June 15-19, 2014, Larnaca, Cyprus JWorkshop in Honor of ...lnf.nsu.ru/en/pdf/present/3_...

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Transcript of June 15-19, 2014, Larnaca, Cyprus JWorkshop in Honor of ...lnf.nsu.ru/en/pdf/present/3_...

  • June 15-19, 2014, Larnaca, Cyprus

    JWorkshop in Honor of Michail Ivanov ’ 70 th Birthday

  • The equation studied The Fourier image of the spatially homogeneous and isotropic Boltzmann equation with a source term has the form:

    ϕt (x , t) + ϕ(x , t)ϕ(0, t) = ∫ 1

    0 ϕ(xs, t)ϕ(x(1− s), t) ds + q̂(x , t).

    Here the function ϕ(x , t) is

    ϕ(x , t) ≡ ϕ(k2/2, t) = ϕ̃(k , t) = 4π k

    ∫ ∞ 0

    v sin(kv)f (v , t) dv .

    where f (v , t) is the distribution function of isotropic in the 3D-space of molecular velocities Similarly, the transform of the isotropic source function q(v , t) is

    ˜̂q(k , t) = 4π k

    ∫ ∞ 0

    v sin(kv)q(v , t) dv ,

  • Determining equation

    A generator of the admitted Lie group is sought in the form

    X = ξ(x , t , ϕ)∂x + η(x , t , ϕ)∂t + ζ(x , t , ϕ)∂ϕ.

    The determining equation for the considered equation is

    Dtψ(x , t)+ψ(0, t)ϕ(x , t)+ψ(x , t)ϕ(0, t)−2 ∫ 1

    0 ϕ(x(1−s), t)ψ(xs, t)ds = 0,

    where Dt is the total derivative with respect to t , and the function ψ(x , t) is

    ψ(x , t) = ζ(x , t , ϕ(x , t))−ξ(x , t , ϕ(x , t))ϕx (x , t)−η(x , t , ϕ(x , t))ϕt (x , t).

  • Assume that the coefficients of the infinitesimal generator X are represented by the formal Taylor series with respect to ϕ:

    ξ(x , t , ϕ) = ∑ l≥0

    ql(x , t)ϕl(x , t),

    η(x , t , ϕ) = ∑ l≥0

    rl(x , t)ϕl(x , t),

    ζ(x , t , ϕ) = ∑ l≥0

    pl(x , t)ϕl(x , t).

    A particular class of solutions is considered. This class is defined by the initial conditions

    ϕ0(x , t) = bxn

    at a given (arbitrary) time t = t0. Here, n = 0,1,2, ....

  • The coefficients of the generator X are

    ξ(x , t , ϕ) = c0x , η(x , t , ϕ) = −c2t+c3, ζ(x , t , ϕ) = (c2+c1x)ϕ

    where c0, c1, c2and c3 are arbitrary constant. Thus, each admitted generator has the form

    X = c0X0 + c1X1 + c2X2 + c3X3,

    where

    X0 = x∂x , X1 = xϕ∂ϕ, X2 = ϕ∂ϕ − t∂t , X3 = ∂t .

    The remaining part of the determining equation becomes

    (c2t − c3)q̂t − c0xq̂x + (c1x + 2c2)q̂ = 0.

  • Equivalence Transformations Let us introduce the operator L:

    Lϕ = ϕt (x , t) + ϕ(x , t)ϕ(0, t)− ∫ 1

    0 ϕ(xs, t)ϕ(x(1− s), t) ds.

    Equivalence transformations corresponding to the generators X0, X1, X2 and X3, are obtained as follows. For example, the transformations corresponding to the generator X0 = x∂x map a function ϕ(x , t) into the function ϕ̄(x̄ , t̄) = ϕ(x̄e−a, t̄), where a is the group parameter. The transformed expression becomes

    L̄ϕ̄ = ϕ̄t̄ (x̄ , t̄) + ϕ̄(x̄ , t̄)ϕ̄(0, t̄)− ∫ 1

    0 ϕ̄(x̄s, t̄)ϕ̄(x̄(1− s), t̄) ds = ϕt̄ (x̄e−a, t̄) + ϕ(x̄e−a, t̄)ϕ(0, t̄)−

    ∫ 1 0 ϕ(x̄e

    −as, t̄)ϕ(x̄e−a(1− s), t̄) ds = ϕt (x , t) + ϕ(x , t)ϕ(0, t)−

    ∫ 1 0 ϕ(xs, t)ϕ(x(1− s), t) ds

    = Lϕ.

    This defines the Lie group of equivalence transformations of the equation

    x̄ = xea, t̄ = t , ϕ̄ = ϕ, ¯̂q = q̂.

  • Similarly, the transformations corresponding: to the generator X3 = ∂t

    x̄ = x , t̄ = t + a, ϕ̄ = ϕ, ¯̂q = q̂.

    to the generator X2 = ϕ∂ϕ − t∂t

    x̄ = x , t̄ = te−a, ϕ̄ = ϕea, ¯̂q = q̂e2a

    to the generator X1 = xϕ∂ϕ

    x̄ = x , t̄ = t , ϕ̄ = ϕexa, ¯̂q = q̂exa.

    Thus the generators defining an equivalence Lie group of the considered equation are

    X e0 = x∂x , X e 1 = xϕ∂ϕ+xq̂∂q̂, X

    e 2 = ϕ∂ϕ−t∂t +xq̂∂q̂, X

    e 3 = ∂t .

  • Let us study the change of a generator

    X = x0X0 + x1X1 + x2X2 + x3X3

    under these equivalence transformations. After the change one gets the generator

    X = x̂0X̂0 + x̂1X̂1 + x̂2X̂2 + x̂3X̂3,

    where

    X̂0 = x̄∂x̄ , X̂1 = x̄ϕ̄∂ϕ̄, X̂2 = ϕ̄∂ϕ̄ − t̄∂t̄ , X̂3 = ∂t̄ .

    The corresponding transformations of the basis generators are

    X e0 : X0 = X̂0,X1 = e −aX̂1,X2 = X̂2,X3 = X̂3;

    X e1 : X0 = X̂0 + aX̂1,X1 = X̂1,X2 = X̂2,X3 = X̂3;

    X e2 : X0 = X̂0,X1 = X̂1,X2 = X̂2,X3 = e −aX̂3;

    X e3 : X0 = X̂0,X1 = X̂1,X2 = X̂2 + aX̂3,X3 = X̂3.

  • Any generator X can be expressed as a linear combination of the basis generators:

    X = x̂0X̂0 + x̂1X̂1 + x̂2X̂2 + x̂3X̂3 = x0X0 + x1X1 + x2X2 + x3X3

    Using the invariance of a generator with respect to a change of the variables, the basis generators Xi (i = 0,1,2,3) and X̂j (j = 0,1,2,3) in corresponding equivalence transformations are related as follows:

    X e0 : x̂1 = x1e −a,

    X e1 : x̂1 = x1 + ax0, X e2 : x̂3 = x3e

    a,

    X e3 : x̂3 = x3 + ax2.

  • For classification an algebraic algorithm was applied,

    I first we study the Lie algebra L4 composed by the generators X0,X1,X2,X3. The commutator table is

    X0 X1 X2 X3 X0 0 X1 0 0 X1 −X1 0 0 0 X2 0 0 0 −X3 X3 0 0 X3 0

    The inner automorphisms are

    A0 : x̂1 = x1ea, A1 : x̂1 = x1 + ax0, A2 : x̂3 = x3ea, A3 : x̂3 = x3 + ax2,

  • I Second, one can notice that the results of using the equivalence transformations corresponding to the generators X e0 ,X

    e 1 ,X

    e 2 ,X

    e 3 are similar to changing

    coordinates of a generator X with regards to the basis change. These changes are similar to the inner automorphisms.

    Really

    A0 : x̂1 = x1ea, X e0 : x̂1 = x1e −a,

    A1 : x̂1 = x1 + ax0, X e1 : x̂1 = x1 + ax0, A2 : x̂3 = x3ea, X e2 : x̂3 = x3e

    a, A3 : x̂3 = x3 + ax2, X e3 : x̂3 = x3 + ax2.

  • Optimal system of subalgebras

    Construction of an optimal system of subalgebras of the Lie algebra L4

    I It is simplified if one notices that L4 = F1 ⊕ F2, where F1 = {X0,X1} and F2 = {X2,X3} are ideals of the Lie algebra L4.

    I This decomposition gives a possibility to apply a two-step algorithm (Ovsiannikov, 1993 and 1994).

    The result of construction of an optimal system of subalgebras is presented in Table 1.

  • Optimal system of subalgebras

    No. Basis No. Basis 1. X0, X1, X2, X3 13. X0 + X3, X1 2. γX0 + X2, X1, X3 14. X1, X3 3. X0, X1, X3 15. X0, X3 4. X0, X1, X2 16. X0, X1 5. X0, X2, X3 17. γX0 + X2 6. X2, X3 18. X1 + X2 7. X2 + X0, X1 + X3 19. X1 − X2 8. X2 + γX0, X3 20. X0 + X3 9. X1 + X2, X3 21. X1 + X3

    10. X1 − X2, X3 22. X0 11. X0, X2 23. X1 12. γX0 + X2, X1 24. X3

  • Obtaining the Function q̂

    Example: Lie algebra {γX2 + 2X0, X3} For this Lie algebra there are two sets of the coefficients ci , (i = 0,1,2,3):

    γX2 + 2X0 : c0 = 2 c1 = 0 c2 = γ c3 = 0; X3 : c0 = 0 c1 = 0 c2 = 0 c3 = 1.

    These sets define the system of equations by substituting the coefficients ci into the remaining equation:

    γ( 1 2

    t q̂t + q̂)− xq̂x = 0, q̂t = 0.

    The general solution of these equations is q̂ = βxγ , where β is constant.

  • Group Classification

    No. q̂(t , x) Generators 1. 0 X0, X1, X2 X3 2. kx2etx X2 + X0, X1 + X3 3. kxγ γX2 + 2X0, X3 4. kt−2 X0, X2 5. t−2Φ(xtα) αX0 + X2 6. t−(x+2)Φ(x) X1 + X2 7. tx−2Φ(x) X1 − X2 8. Φ(xe−t ) X0 + X3 9. ext Φ(x) X1 + X3 10. Φ(t) X0 11. Φ(x) X3

    where α, β, γ and k are constant

  • Representation of invariant solution for q̂ = kx2ext Equation

    ϕt (x , t) + ϕ(x , t)ϕ(0, t) = ∫ 1

    0 ϕ(xs, t)ϕ(x(1− s), t) ds + kx2ext .

    The corresponding admitted Lie algebra of the equation is {X2 + X0, X1 + X3}. An optimal system of subalgebras of this Lie algebra is:

    {X2 + X0}, {X1 + X3}, {X2 + X0, X1 + X3}

    For the subalgebra {X2 + X0} corresponding invariant solution has a representation

    ϕ = t−1r(z), z = xt

    Substituting this representation of invariant solution, we obtain the reduced equation:

    zr ′(z)− r(z) + r(z)r(0)− ∫ 1

    0 r(zs)r(z(1− s)) ds = kz2ez .

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