Renormalized asymptotic solutions of the Burgers equation and the Korteweg-de Vries equation

0.09 MB·English

by  Sergei V. Zakharov

Renormalized asymptotic solutions
of the Burgers equation
and the Korteweg–de Vries equation
S.V. Zakharov
Institute of Mathematics and Mechanics,
Ural Branch of the Russian Academy of Sciences,
16, S.Kovalevskaja street, 620990, Ekaterinburg, Russia
E-mail: [email protected]
5 Abstract. The Cauchy problem for the Burgers equation and the Korteweg–de Vries
1
0 equation is considered. Uniform renormalized asymptotic solutions are constructed in cases
2 of a large initial gradient and a perturbed initial weak discontinuity.

n
a Keywords: Burgers equation, KdV equation, renormalization, weak discontinuity.
J
Mathematics Subject Classification: 35K15, 35Q53.
2
1

] 1 Problem with a large initial gradient
h
p
- A simplest model of the motion of continuum, which takes into account nonlinear effects and
h
t dissipation, is the equation of nonlinear diffusion
a
m
∂u ∂u ∂2u
[ +u = ε , ε > 0, (1.1)
∂t ∂x ∂x2

1
v for the first time presented by J. Burgers [1]. This equation is used in studying the evolution
2
of a wide class of physical systems and probabilistic processes, for example, acoustic waves
4
5 in fluid and gas [2].
2
Let us consider the Cauchy problem
0
.
1 u(x,0,ε,ρ) = Λ(xρ 1), t = 0, x R, ρ > 0, (1.2)
0 − ∈
5
1 where ρ is another small parameter. The initial function Λ is smooth and it has finite limits
:
v
Λ = lim Λ(s).
i ±0
X s
→±∞
r
a We see that the origin (x = 0 and t = 0) must be a singular point in the problem
under consideration. It is known that the behavior of solutions of differential equations with
a small parameter at a higher derivative in a neighborhood of a singular point sometimes
becomes self-similar. Then it is effective to analyze the solution using the renormalization
group method [3]. The success of applying renormalization is confirmed by examples from
hydrodynamics [4], mechanics [5], and asymptotic analysis ofsolutions of nonlinear parabolic
equations [6].
In paper [7], for the solution of a more general quasi-linear parabolic equation
∂u ∂ϕ(u) ∂2u
+ = ε , t > 0,
∂t ∂x ∂x2
1
with condition (1.2) the following asymptotic formula is obtained:
1 ∞ x ρs t ρ 1/4
u(x,t,ε,ρ) = Γ − , Λ(s)ds+O (1.3)
Λ+ Λ ε ε ′ ε
0 − −0 Z (cid:18) (cid:19) (cid:18)(cid:16) (cid:17) (cid:19)
−∞
as ε 0 and ρ/ε 0 uniformly in the strip (x,t) : x R, 0 6 t 6 T , where Γ is the
→ → { ∈ }
solution of the limit problem in the inner variables η = x/ε, θ = t/ε
∂Γ ∂ϕ(Γ) ∂2Γ Λ , η < 0,
+ = 0, Γ(η,0) = −0
∂θ ∂η − ∂η2 (Λ+, η > 0.
0
The procedure of obtaining expression (1.3) can be also seen below in details on the example
of the KdV equation. The difference consists in the scales of the inner variables. For the
solution of the Burgers equation (1.1) as ε 0 and ρ/ε 0 in the strip (x,t) : x R, 0 6
→ → { ∈
t 6 T there holds the asymptotic formula
}
∞ Λ(s) t(Λ+)2 2Λ+x Λ+ρs Λ+t x+ρs
u(x,t,ε,ρ) = ′ Λ+exp 0 − 0 + 0 erfc 0 − +
Z Λ+0 −Λ−0 (cid:20) 0 (cid:18) 4ε 2ε (cid:19) (cid:18) 2√εt (cid:19)
−∞
t(Λ )2 2Λ x Λ ρs x Λ t ρs
+ Λ exp −0 − −0 + −0 erfc − −0 −
−0 4ε 2ε 2√εt ×
(cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
t(Λ+)2 2Λ+x Λ+ρs Λ+t x+ρs
exp 0 − 0 + 0 erfc 0 − +
× 4ε 2ε 2√εt
(cid:20) (cid:18) (cid:19) (cid:18) (cid:19)
+ exp t(Λ−0)2 −2Λ−0x + Λ−0ρs erfc x−Λ−0t−ρs −1ds+O µ1/4 .
4ε 2ε 2√εt
(cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
(cid:0) (cid:1)
2 Renormalized weak discontinuity
In the present paper, we give an asymptotic solution of the Burgers equation
∂u ∂u ∂2u
+u = (2.1)
∂t ∂x ∂x2
with the initial condition
x
u(x,0) = εΛ , t = 0, x R, (2.2)
ε ∈
(cid:16) (cid:17)
where Λ(s) s as s and Λ(s) 0 as s + .
→ − → −∞ → → ∞
By analogy with (1.3) we postulate the expression
∞
R(x,t,ε) = Λ (s)u (x εs,t)ds,
′′ 0
−
Z
−∞
2
where u = 2Ψ /Ψ is the solution of the limit problem with a weak discontinuity [8],
0 x
−
∂u ∂u ∂2u x, x < 0,
0 0 0
+u = , u (x,0) = −
∂t 0 ∂x ∂x2 0 ( 0, x > 0.
Ψ is the solution of the heat equation Ψ = Ψ . Thus, we obtain a renormalized asymptotic
t xx
solution in the form
2 ∞
R(x,t,ε) = Λ (s)lnΨ(x εs,t)ds.
′′′
−ε −
Z
−∞
Theorem 1. Let Λ(s) be a thrice differentiable function and let
dΛ(s)
= 1+O( s 3), s , (2.3)
−
ds − | | → −∞
dΛ(s)
= O(s 3), s + , (2.4)
−
ds → ∞
d2Λ(s)
= O(s 4), s . (2.5)
−
ds2 → ∞
Then the function
2 ∞ d3Λ(s)
R(x,t,ε) = lnΨ(x εs,t)ds
−ε ds3 −
Z
−∞
is an asymptotic solution of the Burgers equation (2.1), where
0
(x σ)2 σ2 ∞ (x σ)2
Ψ(x,t) = exp − + dσ + exp − dσ.
− 4t 4 − 4t
Z (cid:18) (cid:19) Z (cid:18) (cid:19)
0
−∞
Proof. Substituting u = R(x,t,ε) into (2.1), we find
4 ∞ Ψ (x εs,t) ∞
x
R +RR R = Λ (s) − ds Λ (s)lnΨ(x εs,t)ds
t x − xx ε2 ′′′ Ψ(x εs,t) ′′′ − −
Z − Z
−∞ −∞
2 ∞ Ψ2(x εs,t)
Λ (s) x − ds.
′′′
−ε Ψ2(x εs,t)
Z −
−∞
Let us esimate the integrals on the right-hand side. From (2.3)–(2.5) it follows that
1/√ε 1/√ε
Λ (s)ds = O(ε2), sΛ (s)ds = 1+O(ε3/2).
′′′ ′′′
−
Z Z
1/√ε 1/√ε
− −
3
Then we obtain
1/√ε
∞ Ψ (x,t)
Λ (s)lnΨ(x εs,t)ds = Λ (s) lnΨ(x,t) εs x ds+O(ε2) =
′′′ ′′′
− − Ψ(x,t)
Z Z (cid:20) (cid:21)
1/√ε
−∞ −
Ψ (x,t)
= ε x +O(ε2),
Ψ(x,t)
1/√ε
∞ Ψ (x εs,t) Ψ Ψ2
Λ (s) x − ds = ε xx x sΛ (s)ds+O(ε2) =
′′′ ′′′
Ψ(x εs,t) − Ψ − Ψ2
Z − (cid:18) (cid:19) Z
1/√ε
−∞ −
Ψ Ψ2
= ε xx x +O(ε2),
Ψ − Ψ2
(cid:18) (cid:19)
1/√ε
∞ Ψ2(x εs,t) Ψ Ψ Ψ3
Λ (s) x − ds = 2ε x xx x sΛ (s)ds+O(ε2) =
′′′ ′′′
Ψ2(x εs,t) − Ψ2 − Ψ3
Z − (cid:18) (cid:19) Z
1/√ε
−∞ −
Ψ Ψ Ψ3
= 2ε x xx x +O(ε2).
Ψ2 − Ψ3
(cid:18) (cid:19)
Consequently, we conclude that
R +RR R = O(ε).
t x xx
−
Since R(x,0,ε) = εΛ(x/ε), by the continuity of the inverse operator, we can also state
that R(x,t,ε) is an asymptotic approximation of the exact solution of the Cauchy problem
(2.1)–(2.2).
3 Korteweg–de Vries equation
Studying properties of solutions of the Korteweg–de Vries equation
∂u ∂u ∂3u
+u +ε = 0, t > 0, ε > 0, (3.1)
∂t ∂x ∂x3
is of indisputable interest for describing nonlinear wave phenomena [2]. Open questions con-
cerning the behavior of solutions are still the subject of attention for modern researches [9].
Some mathematical results about the solution of the problem in various cases can be found
in [10] and [11].
Consider the Cauchy problem for (3.1) with the initial condition
u(x,0,ε,ρ) = Λ(xρ 1), t = 0, x R, ρ > 0. (3.2)
−
∈
Here, we assume that the initial function Λ is bounded and possesses the first derivative,
which sufficiently fast tends to zero at infinity. It is required to find an asymptotic approx-
imation of the solution u(x,t,ε,ρ) to problem (3.1)–(3.2) in small parameters ε and ρ for
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all x and finite values of t. It is clear that the structure of the asymptotics must depend on
the relation between parameters ε and ρ. We assume the fulfillment of the condition
ρ
µ = 0.
√ε →
In paper [12], a uniformly suitable asymptotic approximation of the solution to problem
(3.1)–(3.2) is constructed using the renormalization method in the following most simple
form. Let us pass to the inner variables
x = √εη, t = √εθ,
sincethisallowsonetotakeintoaccountalltermsinequation(3.1). Asafirstapproximation,
we take the solution of the equation
∂Z ∂Z ∂3Z
+Z + = 0, (3.3)
∂θ ∂η ∂η3
with the initial condition
Λ , η < 0,
Z(η,0) = −0 (3.4)
(Λ+, η > 0,
0
where Λ = lim Λ(s). As a model of collisionless shock waves, problem (3.3)–(3.4) was
±0
s
→±∞
studied by A.V. Gurevich and L.P. Pitaevskii in [13].
Let us construct the expansion of the solution in the following form:
u(x,t,ε,ρ) = Z(η,θ)+µW(η,θ,µ)+O(µα), α > 0, (3.5)
where the addend µW(η,θ,µ) must eliminate the singularity of Z at the initial moment of
time. Then the function W satisfies the linear equation
∂W ∂(ZW) ∂3W
+ + = 0. (3.6)
∂θ ∂η ∂η3
Differentiating equation (3.3) with respect to η, we find that the expression
1 ∂Z(η,θ)
G(η,θ) =
Λ+ Λ ∂η
0 − −0
satisfies equation (3.6). Moreover, G is the Green function, because
∞ 1 ∞
lim G(η,θ)f(η)dη = Z(η,0)f (η)dη = f(0)
θ→+0 Z −Λ+0 −Λ−0 Z ′
−∞ −∞
for any smooth function f with compact support, thus G(η,0) = δ(η).
Let us choose the solution W in the form the convolution with the Green function G
so that the asymptotic approximation would satisfy the initial condition (3.2). As a result,
expansion (3.5) becomes
u(x,t,ε,ρ) = U (x,t,ε,ρ)+O(µα),
0
5
where
µ ∞ ∂Z(η µs,θ)
U (x,t,ε,ρ) = Z(η,θ)+ − [Λ(s) Z(s,0)] ds.
0 Λ+ Λ ∂η −
0 − −0 Z
−∞
Integrating by parts, we obtain the asymptotic approximation
1 ∞ x ρs t
u(x,t,ε,ρ) U (x,t,ε,ρ) = Z − , Λ(s)ds.
≈ 0 Λ+ Λ √ε √ε ′
0 − −0 Z (cid:18) (cid:19)
−∞
A rigorous mathematical justification of this formal representation is out of the framework
of the present paper. It is clear that the behavior of the convolution integral is essentially
determined by the solution of (3.3)–(3.4) with discontinuous initial data that confirms the
importance of the Gurevich–Pitaevskii problem.
Constructing complete asymptotic expansions of the solution near the singular point by
the standardmatching method may beconnected with serious difficulties. Infact, it is neces-
sary to solve the scattering problem for a recurrence system of partial differential equations
with variable coefficients [14]. In addition, the investigation of the shock wave generated
by gradient catastrophe shows that the asymptotics of the solution in a neighborhood of a
singular point may have a multiscale structure [8].
The renormalization approach allows one to construct a uniformly suitable asymptotics
in the whole domain of independent variables avoiding difficulties arising from the matching
procedure.
In particular, for Λ+ = 0 and Λ = a > 0 the following formula was obtained in
0 −0
paper [12]:
x+at x 2at/3
u(x,t,ε,ρ) 2Λ Λ −
≈ ρ − ρ −
(cid:18) (cid:19) (cid:18) (cid:19)
2/3
at x aty a3/2tω(y)
Λ − 2dn2 ,σ(y) +σ2(y) dy,
′
− ρ ρ √ε
Z (cid:18) (cid:19)(cid:20) (cid:18) (cid:19) (cid:21)
1
−
where dn(v,σ) is the elliptic Jacobi function
ϕ(u)
dv
dn(u,m) = 1 msin2ϕ, u = ,
− 1 msin2v
q Z0 −
1 1 2pσ2(1 σ2)K(σ)
ω(y) = y 1+σ2(y) , 1+σ2 − = 3y,
√6 − 3 − E(σ) (1 σ2)K(σ)
(cid:26) (cid:27) − −
K(σ) and E(σ) are complet(cid:2)e elliptic i(cid:3)ntegrals of first and second kind:
π/2 π/2
dv
K(σ) = E(σ) = 1 σ2sin2vdv.
1 σ2sin2v −
Z Z
0 − 0 p
Note that using the chanpge x = ρξ, t = ρτ for (3.1)–(3.2), we obtain the problem
∂u ∂u ∂3u
+u +µ 2 = 0, u = Λ(ξ),
∂τ ∂ξ − ∂ξ3 |τ=0
which formally contains only one parameter. However, in this case one should study asymp-
totics as µ 2 and τ .
−
→ ∞ → ∞
6
4 Perturbation of a weak discontinuity
Now consider the Cauchy problem for the KdV equation
∂u ∂u ∂3u
+u + = 0 (4.1)
∂t ∂x ∂x3
with the initial condition
x
u(x,0,ε) = εΛ , t = 0, x R, (4.2)
ε ∈
(cid:16) (cid:17)
where Λ(s) = sΘ( s)+O(s 2) as s , Θ is the Heaviside function. By analogy with
−
− − → ∞
the case of the Burgers equation we postulate the expression
∞ d2Λ(s)
R(x,t,ε) = Φ(x εs,t)ds (4.3)
ds2 −
Z
−∞
as an asymptotic solution for 0 < t 6 δ < 1, where Φ(x,t) is a smooth solution of the
Faminskii problem [15]
Φ +ΦΦ +Φ = 0, Φ(x,0) = xΘ( x) (4.4)
t x xxx
− −
with an initial weak discontinuity also called a contact discontinuity. It is easy to show that
the function (4.3) exactly satisfies the initial condition (4.2).
Theorem 2. Let Λ(s) be a twice differentiable function and let
dΛ(s)
Λ(s) = sΘ( s)+O(s 2), = Θ( s)+O( s 3), s . (4.5)
− −
− − ds − − | | → ∞
Then the function (4.3) is an asymptotic solution of the Korteweg–de Vries equation (4.1).
Proof. From (4.5) it follows that
1/√ε 1/√ε
Λ (s)ds = 1+O(ε3/2), sΛ (s)ds = O(ε).
′′ ′′
Z Z
1/√ε 1/√ε
− −
Taking into account these relations, using (4.4), and substituting u = R(x,t,ε) into (4.1),
we find
∞ ∞
R +RR +R = Λ (s)Φ(x εs,t)ds Λ (s)Φ (x εs,t)ds
t x xxx ′′ ′′ x
− − −
Z Z
−∞ −∞
∞
Λ (s)Φ(x εs,t)Φ (x εs,t)ds = O(ε3/2).
′′ x
− − −
Z
−∞
Acknowledgments. This work was supported by the Russian Foundation for Basic
Research, project no. 14-01-00322.
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