On a method of exact linearization ordinary differential equations and on a new class of higher order nonlinear evolutionary equations

Lev M. Berkovich

(Samara State University, Samara, Russia) e-mail: berk@ssu.samara.ru

The last 30 years or so have seen an explosion of activity in the study of nonlinear phenomena. The motivation has largely come from the physical sciences and engineering, and also from problems in mathematical physics and mechanics.

Mathematical progress in the treatment of nonlinear differential equations has been swift and broadly based. The most spectacular developments have been in the field of exact analytic solutions of nonlinear equations.

Formulation of problem.

For a linear Ordinary Differential Equations (ODE), or a system of such equations, a linear superposition principle (LSP) allows us to represent the general solution as a linear combination in terms of a finite number of particular solutions.

It makes sense to ask whether this property is restricted to linear ODE’s or whether it can be generalized. This question was posed by several mathematicians in the end of XIX century (S. Lie, [1], E.Vessiot [2], V. Ermakov [3] and others).

The problem can by formulated as follows. Consider a system of first-order ODE’s

\frac{du}{dt} = η(u, t), i.e. \frac{du^{k}}{dt} = η^{k}(u, t), 1 ≤ k ≤ n,  (1)

where u, η ∈ R^{n} or C^{n}.

Question: when can the general solution of this system be expressed as a function of a finite number m of particular solutions

u_{1}(t), u_{2}(t), ⋯, u_{m}(t)  (2)

and a sufficient number n of arbitrary constants c_{1}, c_{2}, ⋯, c_{n}, i.e.

u(t) = F (u_{1}(t), u_{2}(t), ⋯, u_{m}(t), c_{1}, c_{2}, ⋯ c_{n}).  (3)

Theorem (S.Lie [1] ). The system of equations (1) has a fundamental set of solutions (FSS) (2) such that its general solution is given by nonlinear superposition rule (3), if and only if the following conditions hold:

1) The right-hand side of (1) can be (possibly after a transformation) factorized to give

\frac{du}{dt} = Z_{1}(t)ξ_{1}(u) + Z_{2}(t)ξ_{2}(u) + ⋯ + Z_{r}(t)ξ_{r}(u).

2) The vector fields

X_{i} = ∑_{s=1}^{n} ξ_{i}^{s}(u)\frac{∂}{∂u^{s}}

generate a finite-dimensional Lie algebra

[X_{a}, X_{b}] = ∑_{c=1}^{r} C_{ab}^{c}X_{c}, 1 ≤ a, b, c ≤ r.

The number m of solutions in the fundamental set (2) satisfies the inequality

n · m ≥ r.

Example 1. The Riccati equation. The superposition principle for the Riccati equation is well known. Indeed, any four solutions of

\frac{du}{dt} = Z_{1}(t) + Z_{2}(t)u + Z_{3}(t)u^{2},

e.g. u, u_{1}, u_{2}, u_{3}, satisfy the property that their anharmonic ratio is constant:

\frac{u − u_{1}}{u − u_{2}} : \frac{u_{3} − u_{1}}{u_{3} − u_{2}} = c.  (4)

Considering u(t) to be the general solution and u_{1}, u_{2}, u_{3} to be particular solutions, we can solve (4) for u and obtain the superposition law

u(t) = \frac{(u_{1} − u_{3})u_{2}c + u_{1}(u_{3} − u_{2})}{(u_{1} − u_{3})c + (u_{3} − u_{2})},

where c ∈ R is a constant related to the initial value u(t_{0}) of the solution u(t). This direction is being developed by Winternitz et all (see, for example [4]). Ermakov’s nonlinear superposition principle (NSP) (see also [5,6]). Consider a nonlinear equation

y^{(n)} = f(x, y, y′, ⋯, y^{(n−1)})  (5)

be given. Let also the associated linear equation be given by

Y^{(m)} + a_{1}(x)Y^{(m−1)} + ⋯ + a_{m−1}(x)Y′ + a_{m}(x)Y = 0  (6)

and

Y_{1}(x), Y_{2}(x), ⋯, Y_{m}(x)

be particular linear independent solutions of (6).

Definition 1. Equation (5) admits NSP, if its general solution has the form

y = F (Y_{1}(x), ⋯, Y_{m}(x); c_{1}, c_{2}, ⋯, c_{n}).  (5′)

Example 2. Ermakov–Pinney equation (Ermakov (1880), Pinney (1951)). Consider the equation

y′′ + a_{0}(x)y + b_{0}y^{−3} = 0, b_{0} = const.  (7)

and the associated equation

Y′′ + a_{0}(x)Y = 0.  (8)

General solution of (7) has the form:

y(x) = \sqrt{AY_{1}^{2} + BY_{1}Y_{2} + CY_{2}^{2}}, B^{2} − 4AC = const,

where Y_{1}(x), Y_{2}(x) are FSS of (8).

Example 3. Kummer-Schwarz equation

\frac{1}{2} \frac{y′′}{y} − \frac{3}{4}(\frac{y′}{y})^{2} + b_{0}y^{2} = a_{0}(x),

It has the general solution

y(x) = (AY_{1}^{2}(x) + BY_{1}Y_{2} + CY_{2}^{2})^{−1},

where Y_{1}(x), Y_{2}(x) are FSS of (8).

Example 4.

\frac{y′′′}{y} − \frac{3}{2}(\frac{y′′}{y})^{2} ≡ \{y, x\} = 0,

where {, } is the Schwarz derivative. Let Y′′(x) = 0 be the associated equation, and Y_{1} = 1, Y_{2} = x is a FSS. Then

y(x) = \frac{ax + b}{cx + d}, ad − bc = const.

NSP: generalized definition. Definition 2. The equation (5) admits NSP, if its general solution is (5′), where c_{1}, c_{2}, ⋯, c_{n}

are arbitrary const, and a set of particular solutions is either 1) particular solutions of (5), or

2) partial linear independent solutions of associated linear equation (6) , or

3) partial solutions of associated nonlinear equation

Y^{(m)} = g(x, Y, Y′, ⋯, Y^{(m−1)});

4) general solutions fo (5) is an nonlinear function of arbitrary const.

NSP may be extended also to partial differential equations. Example 5. Burgers equation:

u_{t} = −2ν + uu_{x} − νu_{xx} = 0.  (9)

By the transformation (Cole, 1951, and Hopf, 1950)

u = −2ν \frac{θ_{x}}{θ} ,

the equation (9) is reduced to the linear heat equation

θ_{t} − νθ_{xx} = 0.

Factorization of nonlinear ordinary differential equations and linearization

Theorem 1 [7,8]. Consider the equation

y^{(n)} = f(y, y′, ⋯, y^{(n−1)}), (′) = d/dx.  (10)

The invertible transformation

y = v(y)z, dt = u(y)dx,  (11)

where v, u are sufficiently smooth nonvanishing functions in the domain G(x, y), transforms (10) into the linear autonomous form (12),

z^{(n)}(t) + ∑_{k=1}^{n} \binom{n}{k}b_{k}z^{(n−k)}(t) = 0,  (12)

if and only if (10) admits the following factorization through first-order nonlinear differential operators:

∏_{k=1}^{n} (\frac{1}{u}D − \frac{v_{y}y′}{vu} − r_{k}) y = 0, D = \frac{d}{dx},

(commutative factorization), or

∏_{k=n}^{1} [D − \frac{v_{y}y′}{v} − (k − 1)\frac{u_{y}y′}{u} − r_{k}u] y = 0

(noncommutative factorization).

Theorem 2. The transformation (11) reduces the equation (10) to the linear form

z^{(n)}(t) + ∑_{k=1}^{n} \binom{n}{k}b_{k}z^{(n−k)}(t) + c = 0, b_{k}, c = const.  (13)

if and only if (10) can factorized as

∏_{k=1}^{n} [\frac{1}{φ}D − (\frac{1}{yφ} − \frac{φ^{\frac{n^{2}−3n+2}{2n}} exp(∫ fdy)}{∫ φ^{\frac{n^{2}−n+2}{2n}} exp(∫ fdy)dy}) \frac{∂y}{∂x} − r_{k}] y+

+φ^{\frac{−n^{2}+n−2}{2n}} exp(− ∫ fdy) (b_{n} ∫ φ^{\frac{n^{2}−n+2}{2n}} exp(∫ fdy)dy + \frac{c}{β} ) = 0,  (14)

where β is normalizing factor.

The lexicographic form of equation (14) is the following:

y^{(n)} + nf(y)y′y^{(n−1)} + ⋯ + nb_{1}φ(y)y^{(n−1)} + ⋯ +
φ^{\frac{n^{2}+n−2}{2n}} exp(− ∫ fdy) (b_{n} ∫ φ^{\frac{n^{2}−n+2}{2n}} exp(∫ fdy)dy + \frac{c}{β} ) = 0.  (15)

The linearizing transformation is

z = β ∫ φ^{\frac{n^{2}−n+2}{2n}} exp(∫ f(y)dy)dy, dt = φ(y)dx.

At c = 0, the equation has one-parameter families of solutions

∫ \frac{φ^{\frac{n^{2}−3n+2}{2n}} exp(∫ fdy)dy}{∫ φ^{\frac{n^{2}−n+2}{2n}} exp(∫ fdy)dy} = r_{k}x + c_{k},

where r_{k} are different roots of equation

r^{n} + ∑_{k=1}^{n} \binom{n}{k}b_{k}r^{n−k} = 0.

NSP for autonomous equations.

Corollary. Equation (10), represented in the form (14), (15), admits the following general solution:

β ∫ φ^{\frac{n^{2}−n+2}{2n}} exp(∫ fdy)dy = ∑_{k=1}^{n} c_{k}z_{k}(t) + z̃(t),
dt = φ(y)dx,

where z_{k}(t), k = 1, ..., n, are linear independent solutions of linear homogeneous equation (12) and z̃ is a particular solution of nonhomogeneous equation (13).

A new class of nonlinear evolutionary equations (NEE).

Theorem 3. [9] In order for NEE

\frac{∂y}{∂t} = F (y, \frac{∂y}{∂x}, …, \frac{∂^{n}y}{∂x^{n}})  (16)

to be transformed by substitution

y = v(y)z, ds = u(y)dx,

where

z = β ∫ φ^{\frac{n^{2}−n+2}{2n}} exp(∫ fdy)dy, ds = φdx, β is normalizing factor

to the linear evolutionary equation

\frac{∂z}{∂t} = \frac{∂^{n}z}{∂s^{n}} + ∑_{k=1}^{n} \binom{n}{k}b_{k} \frac{∂^{n−k}z}{∂s^{n−k}} , b_{k} = const,  (17)

it is nessesary and sufficient that (16) can be represented in the form

\frac{φ^{\frac{n^{2}−n+2}{2n}} exp(∫ fdy)}{∫ φ^{\frac{n^{2}−n+2}{2n}} exp(∫ fdy)dy} \frac{∂y}{∂t} =

= ∏_{k=1}^{n} [\frac{1}{φ}D − (\frac{1}{yφ} − \frac{φ^{\frac{n^{2}−3n+2}{2n}} exp(∫ fdy)}{∫ φ^{\frac{n^{2}−n+2}{2n}} exp(∫ fdy)dy}) \frac{∂y}{∂x} − r_{k}] y.  (18)

Nonlinear superposition principle for NEE.

Let z_{1}(s, t), …, z_{n}(s, t) be linearly independent partial solutions of (17). Then the total integral of (17) is

z = c_{1}z_{1} + ⋯ + c_{n}z_{n},

where c_{1}, c_{2}, …, c_{n} are arbitrary constants, and this formula is a LSP. We obtain a NSP according to the formulas

z = β ∫ φ^{\frac{n^{2}−n+2}{2n}} exp(∫ fdy)dy = ∑_{k=1}^{n} c_{k}z_{k}, ds = φdx,

where β = const is a normalizing factor.

Let us demonstrate NEE of orders n = 2 and n = 3 of general form belonging to the class (16), (18):

φ^{2}y_{t} = y_{xx} + fy_{x}^{2} + 2b_{1}φy_{x} + b_{2}φexp(− ∫ fdy) ∫ φexp(∫ fdy)dy;

φ^{3}y_{t} = y_{xxx} + 3fy_{x}y_{xx} + (\frac{1}{3} \frac{φ_{yy}}{φ} − \frac{5}{9} \frac{φ_{y}^{2}}{φ^{2}} − \frac{1}{3}f \frac{φ_{y}}{φ} + f^{2} + f_{y}) y_{x}^{3}+

+3b_{1}φ (y_{xx} + (f + \frac{1}{3} \frac{φ_{y}}{φ})y_{x}^{2}) + 3b_{2}φ^{2}y_{x} + b_{3}φ^{5/3}exp(− ∫ fdy) ∫ φ^{4/3}exp(∫ fdy)dy.

Example 6. (Maslov et al., see, for example, [10]) The equation

y_{t} = \frac{h}{2}y_{xx} − \frac{1}{2}y_{x}^{2}  (19)

by the transformation

z = exp(−\frac{y}{h}), ds = \sqrt{\frac{2}{h}}dx

is linearized to

z_{t} = z_{ss}.

Remark. The correspondence principle (in the sense [8]) is a NSP .

The equation (19) by transformation z = exp(−y/h) takes the form

z_{t} = −hz_{xx}.  (20)

Let z_{1}, z_{2} be linearly independed solutions of (20). Then the total integral z = c_{1}z_{1} + c_{2}z_{2}, where c_{1}, c_{2} are arbitrary constants, is a LSP, and the formula

y = −h ln [exp (−\frac{1}{h}y_{1} − \frac{λ_{1}}{h} ) + exp (−\frac{1}{h}y_{2} − \frac{λ_{2}}{h} )] ,

where λ_{1}, λ_{2} are arbitrary constants, and y(x, t) is a total integral, and y_{1}(x, t), y_{2}(x, t) are particular solutions of (19), is a NSP.

Concluding remarks

1) A FSS does not determine the nonlinear equation completely; to determine the equation one has to know both the FSS and the NSP.

Example 7. Equation

y′′ + (−\frac{2}{y} + \frac{1}{y^{2}} ) y^{′2} − y^{2} = 0

by transformation z = exp(−1/y) is reduced to z′′ − z = 0.

F SS :  y_{1} = \frac{1}{x}, y_{2} = \frac{−1}{x},

NSP :  y = −\frac{1}{ln(c_{1}e^{x} + c_{2}e^{−x})} = −\frac{1}{ln (exp(−\frac{1}{y_{1}} + α) + exp(−\frac{1}{y_{2}} + β))},

where c_{1}, c_{2}, α, β are integration constants. The following equation

y′′ − \frac{1}{y}y^{′2} − y^{3} = 0

can be linearized to the form

ÿ − y = 0

by the substitution dt = ydx. It has the same FSS as before, but different from the NSP above. We omit the NSP in this case, since it very bulky.

2) One can linearise an equation by means of different transformations.

Example 8. Consider the equation

y′′ + 3yy′ + y^{3} = 0  (21)

Equation (21) can be linearized by the following substitution

z = y^{2}, dt = ydx

to the form

z̈ + 3zż + z^{3} = 0.

Applying the point transformations

X = x − \frac{1}{y}, Y = \frac{1}{2}x^{2} − \frac{x}{y},

we get another linear equation

Y′′(X) = 0.

3)Factorization may be regarded as a possible method of linearization.

Let equation

y^{(n)} = f(y, y′, ⋯, y^{(n−1)})

admit a factorization

∏_{k=1}^{n}(γD − βy′ − α − r_{k})y = 0, γ = γ(y), β = β(y), α = α(y), r_{k} = const.

This factorization generetes Bäcklund transformation:

\frac{y′}{y} (1 − \frac{β}{γ} y) = \frac{α}{γ} + \frac{z′}{z}

that reduces to a linear equation

∏_{k=1}^{n}(D − r_{k})z = 0

Open problem.

Can one obtain new non-linear compositions laws making use of the following invertible transformations

y = v(x, y)z, dt = u_{1}(x, y)dx + u_{2}(x, y)dy

for nonlinear and nonautonomous equation

y^{(n)} = f(x, y, y′, …, y^{(n−1)})?

Acknowledgment: the author expresses his gratitude to Sergei Kuleshov and Victor Protsak for their help in preparing the text for publication.

References

[1] Lie. S., Vorlesungen über continuerliche Gruppen mit Geometrischen und anderen Anwendungen, Bearbeitet and herausgegeben von Dr. G. Scheffers, Teubuer, Leipig, 1893 (p. 765-804).

[2] Vessiot E., Sur les systems d’équations différentielles du preimer order qui on des systèmes fondamentaux d’intégrales, Ann. sc. École Norm. sup. 10 (1883).

[3] Ermakov V.P. Universitetskie Izvestiya, Kiev, 1880.

[4] Turbiner A., Winternitz P. Solutions of Nonlinear Differential and Difference equations with Superpositions Formulae, Letters in Mathematical phisics, 50, 189-201 (1999).

[5] Berkovich L.M., Transformations of linear differential equations, Kujbyshev University Publ., 1978 (in Russian).

[6] Berkovich L.M., Factorization and transformations of ordinary differential equations, Saratov University Publ., Saratov 1989 (in Russian)

[7] Berkovich L.M., Factorization of Nonlinear Ordinary Differential Equations and Linearization, Doclady Mathemetics, Vol. 60, No 2, (1999) p.278-282. (Doklady Akademii Nauk, Vol. 368, No5, (1999) p. 604-608).

[8] Berkovich L.M., Factorization and transformations of nonlinear differential equations: methods and algorithns, In Intern. Congress on Nonlinear Analysis and its applications, Moscow, Sept. 1-5, 1998, see http://www.acad-ns.newmail.ru/nlaia98 (see also www.ssu.samara.ru/Campus/Teachers/Berkovich).

[9] Berkovich L.M., On a method of exact linearization and on higher order nonlinear evolytionary equations, International Conference “Differential Equations and Related Topics”, dedicated to the Centenary Anniversary of Ivan G. Petrovskii, Books of Abstracts, Moskow, May 22 - 27, 2001, 59-60.

[10] Litvinov G.I., Maslov V.P., Correspondence principle for idempotent calculus and some computer applications, -In: J.Gauwardena (editor), Idempotency. - Publ. of the Newton Institute, Cambridge Univ. Press, Cambridge, 1998, 420-443.