Consider the PDE known as the complex Ginzburg Landau (CGL) equation u₁ = Ru + (1 + iv)Au - (1 + i)ulu² (E9.2) on the 2d periodic domain = [0, 1]². The parameters R, μ and v are real and lie in the range R > 1, \µ| < ∞o and |v|< ∞o. Show that (i) when [u] ≤ √√3, then d₁ ≤ R whereas (ii) when |μ| > √√3 then d₁ ≤ c R where c is independent of v but not µ.
Consider the PDE known as the complex Ginzburg Landau (CGL) equation u₁ = Ru + (1 + iv)Au - (1 + i)ulu² (E9.2) on the 2d periodic domain = [0, 1]². The parameters R, μ and v are real and lie in the range R > 1, \µ| < ∞o and |v|< ∞o. Show that (i) when [u] ≤ √√3, then d₁ ≤ R whereas (ii) when |μ| > √√3 then d₁ ≤ c R where c is independent of v but not µ.
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter4: Vector Spaces
Section4.8: Applications Of Vector Spaces
Problem 83E
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Consider the PDE known as the complex Ginzburg Landau
(CGL) equation
u₁ = Ru + (1 + iv) Au-(1 + iu)ulu²
(E9.2)
on the 2d periodic domain = [0, 1]². The parameters R, μ and
v are real and lie in the range R > 1, μ| <∞o and v| <∞o. Show
that (i) when u ≤ √√3, then d₁ ≤ R whereas (ii) when |μ| >√√3
then dy ≤ c R where c is independent of v but not μ.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa303b5f8-6488-4989-8df3-bdae41015b1e%2Ff2383bd8-0481-4e7b-a229-8e66adb7e9ce%2Fzhrp04s_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3
Consider the PDE known as the complex Ginzburg Landau
(CGL) equation
u₁ = Ru + (1 + iv) Au-(1 + iu)ulu²
(E9.2)
on the 2d periodic domain = [0, 1]². The parameters R, μ and
v are real and lie in the range R > 1, μ| <∞o and v| <∞o. Show
that (i) when u ≤ √√3, then d₁ ≤ R whereas (ii) when |μ| >√√3
then dy ≤ c R where c is independent of v but not μ.
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