€
a∂ 2u
∂t 2+ b
∂ 2u
∂t∂ x+ c
∂ 2u
∂ x 2+ d
∂u
∂t+ e
∂u
∂ x+ f u = g(x, t)
Δ = b2 − 4ac
Classification of 2nd order PDEs
Classification of 2nd order PDEs
€
Δ > 0 ⇒ hyperbolic equations
∂ 2u
∂t 2− c0
2 ∂ 2u
∂ x 2= h1 t, x,u,
∂u
∂t,∂u
∂ x
⎛
⎝ ⎜
⎞
⎠ ⎟ canonical form
e.g. ∂ 2u
∂t 2− c0
2 ∂ 2u
∂ x 2= 0 →
∂u
∂t+ c0
∂u
∂ x= 0
∂ 2u
∂t∂ x= h2 t, x,u,
∂u
∂t,∂u
∂ x
⎛
⎝ ⎜
⎞
⎠ ⎟ characteristic form
Classification of 2nd order PDEs
€
Δ =0 ⇒ parabolic equations
∂ 2u
∂ x 2= h3 t,x,u,
∂u
∂t,∂u
∂ x
⎛
⎝ ⎜
⎞
⎠ ⎟ e.g.
∂u
∂t= μ
∂ 2u
∂ x 2
Δ < 0 ⇒ elliptic equations
∂ 2u
∂t 2+
∂ 2u
∂ x 2= h4 t, x,u,
∂u
∂t,∂u
∂ x
⎛
⎝ ⎜
⎞
⎠ ⎟
e.g. ∇ 2u = 0
e.g.∂u
∂t= μ∇ 2u
Homogeneous state and its stability€
∂u
∂t= μ
∂ 2 u
∂ x 2
u = 0stable and attracting if μ > 0
unstable if μ < 0
⎧ ⎨ ⎩
for μ = 0 there is a bifurcation
An important case:
the heat equation in 1D
€
∂u
∂t= μ
∂ 2u
∂ x 2
u x, t( ) =1
2πˆ u (k)
−∞
∞
∫ exp ikx − μk 2t( ) dk
ˆ u (k) = u(x, 0)−∞
∞
∫ exp −ikx( ) dx
ˆ u (k, t) = ˆ u (k) exp −μk 2t( ) = ˆ u (k) ˆ S (k)
the heat equation in 1D (Fourier approach)
€
S(x, t) =1
2πˆ S (k, t)
−∞
∞
∫ exp ikx( )dk
S(x, t) =1
4π μ texp −
x 2
4μ t
⎛
⎝ ⎜
⎞
⎠ ⎟
ˆ u (k) ˆ S (k, t) ⇒ u(x,0)∗S(x, t)
u(x, t) =1
4π μ texp −
x − y( )2
4μ t
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟u(y,0) dy
−∞
∞
∫
The heat equation in 1D
€
u(x,0) =δ(x)
uG (x, t) =1
4π μ texp −
x − y( )2
4 μ t
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟δ(y) dy
−∞
∞
∫
uG (x, t) =1
4π μ texp −
x 2
4 μ t
⎛
⎝ ⎜
⎞
⎠ ⎟
Green’s function solution of the heat equation
€
∂u
∂t= μ
∂ 2u
∂ x 2
u = u(z = x − ct)
c∂u
∂z+ μ
∂ 2u
∂z2= 0 ⇒ u = A + Bexp −
cz
μ
⎛
⎝ ⎜
⎞
⎠ ⎟
B = 0
The heat equation does not havetravelling wave solutions
€
∂u
∂t= f (u) + μ
∂ 2u
∂ x 2
Reaction-diffusion equations
€
∂ ru
∂t=
r f (
r u ) + μ∇ 2 r
u
in 1D, for a scalar field
€
∂u
∂t= ν u + μ
∂ 2u
∂ x 2
u(x, t) =1
2πˆ u (k)exp ikx − iω t( )
−∞
∞
∫ dk
u(x, t) =1
2πˆ u (k)exp ikx( )
−∞
∞
∫ exp ν − μ k 2( ) t[ ] dk
instability (growth) for k <ν
μ
⎛
⎝ ⎜
⎞
⎠ ⎟
12
The simplest case:the KISS model for plankton blooms
€
∂u
∂t= ν u 1−
u
K
⎛
⎝ ⎜
⎞
⎠ ⎟+ μ
∂ 2u
∂ x 2
u → ˜ u =u
K, t → ˜ t = ν t , x → ˜ x =
ν
μ
⎛
⎝ ⎜
⎞
⎠ ⎟
12
x
∂u
∂t=u 1− u( ) +
∂ 2u
∂ x 2
Need for a saturation mechanism:the Fisher equation
The equation is now nonlinear…
€
∂u
∂t−u 1− u( ) −
∂ 2u
∂ x 2= 0
∂
∂tˆ u (k)
−∞
∞
∫ exp ikx − iω t( )dk ⎧ ⎨ ⎩
⎫ ⎬ ⎭−
∂ 2
∂ x 2ˆ u (k)
−∞
∞
∫ exp ikx − iω t( )dk ⎧ ⎨ ⎩
⎫ ⎬ ⎭
− ˆ u (k)−∞
∞
∫ exp ikx − iω t( )dk ⎧ ⎨ ⎩
⎫ ⎬ ⎭
+ ˆ u (k)−∞
∞
∫ ˆ u (k ')exp i k + k '( )x − i ω +ω'( ) t[ ] dk'dk ⎧ ⎨ ⎩
⎫ ⎬ ⎭= 0
Fourier decomposition does not help now
Fisher equation: homogeneous solution(the logistic equation)
€
∂u
∂t=u 1− u( ) +
∂ 2u
∂ x 2
∂
∂ x= 0
d u
dt=u 1− u( )
the logistic equation
€
d u
dt=u 1− u( )
fixed points (stationary states)
u = 0 , u =1
linear stability of the fixed points
exact solution :
u(t) =u0 exp t
1+ u0 exp t −1( )
back to the Fisher equation: travelling waves
€
∂u
∂t=u 1− u( ) +
∂ 2u
∂ x 2, U(z) = u(x − c t)
d2U
dz2+ c
dU
dz+ U 1−U( ) = 0
dU
dz= V
dV
dz= −cV −U 1−U( )
⎧
⎨ ⎪ ⎪
⎩ ⎪ ⎪
dynamical systems in a 2d phase space
€
dU
dz= f U,V( )
dV
dz= g U,V( )
⎧
⎨ ⎪ ⎪
⎩ ⎪ ⎪
types of fixed points and their linearized stability
bifurcations : pitchfork, saddle - node, Hopf
€
dU
dz= f U,V( )
dV
dz= g U,V( )
⎧
⎨ ⎪ ⎪
⎩ ⎪ ⎪
fixed points : f U ,V ( ) = g U ,V ( ) = 0
small perturbation : U = U + u , V = V + v
du
dz= f U + u,V + v( ) ≈
∂ f
∂U
⎛
⎝ ⎜
⎞
⎠ ⎟
U ,V
u +∂ f
∂V
⎛
⎝ ⎜
⎞
⎠ ⎟
U ,V
v
dv
dz= g U + u,V + v( ) ≈
∂g
∂U
⎛
⎝ ⎜
⎞
⎠ ⎟
U ,V
u +∂g
∂V
⎛
⎝ ⎜
⎞
⎠ ⎟
U ,V
v
⎧
⎨ ⎪ ⎪
⎩ ⎪ ⎪
€
du
dz= f U + u,V + v( ) ≈
∂ f
∂U
⎛
⎝ ⎜
⎞
⎠ ⎟
U ,V
u +∂ f
∂V
⎛
⎝ ⎜
⎞
⎠ ⎟
U ,V
v
dv
dz= g U + u,V + v( ) ≈
∂g
∂U
⎛
⎝ ⎜
⎞
⎠ ⎟
U ,V
u +∂g
∂V
⎛
⎝ ⎜
⎞
⎠ ⎟
U ,V
v
⎧
⎨ ⎪ ⎪
⎩ ⎪ ⎪
d
dz
u
v
⎛
⎝ ⎜
⎞
⎠ ⎟=
∂ f
∂U
∂ f
∂V∂g
∂U
∂g
∂V
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟
U ,V
u
v
⎛
⎝ ⎜
⎞
⎠ ⎟
det
∂ f
∂U
∂ f
∂V∂g
∂U
∂g
∂V
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟
U ,V
− λ1 0
0 1
⎛
⎝ ⎜
⎞
⎠ ⎟
⎧
⎨ ⎪
⎩ ⎪
⎫
⎬ ⎪
⎭ ⎪= 0
€
d
dz
u
v
⎛
⎝ ⎜
⎞
⎠ ⎟=
∂ f
∂U
∂ f
∂V∂g
∂U
∂g
∂V
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟
U ,V
u
v
⎛
⎝ ⎜
⎞
⎠ ⎟
Tr
∂ f
∂U
∂ f
∂V∂g
∂U
∂g
∂V
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟
U ,V
=∂ f
∂U+
∂g
∂V= λ 1 + λ 2
dissipative and conservative dynamical systems
€
real eigenvalues
λ +,λ− < 0 stable node
λ + > 0 , λ− < 0 saddle point
λ +,λ− > 0 unstable node
complex conjugate eigenvalues : λ + = λ R + iλ I , λ− = λ R − iλ I
λ R < 0 stable focus
λ R > 0 unstable focus
pure imaginary eigenvalues : λ + = iλ I , λ− = −iλ I center
€
pitchfork bifurcation
dU
dz= α U −U 3 supercritical
dU
dz= α U + U 3 subcritical
types of bifurcation:normal form
€
saddle - node bifurcation
dU
dz= α −U 2 supercritical
dU
dz= α −U 2 subcritical
types of bifurcation:normal form
€
Hopf bifurcation (supercritical)
dU
dz= α U −V −U U 2 + V 2
( )
dV
dz= U + α V −U U 2 + V 2
( )
radial form
dρ
dz= ρ α − ρ 2
( )
dθ
dz=1
types of bifurcation: normal form
Fisher equation: travelling wave
€
dU
dz= V
dV
dz= −cV −U 1−U( )
⎧
⎨ ⎪ ⎪
⎩ ⎪ ⎪
fixed points
U,V( ) = (0,0) ; U,V( ) = (1,0)
(0,0) → λ ± =1
2−c ± c 2 − 4( )
12 ⎡
⎣ ⎢ ⎤ ⎦ ⎥ ⇒
stable node if c 2 ≥ 4
stable focus if c 2 < 4
⎧ ⎨ ⎩
(1,0) → λ ± =1
2−c ± c 2 + 4( )
12 ⎡
⎣ ⎢ ⎤ ⎦ ⎥ ⇒ saddle point
Fisher equation: phase-plane picture
€
dV
dU=
−cV −U 1−U( )V
Fisher equation: travelling wave
http://commons.wikimedia.org/wiki/File:Travelling_wave_for_Fisher_equation.svg
z
c ≥ 2
Fisher equation: travelling wavegeneral initial condition
€
u(x,0) =1 x < x1
u(x,0) = 0 x ≥ x2
u → U(x − 2t)
u(x,0) ≈ Aexp −ax( ) x → ∞
u(x, t) ≈ Aexp −a x − ct( )[ ] x → ∞
c = a +1
aif 0 < a ≤1 ; c = 2 if a >1
Fisher equation: travelling wave
asymptotic form
stability of the travelling wave€
dV
dU=
−cV −U 1−U( )V
no general solution ∀c
expand in ε =1
c 2
A different form of nonlinearity…
€
∂u
∂t+ α u
∂u
∂ x= 0
∂
∂tˆ u (k)
−∞
∞
∫ exp ikx − iω t( )dk ⎧ ⎨ ⎩
⎫ ⎬ ⎭
+ α ˆ u (k)−∞
∞
∫ exp ikx − iω t( )dk ⎧ ⎨ ⎩
⎫ ⎬ ⎭
∂
∂ xˆ u (k ')
−∞
∞
∫ exp ik ' x − iω t( )dk' ⎧ ⎨ ⎩
⎫ ⎬ ⎭= 0
Again, Fourier decomposition does not help
Heuristic, just to understand…
€
∂u
∂t+ α u
∂u
∂ x= 0
ω(k) = c k =α uk
c = α u
This is an hyperbolic equation
Singularity in finite time
The Burgers equation
€
∂u
∂t+ α u
∂u
∂ x= μ
∂ 2 u
∂ x 2
Shock solutions
the Cole-Hopf transformation
€
∂u
∂t+ α u
∂u
∂ x= μ
∂ 2 u
∂ x 2
u = −2μϕ x
ϕ
u =ψ x → ψ t +1
2ψ x( )
2= μ ψ xx
ψ = −2μ logϕ
ϕ t = μϕ xx
ϕ x,0( ) = exp −1
2μu η ,0( ) dη
0
x
∫ ⎡
⎣ ⎢
⎤
⎦ ⎥
u =
x −η
texp −
G
2μ
⎛
⎝ ⎜
⎞
⎠ ⎟dη
−∞
∞
∫
exp −G
2μ
⎛
⎝ ⎜
⎞
⎠ ⎟dη
−∞
∞
∫; G η ;x, t( ) = u η ',0( ) dη '
0
η
∫ +x −η( )
2
2 t
The Burgers equation:
shock structure
general solution and confluence of shocks
test for numerical methods
stochastically-forced Burgers eq.
The Korteweg-de Vries equation
€
∂u
∂t+ α u
∂u
∂ x+
∂ 3 u
∂ x 3= 0
Travelling wave solutions:“solitary waves”
on the periodic: “cnoidal waves”
€
u x, t( ) = 3κ 2 sech2 κ x −κ 3t −θ0
2
⎛
⎝ ⎜
⎞
⎠ ⎟
The inverse scattering transformfor the Korteweg-de Vries equation
€
u x, t( ) =ψ xx
ψ+ λ
⇓
ψ xx + λ − u x, t( )[ ] ψ = 0
€
∂u
∂t− 6u
∂u
∂ x+
∂ 3 u
∂ x 3= 0