Exact Solutions of a Nonlinear Diffusion Equation with Absorption and Production
- https://doi.org/10.2991/jnmp.k.200922.008How to use a DOI?
- Nonlinear diffusion equation, turbulent kinetic energy, exact solutions
We provide closed form solutions for an equation which describes the transport of turbulent kinetic energy in the framework of a turbulence model with a single equation.
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The Partial Differential Equation (PDE) 
In order to search for closed form solutions, it is better to first convert (1.1) to an algebraic equation,
Therefore the final algebraic equation which we will study is (after rescaling t) [10, Eq. (3.24)]
Since there could exist physical systems not requiring the positivity of b, c, d, we will also mention a few solutions with b, c, d not all positive.
The paper is organized as follows. In Section 2 we first recall the ingenious, however not generalizable, method which has allowed Maire to obtain a solution matching all the physical constraints in (1.3).
Section 3 is devoted, using the method of infinitesimal Lie point symmetries, to the construction of all the reductions of the PDE (1.3) to an Ordinary Differential Equation (ODE), then to their integration.
In Section 4, we study the local behaviour of the field w(x, t) near its movable singularities, a prerequisite to the search for closed form solutions.
The next Section 5 is devoted to a search for new exact solutions based on the singularity structure.
2. METHOD OF GALAKTIONOV AND POSASHKOV
The PDE (1.3) belongs to the class
Then Galaktionov and Posashkov  observed that the (kind of “adiabatic”) assumption
The most physically relevant solution [10, Eq. (3.37)],
The second solution is stationary,
Finally, the fourth solution is characterized by a second order nonlinear ODE,
If one relaxes the constraint b > 0, there also exists a fifth solution,
Those based on the symmetries of the PDE, which generate reductions to ODEs;
Those based on the movable singularities of the PDE, which (after a double study, local then global) generate closed form solutions w(x, t).
3. SOLUTIONS OBTAINED BY SYMMETRIES
For generic values of b, c, d, the only Lie point symmetries of the PDE (1.3) are arbitrary translations of both x and t. The resulting characteristic system,
One must distinguish β ≠ 0 (the reduction preserves the differential order two) and β = 0 (the reduction lowers it to one).
For β ≠ 0, since the positive parameter b is never equal to −(1 − 1/n), with n some signed integer, the general solution W(ξ) is multivalued  and generically cannot be obtained in closed form. A notable exception is β ≠ 0 and α = 0 (stationary wave),
For K = 0, the general solution is physically acceptable and has already been obtained, see Eq. (2.5), and for K ≠ 0 the solution is given implicitly by the quadrature
The only cases of invertibility of this quadrature are −2/b = 1, 2, 3, 4, yielding expressions W(x) trigonometric (b = −2, −1) or elliptic (b = −2/3, −1/2), which however all violate the physical requirement b > 0.
For β = 0 one obtains the front independent of x, Eq. (2.6).
4. STRUCTURE OF SINGULARITIES
Any closed form solution depends on arbitrary functions or constants, such as x0, t0 in (2.4), which may define movable singularities. For instance, the solution (2.4) definies two families of movable singularities: on one side the movable poles of w located at the points , on the other side movable poles of 1/w (movable zeroes of w) located on the singular manifold φ (x, t) = 0 defined by
A prerequisite to the systematic search for solutions is therefore the determination of the structure of the movable singularities of (1.3),
Since the highest derivative term wwxx displays the singularity w = 0, one must also study the movable zeros of w (movable poles of w−1 = f ),
This is a classical computation [6, §4.4.1], whose results are the following.
The PDE (1.3) admits two types of movable singularities.
If φx = 0 (the singular manifold is then said characteristic), the highest derivative term wwxx does not contribute to the leading order, and w (as well as f ) presents one family of movable simple poles,(4.4)(4.5)
If φx ≠ 0 (noncharacteristic singular manifold), w has no movable poles and w−1 = f presents two families of movable simple poles,(4.6)
To finish this local analysis, one must then compute the Fuchs indices of the linearized equation of (1.3) in the neighborhood of φ = 0. Indeed, a necessary condition of singlevaluedness is that all Fuchs indices be integers of any sign. For the families (4.4) or (4.5), the unique Fuchs index is i = −1. For each of the two families (4.6), one finds
In order to build solutions of this kind of nonintegrable PDE, the various methods based on the singularity structure are reviewed in summer school lecture notes , where the proper original references can be found. Let us now investigate a few of them.
5.1. Characteristic One-family Truncation of w
The truncation (5.1) with χx = 0 defines the system
The value of χ results by integrating the Riccati system (5.4),
5.2. Characteristic One-family Truncation of f
The truncation (5.2) with χt = 0 defines the system
After integration of the Riccati system (5.4),
To conclude, these characteristic truncations yield nothing new.
5.3. Noncharacteristic One-family Truncation of f
In the case b ≠ −1/2, the algebraic (i.e. nondifferential) elimination of F0,t, F1,t, Sx yields the much simpler equivalent system,
This system (5.17) is solved in three steps:
integration of the ODE in F1 defined by X = 0, which introduces at most two arbitrary functions of t;
determination of F0 by solving the overdetermined system (E1 = 0, E2 = 0);
knowing the values of (S, C), integration of the Riccati system (5.4) for χ.
In the unphysical case b = −1/2, we could not find an equivalent system as simple as (5.17).
Let us now perform the above mentioned three steps.
5.3.1. Values of F1
All singlevalued solutions of the ODE for F1 are obtained by two methods: for the general solution, by looking in the classical exhaustive tables [8,11]; for particular solutions, by looking for Darboux polynomials. One thus finds exactly seven solutions, five of them with a negative b (unphysical for the diffusion problem, but possibly admissible for other systems) and two with an arbitrary value of b,
For b = −1/2, the obtained solution is a particular case of the solution, which we could not obtain, resulting from the system (5.15).
For b = −1 and c ≠ 0, the ODE for F1 has no singlevalued solution.
Let us next determine F0. By the elimination of F1 between E1 = 0 and E2 = 0, the value of F0 is the root of a sixth degree polynomial whose coefficients are polynomial in F1 and its derivatives. However, this computation is only tractable for the two tanh solutions (which only depend on g(t)), and one finds constant values for F0 and g′(t),
One first notices that, in its homogeneous part, the simple pole of F1 with residue r = −1 is a Fuchsian singularity for the EDO in F0, whose Fuchs indices i, the roots of
5.3.2. Solution, case of the first tanh value of F1
5.3.3. Solution, case of the second tanh value of F1
Solving (E1, E2) for (F0(x, t), g(t)) is again quite easy,
The ODE for χ−1 is a Lamé equation in its Riccati form, whose solution is singlevalued for b = −1 + 1/n, , multivalued otherwise. For the present diffusion problem, this new solution leaves b, c, d unconstrained and can indeed be used to test the validity of numerical schemes.
5.3.4. Case b = −2
A computation similar to the above one yields
5.3.5. Solutions, case b = −4/3
The resolution of the four cases b = −4/3, −4/5, −1/2, −2/3 follows the same pattern, so we only detail it for b = −4/3 [Eq. (5.18)].
For this value b = −4/3, the ODE E2 = 0 possesses the general solution
As already argued in Subsection 5.3.1, since the relation between F0 and ekx−g(t) is necessarily algebraic, the two functions g+ and g− must vanish. Equation E1 = 0 then yields the necessary and sufficient conditions,
The system (5.17) therefore has for solution, in the first case ω1 ≠ 0, Ω1 = 0,
Each of these two new solutions is outside the class (2.4) and depends on a single arbitrary constant (k0 in the first one, t0 in the second one).
5.3.6. Case b = −4/5
whose coefficients Rj are rational in . One then proves that, since d is nonzero, the discriminant must vanish, thus reducing F1 and F0 to simply periodic functions,
Equation E1 = 0 then generates the constraints
5.3.7. Case b = −1/2
and in the trigonometric subcase ,
Equation E1 = 0 then generates the constraints g′ = 0 in the elliptic subcase, and ε = −1, g′2 = 9d/2 in the trigonometric subcase, restricting the solution to (5.23) with b = −1/2.
5.3.8. Case b = −2/3
This case also happens to be a particular case of (5.23).
By a systematic investigation, we have obtained several new solutions of this diffusion problem. Two of them [Subsection 5.3.3, Eq. (2.7)] match all the physical constraints b > 0, c > 0, d > 0 and can serve to calibrate and validate the numerical schemes.
Finally, in this short paper we have only investigated those solutions w which take into account one of the two movable poles. Taking account of both poles via the two-singular manifold method (see [6, §18.104.22.168] and references therein) should certainly generate additional solutions.
Remark. As suggested by the two new solutions Eqs. (5.39) and (5.40), the class w equal to a second degree polynomial in cosh(kx) with time-dependent coefficients could also generate physically interesting solutions.
CONFLICTS OF INTEREST
The author declares no conflicts of interest.
The author gratefully acknowledges the support of LRC Méso and is happy to thank B.-J. Gréa, A. Llor and R. Motte for suggesting this interesting and challenging problem.
Cite this article
TY - JOUR AU - Robert Conte PY - 2020 DA - 2020/12 TI - Exact Solutions of a Nonlinear Diffusion Equation with Absorption and Production JO - Journal of Nonlinear Mathematical Physics SN - 1776-0852 UR - https://doi.org/10.2991/jnmp.k.200922.008 DO - https://doi.org/10.2991/jnmp.k.200922.008 ID - Conte2020 ER -