PROCEEDINGS OF THE FOURTH INTERNATIONAL CONFERENCE ON DYNAMICAL SYSTEMS AND DIFFERENTIAL EQUATIONS May 24 – 27, 2002, Wilmington, NC, USA
pp. 477–481
CONSERVATION LAWS AND INVARIANT SOLUTIONS FOR SOIL WATER EQUATIONS
C. M. Khalique and G. S. Pai International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences, University of North-West, Private Bag X 2046, Mmabatho 2735, SOUTH AFRICA Abstract. We determine conservation laws for a class of soil water equations and associate these, where possible, with Lie symmetry generators. One cannot invoke Noether’s theorem here as there is no Lagrangian for these equations. We also obtain exact solutions for such a class of equations. These solutions are invariant under a three-dimensional subalgebra of the symmetry Lie algebra.
1. Introduction. The nonlinear partial differential equation C(ψ)ψt = (K(ψ)ψx )x + (K(ψ)(ψz − 1))z − S(ψ),
(1)
represents a model to simulate soil water infiltration, redistribution, and extraction in a bedded soil profile overlaying a shallow water table and irrigated by a line source drip irrigation system. Here ψ is soil moisture pressure head, C(ψ) is specific water capacity, K(ψ) is unsaturated hydraulic conductivity, S(ψ) is a sink or source term, t is time, x is the horizontal and z is the vertical axis which is considered positive downward (see [1, 2]). Baikov et al [3] performed the group classification of equation (1) with respect to admitted transformation groups. Some exact/asymptotic invariant solutions of equation (1) invariant under two-parameter symmetry groups were obtained by Baikov and Khalique [4]. A special case of (1) was considered for conservation laws in [5]. In this paper we determine conserved vectors by the direct method for soil water equations and then associate symmetries with conservation laws. We also produce new conservation laws by acting with the symmetry of equation (1) on a known conservation law. Finally we obtain exact solutions for equation (1) which are invariant under a three-dimensional subalgebra of the symmetry Lie algebra of equation (1) which was not considered before. From a modelling point of view one can observe that equation (1) itself follows from assuming, within the modelling principles, a conservation law for water content, C(ψ)ψt = θt = divergence of V , where θ is the volumetric water content and V is the Darcian water flux. We adopt the notation, definitions and results given in Kara and Mahomed [6, 7] and Ibragimov et al [8]. 1991 Mathematics Subject Classification. Primary: 35L65, 70S10, 34C14 ; Secondary: 34C20. Key words and phrases. Lie point symmetries, Conservation Laws, Soil Water Equations; ∗ Corresponding author : C. M. Khalique.
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2. Conservation laws for soil water equations. In this section we construct conservation laws for equation (1) for some particular type of coefficients C(ψ), K(ψ) and S(ψ) and associate symmetries where possible. We note that we cannot appeal to the classical Noether theorem [9] here as soil water equations being of evolution type do no possess a Lagrangian. The Lie algebra of the Lie transformation group admitted by equation (1) for arbitrary functions C(ψ), K(ψ) and S(ψ) (called the principal Lie algebra, see, e.g. Baikov et al [3]) is the three dimensional Lie algebra spanned by the operators which generate translations along the t, x and z−axis, namely ∂ ∂ ∂ , X3 = , X1 = , X2 = ∂t ∂x ∂z respectively. Hence the principal Lie group of equation (1) is the three-parameter group of translations. We consider the case when K(ψ) = 1, C(ψ) = ψ σ , where σ is an arbitrary constant and S(ψ) = 0. In this case equation (1) has the form ψt = ψ −σ {ψxx + ψzz } .
(2)
According to the classification result in Baikov et al [3], equation (2) admits a sixdimensional Lie algebra L6 obtained by an extension of the principal Lie algebra Lp by the following three operators: ∂ ∂ −x , ∂x ∂z ∂ ∂ = σt + ψ , ∂t ∂ψ
X4
=z
X5 and
∂ ∂ ∂ + σz − 2ψ . ∂x ∂z ∂ψ We now construct a conserved vector (T 1 , T 2 , T 3 ) of equation (2) with T 1 , T 2 and T 3 being functions of t, x, z, ψ, ψt , ψx and ψz by invoking the equation X6 = σx
Dt T 1 + Dx T 2 + Dz T 3 = 0 on the solutions of equation (2). The solution of the determining equation for the conservation law is ψ σ+1 T 1 = f (x, z) , σ 6= −1, σ+1 T 2 = −f (x, z)ψx + ψfx (x, z), T3
= −f (x, z)ψz + ψfz (x, z),
(3)
where the function f (x, z) satisfies fxx + fzz = 0.
(4)
For the case σ = −1, we obtain T 1 = f (x, z) ln |ψ| and T 2 and T 3 are the same as given in (3) and f satisfies equation (4). We consider only the case σ 6= −1 for association of symmetries. The case σ = −1 can be treated in the same manner. We now appeal to the relations X(T i ) + T i Dj (ξ j ) − T j Dj (ξ i ) = 0 ,
i = 1, 2, 3,
in [7] in order to associate the symmetries with the conserved vectors (3).
CONSERVATION LAWS & INVARIANT SOLUTIONS FOR SOIL WATER EQUATIONS 479
1. For X1 the associated conserved components are given by (3) where f solves (4). 2. For X2 the associated conserved components are T1
=
T2 T3
= =
ψ σ+1 , σ 6= −1, σ+1 −(f0 + f1 z)ψx , −(f0 + f1 z)ψz + f1 ,
(f0 + f1 z)
where f0 and f1 are arbitrary constants. 3. For X3 the associated conserved components are ψ σ+1 , σ 6= −1, σ+1 −(f0 + f1 x)ψx + f1 , −(f0 + f1 x)ψz ,
T1
= (f0 + f1 x)
T2 T3
= =
where f0 and f1 are arbitrary constants. 4. For X4 the associated conserved components are ψ σ+1 , σ 6= −1, σ+1 2d0 xψ T 2 = −[d0 ln(x2 + z 2 ) + d1 ]ψx + 2 , x + z2 2d0 zψ T 3 = −[d0 ln(x2 + z 2 ) + d1 ]ψz + 2 , x + z2 where d0 and d1 are arbitrary constants. 5. For X5 there is no associated nontrivial conservation law. 6. For X6 also there is no associated nontrivial conservation law. We now produce new conservation laws by acting with a symmetry of equation (2) on a known conservation law. Since [X2 , X4 ] = −X3 , we invoke the result in [7] and act with X2 on the conservation law associated with X4 . The new conserved components are µ ¶ σ+1 2d0 x ψ T∗1 = , σ 6= −1, 2 2 xµ + z σ+1 µ ¶ ¶ 2d0 x 2d0 4d0 x2 T∗2 = − ψ + − ψ, x 2 2 2 2 (x2 + z 2 )2 µx + z ¶ µx + z ¶ 2d0 x 4d0 xz T∗3 = − ψz + ψ, x2 + z 2 (x2 + z 2 )2 T1
=
[d0 ln(x2 + z 2 ) + d1 ]
where d0 is an arbitrary constant. Now since [X3 , X4 ] = X2 , we act with X3 on the conservation law associated with X4 and obtain another conservation law, viz., µ ¶ σ+1 2d0 x ψ T†1 = , σ 6= −1, xµ2 + z 2 ¶ σ+1 µ ¶ 4d0 xz 2d0 z ψ − ψ, T†2 = − x 2 2 2 2 2 µx + z ¶ µ (x + z ) ¶ 2d0 z 2d0 4d0 z 2 T†3 = − ψ + − ψ, z x2 + z 2 x2 + z 2 (x2 + z 2 )2 where d0 is an arbitrary constant. 3. Invariant solutions under three-dimensional subalgebra. In this section we look for solutions invariant under a three-dimensional subalgebra of the symmetry Lie algebra of equation (1). Equation (1) is then reduced to an algebraic equation which is then solved to obtain the exact solution.
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The operators X4 , X5 and X6 form a three-dimensional subalgebra of the symmetry Lie algebra. We construct an invariant solution of equation (2) which is invariant under these three operators. Firstly we solve X4 I = 0. Three invariants t, x2 + z 2 and ψ are obtained. We now introduce four new variables y 1 , y 2 , y 3 and y 4 and choose the first three to be the invariants. The fourth is chosen such that the Jacobian of the four variables is nonzero. That is y 1 = x,
y 2 = t,
y 3 = x2 + z 2 ,
y 4 = ψ.
Now we seek solutions of the form I = I(y 1 , y 2 , y 3 , y 4 ) and rewrite the action of X4 , X5 and X6 in the space of y 1 ,y 2 ,y 3 and y 4 . This gives ∂ X4 = z 1 ∂y X5 = σy 2
∂ ∂ + y4 4 ∂y 2 ∂y
and
∂ ∂ ∂ + 2σy 3 3 − 2y 4 4 . 1 ∂y ∂y ∂y From X4 I = 0 we see that our solution I is independent of y 1 . That is I = I(y 2 , y 3 , y 4 ) and so ∂ ∂ X5 = σy 2 2 + y 4 4 ∂y ∂y and ∂ ∂ X6 = 2σy 3 3 − 2y 4 4 . ∂y ∂y X6 = σy 1
We now find solutions in the space of three variables. The equation X5 I = 0 gives y2 the invariants y 3 and 4 σ . Again we write these invariants as new variables and (y ) select one other variable such that the three functions give a nonzero Jacobian. That is, y2 v1 = y4 , v2 = y3 , v3 = 4 σ . (y ) Our solutions now are of the form I = I(v 1 , v 2 , v 3 ). Writing the action of X5 and X6 in the space of v 1 , v 2 and v 3 , we obtain ∂ X5 = v 1 1 ∂v and ∂ ∂ X6 = v 2 2 + v 3 3 . ∂v ∂v From X5 I = 0 we see that I is independent of v 1 . That is I = I(v 2 , v 3 ). Now X6 I = 0 gives the invariant
v2 . Thus v3 v2 = C, v3
CONSERVATION LAWS & INVARIANT SOLUTIONS FOR SOIL WATER EQUATIONS 481
where C is a constant to be determined. The above equation can be written as x2 + z 2 σ ψ =C t which is an algebraic equation. Thus µ ¶ σ1 t ψ=C . x2 + z 2 Substituting ψ in equation (2) we find our constant C to be µ ¶ σ1 4 C= . σ Hence µ ¶ σ1 4t ψ= σ(x2 + z 2 ) is a special solution of equation (2) which is invariant under a three-parameter group. 4. Acknowledgement. The authors wish to thank F. M. Mahomed and R. K. Gazizov for helpful discussions. They also thank the referee for useful comments. Financial support from the National Research Foundation of South Africa and the University of North-West FAST Faculty Research Committee is highly acknowledged. REFERENCES [1] G. Vellidis, A. G. Smajstrla and F. S. Zazueta, Soil water redistribution and extraction patterns of drip-irrigated tomatoes above a shallow water table, , Transactions of the American Society of Agricultural Engineers (5) 33 (1990), 1525–1530. [2] G. Vellidis and A. G. Smajstrla, Modelling of soil water redistribution and extraction patterns of drip-irrigated tomatoes above a shallow water table, Transactions of the American Society of Agricultural Engineers (1) 35 (1992), 183–191. [3] V. A. Baikov, R. K. Gazizov, N. H. Ibragimov and V. F. Kovalev, Water Redistribution in Irrigated Soil Profiles: Invariant Solutions of the Governing Equation, Nonlinear Dynamics 13 (1997), 395–409. [4] V. A. Baikov and C. M. Khalique, Some invariant solutions for unsaturated flow models with plant root extraction, Quaestiones Mathematicae 24 (2001), 9–19. [5] A. H. Kara and C. M. Khalique, Conservation laws and associated symmetries for some classes of soil water motion equations, Int. J. Non-Linear Mech. 36 (2001), 1041–1045. [6] A. H. Kara and F. M. Mahomed, The relationship between symmetries and conservation laws, International Journal of Theoretical Physics (1) 39 (2000), 23–40. [7] A. H. Kara and F. M. Mahomed, A basis of conservation laws for partial differential equations, J. Nonlinear Math. Phys. 9 (2002), 60–72. acklund and Noether Symmetries [8] N. H. Ibragimov, A. H. Kara and F. M. Mahomed, Lie-B¨ with Applications, Nonlinear Dynamics 15 (1998), 115–136. [9] E. Noether, Invariante Variationsprobleme, K¨ onig Gesell Wissen G¨ ottingen, Math-Phys Kl Heft 2 (1918), 235.
Received September 2002; in revised April 2003. E-mail address:
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