 2 x xx and t 2 t Consequently, (6) reduces to the linear heat equation t xx The relation between the Burgers and the heat equation was already mentioned in an earlier book [11], but the former had not been recognized as physically relevant; hence, the importance of this connection was seemingly not noticed at the time. Using the transformation of Equation (1), known as the Cole–Hopf transformation, it is easy to solve the initial value problem for this equation. Recently, a generalization of the Cole– Hopf transformation has been successfully used to linearize the boundary value problem for the Burgers equation posed on the semiline x 0 [12]. Many solutions of equation (9) are well known in the literature [13][23]
2. Fermionic extension of the Burgers equation Let us consider the fermionic extension of the Burgers equation (also called
(8)
(9)
supersymmetric Burgers equation) [1], ft f xx (bf ) x R bt bxx bbx f x f
(10)
Note that system (10) is the Burgers equation itself if the fermionic field f ( x t ) is set to zero. System (10) models an unusual physical phenomenon. Assume that at any point x R there are two types of a physical value, the bosonic field with density b( x t ) and the “invisible” fermionic field with density w( x t ) , and let the dynamics of these two fields be described by system (10) (or equation (11) below). Suppose that the initial numeric values w( x 0) and b( x 0) of the fermionic and bosonic densities, respectively, coincide at t 0 . If 0 , then densities will coincide for all t 0 and the corresponding integral quantities
w( x t ) dx and
b( x t ) dx will be conserved in time. In this case
equation (10) reduces to f t f xx (bf ) x
bt bxx bbx
As we remarked in the previous section, the bosonic component b( x t ) in (11) is linearized by the Cole–Hopf substitution b 2q 1qx where the function q ( x t ) is a solution of the heat equation qt qxx . From (12) it follows that the Burgers equation is the factor of the heat equation with respect to (briefly, w.r.t.) its scaling symmetry. This scheme is of general nature and can be used for constructing new equations from homogeneous systems. If 0 , then the feed-back is switched on in (10). A ripple in the fermionic space is
the cause for the bosonic integral longer conserved unless
(11)
(12)
b( x t ) dx to change. Indeed, this quantity is no
w( x t ) const. or, generally, unless the condition
wxx ( x t ) wx ( x t ) dx const. holds for all t 0 . We see that the reaction of the
fermionic component on the bosonic field depends on the incline wx f and curvature wxx f x but not on the density w( x t ) . The fermionic component in (10) is linear w.r.t. the field f ( x t ) , and hence the superposition principle is valid for it. The quantity
f ( x t )dx const
is an integral of motion for (10). The fermionic variable w( x t ) is assigned to the conserved current Dt ( f ) Dx ( f x bf ) .
3. Traveling wave solutions to fermionic extension of the Burgers equation In this section we obtain soliton solutions to system (10).
3.1. One-soliton solutions. Let
(13)
kx t 0 k 0 0 arbitrary constant f f ( x t ) f ( ) and b b( x t ) b( ) where k and are some constants to be determined. Substituting (14) in (10) yields k 2 f ( ) kb( ) f ( ) kb ( ) f ( ) f ( ) 0 2 k b ( ) kb( )b ( ) b ( ) kf ( ) f ( ) 0 Integrating both equations w.r.t. gives ( kb( )) f ( ) k 2 f ( ) C 2 2 2 2b( ) kb ( ) kf ( ) 2k b ( ) D where C and D are the constants of integration. Solving the first equation of (16) w.r.t. b( ) we get
C f ( ) k 2 f ( ) kf ( ) Now, we replace (17) in the second equation of (16) to obtain C 2 4Ck 2 f ( ) Dkf ( ) 2 2k 4 f ( ) f ( ) b( )
3k 4 f ( ) 2 k 2 f ( ) 4 2 f ( ) 2 0 Thus, in order to solve the supersymmetric Burgers equation (11), it is enough to solve equation (18). To this end, we will suppose that a solution to (18) has the form q f ( ) p q 0 r 0 1 r exp( ) Substituting (19) into (18) we obtain a polynomial equation in the variable exp( ) .
Equating to zero the coefficients of i ( i 01 2 3 ) yields an algebraic system. This system reads C 2 p 2 k 2 p 2 2 Dk 0 2C 2 2Ck 2 q kp 2kp 2 ( p q ) q D k 3 2 Dp p 2 (2 p q ) 0 2 2 2 2 4 2 2 2 2 2 2 2 6C 8Ck q Dk 6 p 6 pq q k q 6k p ( p q ) 6 p 6 pq q 0 2C 2 2Ck 2 q ( p q) k 4 q 2k 2 p ( p q ) 2 2 (2 p q ) Dk (2 p q ) 0 C 2 ( p q ) 2 k 2 ( p q ) 2 2 Dk 0
(20)
Solving system (20) with the aid of either Mathematica 7 or Maple 13 and using (14) and (17) we receive following solutions to system (10) :
k k, p
k 2 k 4 4 Ck 2k
, q
k
, r r , k 4 Dk 2Ck :
(14)
(15)
(16)
(17)
(18)
(19)
k 2 k 4 4Ck k f ( ) 2k 1 r exp C p k k 2 p ( p q ) Cq b ( ) kp 1 r exp kp ( p q pr exp ) 4 kx t 0 kx k Dk 2Ck t 0
k k, p
k 2 k 4 4 Ck 2k
, q
k
, r r , k 4 Dk 2Ck :
k f ( ) p 1 r exp k k 2 p ( p q ) Cq b( ) C p kp 1 r exp kp ( p q pr exp ) 4 kx t 0 kx k Dk 2Ck t 0
k k, p
k 2
, q
k
(21)
, rr, C
3k 3 4
,
5 k 4 2 Dk 2
(22)
:
k k f ( ) 2 1 r exp k k 2 p ( p q ) Cq b( ) C p 1 r exp kp ( p q pr exp ) kp 5k 4 2 Dk kx t 0 kx t 0 2
(23)
k k , p 2 k , q
k
, r r , C 43k , 3
5 k 4 2 Dk 2
:
k k f ( ) 2 1 r exp k k 2 p ( p q ) Cq b( ) C p 1 r exp kp ( p q pr exp ) kp 5k 4 2 Dk kx t 0 kx t 0 2
Previous solutios are valid for 0 . If 0 we always may choose C and D in order to obtain real-valued solutions from (21) - (24). On the other hand, if 0 we take C D 0 and we replace k with k 1 to obtain real-valued solutions from (21) and (22). To obtain real-valued solutions from (23) and (24) in the case when 0 we set D 0 and we replace k with k 1 . Now, let us assume that 0 . In this case we only get trivial solutions from system (20). Equation (18) takes the form C 2 4Ck 2 f ( ) Dkf ( ) 2 2k 4 f ( ) f ( ) 3k 4 f ( ) 2 2 f ( ) 2 0 We seek solutions to equation (25) in the form f ( ) p q tanh 2 ( ) r coth 2 ( ) where qr 0 Substituting (26) into (25) we obtain a polynomial equation in the variable exp( ) . Equating to zero the coefficients of i ( i 01 2 3 ) yields an algebraic system. Solving it and making use of (14) and (17) we receive following solutions to system (10) in the case 0 , i.e. ft f xx (bf ) x R bt bxx bbx
(24)
(25) (26)
(27)
k k , p q , q q , r 0 , C 0 , 4k 4 Dk :
f ( ) q sec h 2 1 4k 4 Dk 2k tanh b( ) k 4 kx t 0 kx 4k Dkt 0
(28)
k k , p 2q , q q , r q , C 0 , 16k 4 Dk :
f ( ) 4q csc h 2 (2 ) 1 16k 4 Dk 4k coth(2 ) b( ) k 4 kx t 0 kx 16k Dkt 0
(29)
3.2. Two-soliton solutions. We seek two-soliton solutions to system (10) in the form f (t x) A x log(1 exp(1 ) exp( 2 )) B b(t x) C x log(1 exp(1 ) exp( 2 )) D k x t k x t k k 0 1 2 2 2 1 2 1 2 1 1
(30)
Substituting (30) into (10) we obtain a polynomial equation in the variables 1 exp(1 ) and 2 exp( 2 ) . Equating to zero the coefficients of 1i2j ( i , j 01 2 3 ) yields an algebraic system. Solving it following solutions are obtained : BB, D D, A 1 , C 1, k1 k1 , k2 k2 ,
1 k1 ( D k1 B ) , 2 k2 ( D k2 B ) , 1 k1 exp(k1 x k1 ( D k1 B )t ) k2 exp(k2 x k2 ( D k2 B )t ) B f (t x) 1 exp(k1 x k1 ( D k1 B )t ) exp(k2 x k2 ( D k2 B )t ) b(t x) k1 exp(k1 x k1 ( D k1 B )t ) k2 exp(k2 x k2 ( D k2 B )t ) D 1 exp(k1 x k1 ( D k1 B )t ) exp(k2 x k2 ( D k2 B )t ) (31)
3.3. Three-soliton solutions. We seek three-soliton solutions to system (10) in the form f (t x) A x log(1 exp(1 ) exp( 2 ) exp(3 )) B b(t x) C x log(1 exp(1 ) exp( 2 ) exp(3 )) D k x t k x t k x t k 0 (i 1 2 3) 1 2 2 2 3 3 3 i i 1 1
Substituting (30) into (10) we obtain a polynomial equation in the variables 1 exp(1 ) , 2 exp( 2 ) and 3 exp(3 ) . Equating to zero the coefficients of
1i2j3k ( i , j , k 01 2 3 ) yields an algebraic system. Solving it following solutions are obtained :
(32)
A
1
, B B , C 1 , D B k3 k33 , k1 k1 , k2 k2 , k3 k3 ,
1 k1 (k1 k3 ) k1 k , 2 k2 (k2 k3 ) k2 , 3
3
3 3 : 1 k1 exp(1 ) k2 exp( 2 ) k3 exp(3 ) f (t x) B 1 exp(1 ) exp( 2 ) exp(3 ) k exp(1 ) k2 exp( 2 ) k3 exp(3 ) b(t x) 1 B k3 3 1 exp(1 ) exp( 2 ) exp(3 ) k3 k x k k k 3 t k x k k k 3 t k x t 1 1 3 2 2 2 2 3 3 3 3 1 1 k3 k3
3.4. Four-soliton solutions We seek four-soliton solutions to system (10) in the form f (t x) A x log(1 exp(1 ) exp( 2 ) exp(3 ) exp( 4 )) B b(t x) C x log(1 exp(1 ) exp( 2 ) exp(3 ) exp( 4 )) D k x t k x t k x t k x t k 0 (i 1 2 3 4) i i 1 2 2 2 3 3 3 4 4 4 1 1 (34)
Substituting (34) into (10) we obtain a polynomial equation in the variables 1 exp(1 ) , 2 exp( 2 ) , 3 exp(3 ) and 4 exp( 4 ) . Equating to zero the coefficients of 1i2j3k4l ( i , j , k , l 01 2 3 ) yields an algebraic system. Solving it following solutions are obtained :
A
1
, B Dk4k k44 , C 1 , D D , k1 k1 , k2 k2 , k3 k3 , 2
4
k k4 k4 , 1 k1 (k1 k4 ) k1k4 4 , 2 k2 (k2 k4 ) k2k4 4 , 3 k3 (k3 k4 ) 3k4 4 ,
4 4 :
(33)
1 k1 exp(1 ) k2 exp( 2 ) k3 exp(3 ) k4 exp( 4 ) Dk4 k42 4 f ( t x ) 1 exp( ) exp( ) exp( ) exp( ) k 1 2 3 4 4 k exp(1 ) k2 exp( 2 ) k3 exp(3 ) k4 exp( 4 ) D b(t x) 1 1 exp(1 ) exp( 2 ) exp(3 ) exp( 4 ) 1 k1 x k1 k1 k4 4 t 2 k2 x k2 k2 k4 4 t k4 k4 3 k3 x k 3 k 3 k 4 4 t 4 k 4 x 4 t k4 (35)
Other methods for solving nonlinear pde’s may be found in [14], [15], [6],..., [39], [31].
4. Conclusions We have obtained traveling wave solutions (one-soliton, two-soliton, three-soliton and four-soliton solutions) to supersymmetric Burgers equation by using elementary methods. We think that results in this work may help in studying the phenomenom described by this equation. In our opinion solutions we obtained here are new in the open literature.
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by using the generalized tanh method, Scientia et Technica, Universidad Tecnológica de Pereira, Risaralda–Colombia Vol 13, No. 035 (2007) pp. 409–410. [29] Salas Alvaro H., Symbolic Computation of solutions for a forced Burgers equation, Applied mathematics and Computation, Volume 204, Issue 1, 1 October 2008, Pages 257-260 [30] Salas Alvaro H., Computing exact solutions to a generalized Lax seventh–order forced KdV equation (KdV7), Applied Mathematics and Computation, march 2010,in press. [31] Gómez C.A., Salas Alvaro H., Exact solutions for the generalized BBM equation with variable coefficients, Mathematical Problems in Engineering, 2010 (in press). [32] Salas Alvaro H., Castillo Hernández Jairo Ernesto & Escobar Lugo José Gonzalo, About the seventh–order Kaup–Kupershmidt equation and its solutions, http://arxiv.org, arXiv:0809.2870, September 2008. [33] Salas Alvaro H. & Gómez C.A. & Castillo Hernández Jairo Ernesto, New solutions for the modified generalized Degasperis–Procesi equation, Applied Mathematics and Computation, Volume 215, Issue 7, 1 December 2009, Pages 26082615. [34] Salas Alvaro H., Some exact solutions for the Caudrey–Dodd–Gibbon equation, http://arxiv.org, arXiv:0805.2969, May 2008. [35] Salas Alvaro H., Gómez C.A., Exact solutions to a KdV equation with variable coefficients and forcing term, Mathematical Problems in Engineeering, august, 2009, in press. [36] Salas Alvaro, Some solutions for a type of generalized Sawada–Kotera equation, Applied Mathematics and Computation, Volume 196, Issue 2, 1 March 2008, pp. 812817. [37] Salas Alvaro H., Gómez C.A., El software Mathematica en la búsqueda de soluciones exactas de ecuaciones diferenciales no lineales en derivadas parciales mediante el uso de la ecuación de Riccati. Memorias del Primer Seminario Internacional de Tecnologías en Educación Matemática, Universidad Pedagógica Nacional, Santafé de Bogotá, Colombia 1 (2005) 379-387. [38] Salas Alvaro H., Gómez C.A., Palomá L.L. Computing exact solutions to reduced Ostrovsky equation, International Journal of Applied Mathematics (IJAM), march 2010, in press. [39] Salas Alvaro H., Exact solutions to coupled sine–Gordon equattion, Nonlinear Analysis : Real World Applications, march 2010, in press. [40] Salas Alvaro H., Gómez C.A., Application of the Cole–Hopf transformation for finding exact solutions for several forms of the seventh-order KdV equation (KdV7), Mathematical Problems in Engineering, 2010 (in press).
