Simulation Examples

Example 1: Cosine Wave Initial Condition

λ, T, Ω = params(λ = 0.8im)

xᵣ = 0=>20
box = Box(xᵣ, T, dx=5e-3, Nₜ = 256, n_periods = 1)

coeff = [1e-4]
ψ₀, A₀ = ψ₀_periodic(coeff, box, Ω)

sim = Sim(λ, box, ψ₀, T4A_TJ!)

solve!(sim)
compute_IoM!(sim)
surface(sim)
heatmap(sim, :ψ̃)
plot(sim, :IoM)

[ Info: Passed λ=0.0 + 0.8im, computed T = 5.23598775598299 and Ω = 1.1999999999999997
[ Info: Initializing Box with 1 period(s) and dx = 0.005, Nₜ = 256.
[ Info: Longitudinal range is [0, 20], transverse range is [-2.617993877991495, 2.617993877991495)
[ Info: Done computing t, x, ω
[ Info: Initializing periodic ψ₀
[ Info: Computing A₀ to preserve normalization.
[ Info: Computed A₀ = 0.99999999
[ Info: ψ₀ = 0.99999999 + 2 × 0.0001 × cos(1 × 1.1999999999999997 t)
[ Info: Passed λ=0.0 + 0.8im, computed T = 5.23598775598299 and Ω = 1.1999999999999997
[ Info: Generating FFT plans
[ Info: Starting evolution
[ Info: Computing Spectrum
[ Info: Computation Done!
[ Info: Computing integrals of motions
[ Info: Integrals of motion computed.
/home/runner/.julia/packages/GR/G9I5v/src/../deps/gr/bin/gksqt: error while loading shared libraries: libQt5Widgets.so.5: cannot open shared object file: No such file or directory
connect: Connection refused
GKS: can't connect to GKS socket application

GKS: Open failed in routine OPEN_WS
GKS: GKS not in proper state. GKS must be either in the state WSOP or WSAC in routine ACTIVATE_WS
/home/runner/.julia/packages/GR/G9I5v/src/../deps/gr/bin/gksqt: error while loading shared libraries: libQt5Widgets.so.5: cannot open shared object file: No such file or directory
connect: Connection refused
GKS: can't connect to GKS socket application

GKS: Open failed in routine OPEN_WS
GKS: GKS not in proper state. GKS must be either in the state WSOP or WSAC in routine ACTIVATE_WS
/home/runner/.julia/packages/GR/G9I5v/src/../deps/gr/bin/gksqt: error while loading shared libraries: libQt5Widgets.so.5: cannot open shared object file: No such file or directory
connect: Connection refused
GKS: can't connect to GKS socket application

GKS: Open failed in routine OPEN_WS
GKS: GKS not in proper state. GKS must be either in the state WSOP or WSAC in routine ACTIVATE_WS

Example 2: Soliton Initial Condition

λ = 0.75im

T = 20
xᵣ = 0=>20
box = Box(xᵣ, T, dx=1e-2, Nₜ = 256, n_periods = 1)

ψ₀ = Array{Complex{Float64}}(undef, box.Nₜ)
ψ₀ .= 2*imag(λ)./cosh.(2*imag(λ).*box.t)

sim = Sim(λ, box, ψ₀, T4A_TJ!)

solve!(sim)
surface(sim)

[ Info: Initializing Box with 1 period(s) and dx = 0.01, Nₜ = 256.
[ Info: Longitudinal range is [0, 20], transverse range is [-10.0, 10.0)
[ Info: Done computing t, x, ω
[ Info: Passed λ=0.0 + 0.75im, computed T = 4.749641646894903 and Ω = 1.3228756555322954
[ Info: Generating FFT plans
[ Info: Starting evolution
[ Info: Computing Spectrum
[ Info: Computation Done!
/home/runner/.julia/packages/GR/G9I5v/src/../deps/gr/bin/gksqt: error while loading shared libraries: libQt5Widgets.so.5: cannot open shared object file: No such file or directory
connect: Connection refused
GKS: can't connect to GKS socket application

GKS: Open failed in routine OPEN_WS
GKS: GKS not in proper state. GKS must be either in the state WSOP or WSAC in routine ACTIVATE_WS

Example 3: Nonlinear Talbot Carpet (Pruning)

λ, T, Ω = params(a = 0.36)

xᵣ = 0=>60
box = Box(xᵣ, T, dx=1e-2, Nₜ = 512, n_periods = 5)

coeff = [(2.7 + 4.6im)*1e-2]
ψ₀, A₀ = ψ₀_periodic(coeff, box, Ω)

sim = Sim(λ, box, ψ₀, T4A_TJ!, β=10.0)

solve!(sim)
heatmap(sim)
plot(sim, :ψ̃)

[ Info: Passed a = 0.36, computed λ = 0.0 + 0.848528137423857im, T = 5.937052058618629 and Ω = 1.0583005244258363
[ Info: Initializing Box with 5 period(s) and dx = 0.01, Nₜ = 512.
[ Info: Longitudinal range is [0, 60], transverse range is [-14.842630146546572, 14.842630146546572)
[ Info: Done computing t, x, ω
[ Info: Initializing periodic ψ₀
[ Info: Computing A₀ to preserve normalization.
[ Info: Computed A₀ = 0.9971509414326399
[ Info: ψ₀ = 0.9971509414326399 + 2 × 0.027000000000000003 + 0.046im × cos(1 × 1.0583005244258363 t)
[ Info: Passed λ=0.0 + 0.848528137423857im, computed T = 5.937052058618629 and Ω = 1.0583005244258363
[ Info: Generating FFT plans
[ Info: Starting evolution
[ Info: Computing Spectrum
[ Info: Computation Done!
/home/runner/.julia/packages/GR/G9I5v/src/../deps/gr/bin/gksqt: error while loading shared libraries: libQt5Widgets.so.5: cannot open shared object file: No such file or directory
connect: Connection refused
GKS: can't connect to GKS socket application

GKS: Open failed in routine OPEN_WS
GKS: GKS not in proper state. GKS must be either in the state WSOP or WSAC in routine ACTIVATE_WS
/home/runner/.julia/packages/GR/G9I5v/src/../deps/gr/bin/gksqt: error while loading shared libraries: libQt5Widgets.so.5: cannot open shared object file: No such file or directory
connect: Connection refused
GKS: can't connect to GKS socket application

GKS: Open failed in routine OPEN_WS
GKS: GKS not in proper state. GKS must be either in the state WSOP or WSAC in routine ACTIVATE_WS

Example 4: Darboux Transformation Initial Condition

λ₁ = 0.98im
λ, T, Ω = params(λ = λ₁)

xᵣ = 0=>20
box = Box(xᵣ, T, dx=5e-3, Nₜ = 512, n_periods = 1)

λ = λ_maximal(λ₁, 5) # array of 5 eigenvalues
xₛ = [0.0, 0.0, 0.0, 0.0, 0.0]
tₛ = [0.0, 0.0, 0.0, 0.0, 0.0]
ψ₀ = ψ₀_DT(λ, tₛ, xₛ, -10, box)

sim = Sim(λ₁, box, ψ₀, T4A_TJ!)

solve!(sim)
compute_IoM!(sim)
heatmap(sim)

[ Info: Passed λ=0.0 + 0.98im, computed T = 15.787097084991364 and Ω = 0.3979949748426484
[ Info: Initializing Box with 1 period(s) and dx = 0.005, Nₜ = 512.
[ Info: Longitudinal range is [0, 20], transverse range is [-7.893548542495682, 7.893548542495682)
[ Info: Done computing t, x, ω
[ Info: Initializing Box with 1 period(s) and dx = 0.0, Nₜ = 512.
[ Info: Longitudinal range is [-10, -9.99999], transverse range is [-7.893548542495682, 7.893548542495682)
[ Info: Done computing t, x, ω
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (5,1)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (4,1)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (3,1)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (2,1)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (1,1)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (1,2)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (2,2)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (1,3)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (3,2)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (2,3)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (1,4)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (4,2)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (3,3)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (2,4)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (1,5)
[ Info: Passed λ=0.0 + 0.98im, computed T = 15.787097084991364 and Ω = 0.3979949748426484
[ Info: Generating FFT plans
[ Info: Starting evolution
[ Info: Computing Spectrum
[ Info: Computation Done!
[ Info: Computing integrals of motions
[ Info: Integrals of motion computed.
/home/runner/.julia/packages/GR/G9I5v/src/../deps/gr/bin/gksqt: error while loading shared libraries: libQt5Widgets.so.5: cannot open shared object file: No such file or directory
connect: Connection refused
GKS: can't connect to GKS socket application

GKS: Open failed in routine OPEN_WS
GKS: GKS not in proper state. GKS must be either in the state WSOP or WSAC in routine ACTIVATE_WS