Nonlinear transport phenomena: models, method of solving and unusual features (4)

Nonlinear transport phenomena:models, methods of solving and unusual

features: Lecture 4

Vsevolod Vladimirov

AGH University of Science and technology, Faculty of AppliedMathematics

Krakow, August 6, 2010

KPI, 2010, L4 Nonlinear transport phenomena, KdV equation 1 / 9

Historical background: from Scott Russel’s observationto KdV equation discovery

One of the most advanced mathematical theory dealing with wave patterns’formation and evolution is the soliton theory. The origin of this theory goesback to Scott Russell’s description of the solitary wave movement in thesurface of channel filled with water.

It was the ability of the wave to move quite a long distance without anychange of shape which stroke the imagination of the first chronicler of thisphenomenon.

Scott Russell was the first person describing the solitary wave in a scientificpaper. Unfortunately, his report did not involve any impact in these days.

Nevertheless, the scientist seemed to be aware of finding out something

unusual. Later on he reconstructed solitary waves in the artificial channel

and has observed certain dynamical features of solitons, rediscovered in the

second half of the XX century.





























In 1895 Korteveg and de Vries put forward their famousequation, describing long nonlinear waves’ evolution on ashallow water. In dimensionless variables, they can be describedas follows: form

ut + β uux + uxxx = 0, (1)

Korteveg and de Vries found the analytical solution to thisequation, corresponding to the solitary wave:

u =12 a2

βsech2

[a(x− 4 a2 t)

]. (2)

Both the already mentioned report by Scott Russell as well asthe model suggested to explain his observation did not involve aproper impact till the middle of 60-th of the XX century whenthere have been established a number of outstanding features ofequation (1).



ut + β uux + uxxx = 0, (1)


u =12 a2

βsech2

[a(x− 4 a2 t)

]. (2)




ut + β uux + uxxx = 0, (1)


u =12 a2

βsech2

[a(x− 4 a2 t)

]. (2)



Unusual features of the solutions of the KdV equation:

1. ”elastic” collision of two solitons (nonlinear analog ofsuperposition principle)

2. existence of the soliton set within the smooth Cauchydata.

THESE FEATURES ARE NOW RECOGNIZED ASCONSEQUENCES OF COMPLETE INTEGRABILITYOF THE KdV EQUATION












Miura’s transformation and the Sturm-Lioville problem

Using the proper change of variables

u = Au, x = B x, t = C t,

we can present any KdV-type equation

ut + αuux + β uxxx = 0

into the standard form

ut − 6 u ut + ux x x = 0. (3)

(in what follows, we omit bars over the variables)Lemma 1. The Miura’s transformation

u = v2 + vx

transforms the equation (3) into the modified KdV equation

vt − 6 v2 vx + vx x x = 0. (4)

Proof: inserting ansatz to KdV we get(2v + ∂

∂ x) (vt − 6 v2 vx + vxxx) = 0.KPI, 2010, L4 Nonlinear transport phenomena, KdV equation 5 / 9



u = Au, x = B x, t = C t,




ut − 6 u ut + ux x x = 0. (3)


u = v2 + vx


vt − 6 v2 vx + vx x x = 0. (4)





u = Au, x = B x, t = C t,




ut − 6 u ut + ux x x = 0. (3)


u = v2 + vx


vt − 6 v2 vx + vx x x = 0. (4)





u = Au, x = B x, t = C t,




ut − 6 u ut + ux x x = 0. (3)


u = v2 + vx


vt − 6 v2 vx + vx x x = 0. (4)





u = Au, x = B x, t = C t,




ut − 6 u ut + ux x x = 0. (3)


u = v2 + vx


vt − 6 v2 vx + vx x x = 0. (4)



Lemma 2. The transformation

v =ψx

ψ

transforms the initial problem into the linear equation

ψx x − u(t, x)ψ = 0. (5)

Lemma 3.KdV equation ut − 6uux + uxxx = 0 is invariant with respect tothe following transformation:

t′ = t, x′ = x− 6λ t, u′ = u+ λ.

.Proof.

1. ∂∂ t

= ∂∂ t′ − 6λ ∂

∂ x′ ,∂

∂ x′ = ∂∂ x.

2.

∂ u′

∂ t′− 6λ

∂ u′

∂ x′ − 6 (u′ − λ)∂ u′

∂ x′ +∂3 u′

∂ x′3 =

=∂ u′

∂ t′− 6u′ ∂ u

′

∂ x′ +∂3 u′

∂ x′3 = 0.



v =ψx

ψ


ψx x − u(t, x)ψ = 0. (5)


t′ = t, x′ = x− 6λ t, u′ = u+ λ.

.Proof.

1. ∂∂ t

= ∂∂ t′ − 6λ ∂

∂ x′ ,∂

∂ x′ = ∂∂ x.

2.

∂ u′

∂ t′− 6λ

∂ u′

∂ x′ − 6 (u′ − λ)∂ u′

∂ x′ +∂3 u′

∂ x′3 =

=∂ u′

∂ t′− 6u′ ∂ u

′

∂ x′ +∂3 u′

∂ x′3 = 0.



v =ψx

ψ


ψx x − u(t, x)ψ = 0. (5)


t′ = t, x′ = x− 6λ t, u′ = u+ λ.

.Proof.

1. ∂∂ t

= ∂∂ t′ − 6λ ∂

∂ x′ ,∂

∂ x′ = ∂∂ x.

2.

∂ u′

∂ t′− 6λ

∂ u′

∂ x′ − 6 (u′ − λ)∂ u′

∂ x′ +∂3 u′

∂ x′3 =

=∂ u′

∂ t′− 6u′ ∂ u

′

∂ x′ +∂3 u′

∂ x′3 = 0.



v =ψx

ψ


ψx x − u(t, x)ψ = 0. (5)


t′ = t, x′ = x− 6λ t, u′ = u+ λ.

.Proof.

1. ∂∂ t

= ∂∂ t′ − 6λ ∂

∂ x′ ,∂

∂ x′ = ∂∂ x.

2.

∂ u′

∂ t′− 6λ

∂ u′

∂ x′ − 6 (u′ − λ)∂ u′

∂ x′ +∂3 u′

∂ x′3 =

=∂ u′

∂ t′− 6u′ ∂ u

′

∂ x′ +∂3 u′

∂ x′3 = 0.



v =ψx

ψ


ψx x − u(t, x)ψ = 0. (5)


t′ = t, x′ = x− 6λ t, u′ = u+ λ.

.Proof.

1. ∂∂ t

= ∂∂ t′ − 6λ ∂

∂ x′ ,∂

∂ x′ = ∂∂ x.

2.

∂ u′

∂ t′− 6λ

∂ u′

∂ x′ − 6 (u′ − λ)∂ u′

∂ x′ +∂3 u′

∂ x′3 =

=∂ u′

∂ t′− 6u′ ∂ u

′

∂ x′ +∂3 u′

∂ x′3 = 0.


Corollary.The problem of solving the Cauchy problem

ut − 6uux + uxxx = 0, u(0, x) = ϕ(x)

is equivalent to the solution of the Sturm-Lioville problem

λψ = −ψx x + ϕ(x)ψ

for the Schrodinger operator L = − d2

d x2 + ϕ(x).

IN THIS SENSE THE KdV EQUATION IS SAID TO BECOMPLETELY INTEGRABLE


The Rosenay-Hyman generalization to KdV

K(m, n) = ut + (um)x + (um)x x x = 0, m ≥ 2, n ≥ 2

possesses a one-parameter family of the TW solutions withcompact supports.For m = n = 2 the compactly supported solution has the form

u(t, x) =

{{4 V3 cos

[18 (x− V t)

]}2 if |x− V t| ≤ 4π0 otherwise

The members of K(m, n) hierarchy are not, generally speaking,completely integrable (for they do not pass the tests forcomplete integrability).

Nevertheless they inherit many features of the KdV equation!!!





u(t, x) =

{{4 V3 cos

[18 (x− V t)








u(t, x) =

{{4 V3 cos

[18 (x− V t)








u(t, x) =

{{4 V3 cos

[18 (x− V t)








u(t, x) =

{{4 V3 cos

[18 (x− V t)








u(t, x) =

{{4 V3 cos

[18 (x− V t)





An open question

The solutions to KdV equation possess unusual features,because:

I solution to KdV equation is equivalent to the solution ofthe linear Schrodinger equation;

I KdV equation possesses an infinite set of so calledconservation laws

A common member of the K(m, n) hierarchy

I is not equivalent to any linear equation;I does not possess an infinite set of conservation laws.

Nevertheless, solutions to K(m, n) equations demonstratefeatures very similar to those of the KdV equation.

WHY?


An open question







WHY?


An open question







WHY?


An open question







WHY?


An open question







WHY?


An open question







WHY?


An open question







WHY?


Education

Nonlinear transport phenomena: models, method of solving and unusual features (4)