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janhardo

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  > ....

janhardo 745
May 05 2025
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restart; with(plots); with(LinearAlgebra); printf("Step 1: Define the variables: wave number l and parameter a.\n"); l := 'l'; a := 'a'; printf("Step 2: Define the MI gain function G(l,a) = 2 * Im(sqrt(l^4 - 2*a*l^2)).\n"); G := proc (l, a) options operator, arrow; 2*Im(sqrt(l^4-2*a*l^2)) end proc; printf("Step 3: Create a 3D plot over a range of l and a.\n"); gainPlot := plot3d(G(l, a), l = -3 .. 3, a = .1 .. 2, axes = boxed, labels = ["l", "a", "G(l)"], title = "3D MI Gain Spectrum", grid = [50, 50], style = surfacecontour, shading = zhue); printf("Step 4: Display the 3D plot.\n"); gainPlot

Step 1: Define the variables: wave number l and parameter a.
Step 2: Define the MI gain function G(l,a) = 2 * Im(sqrt(l^4 - 2*a*l^2)).
Step 3: Create a 3D plot over a range of l and a.
Step 4: Display the 3D plot.

  

 
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Download modulatio_instability_plot_of_eq_26_paper_3D_plotmprimes5-5-2025.mw

@salim-barzani  Another parameter n...

janhardo 745
May 05 2025
0 0

@salim-barzani 
Another parameter name for l  .. must  be k in first  plot ?

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janhardo 745
May 05 2025
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restart; with(plots); with(LinearAlgebra); printf("Step 1: Define the wave number variable l and parameter range for plotting.\n"); l := 'l'; lrange := -3 .. 3; printf("Step 2: Define values for the parameter a: a = 1, 0.5, and 0.3.\n"); a1 := 1; a2 := .5; a3 := .3; printf("Step 3: Define the growth rate function G(l) = 2 * Im(sqrt(l^4 - 2*a*l^2)).\n"); G := proc (l, a) options operator, arrow; 2*Im(sqrt(l^4-2*a*l^2)) end proc; printf("Step 4: Create individual plots for each value of a.\n"); plot1 := plot(G(l, a1), l = lrange, color = red, thickness = 2); plot2 := plot(G(l, a2), l = lrange, color = blue, thickness = 2); plot3 := plot(G(l, a3), l = lrange, color = green, thickness = 2); printf("Step 5: Overlay all plots together with proper labels, legend, and styling.\n"); overlay := display([plot1, plot2, plot3], title = "Overlay G(l) for a = 1, 0.5, 0.3", labels = ["Wave number l", "Growth rate G(l)"], legend = ["a = 1", "a = 0.5", "a = 0.3"], axes = boxed, size = [600, 500]); printf("Step 6: Display the final combined plot.\n"); overlay

Step 1: Define the wave number variable l and parameter range for plotting.
Step 2: Define values for the parameter a: a = 1, 0.5, and 0.3.
Step 3: Define the growth rate function G(l) = 2 * Im(sqrt(l^4 - 2*a*l^2)).
Step 4: Create individual plots for each value of a.
Step 5: Overlay all plots together with proper labels, legend, and styling.
Step 6: Display the final combined plot.

  

 
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Download modulatio_instability_plot_of_eq_26_papermprimes5-5-2025.mw

howgeneratesytems_of_equation_solvemprim...

janhardo 745
May 05 2025
0 0

howgeneratesytems_of_equation_solvemprimes5-5-2025.mw

lieariztion_result_and_absolute_value_ha...

janhardo 745
May 05 2025
0 0

lieariztion_result_and_absolute_value_handlingmprimes_5-5-2025.mw

@salim-barzani Probaly not a NLSE p...

janhardo 745
May 04 2025
0 0

@salim-barzani 
Probably not a NLSE pde, so another techniques needed for solving ?
With a example of this pde , ai can maybe decipher the code and explains its working

@salim-barzani Here it is done for ...

janhardo 745
May 04 2025
0 0

@salim-barzani 
Here it is done for NLSE pde, so yes how it works immediately other pde?
Finding coeffiecent is a hassle, but in this procedure i could make progress finding them with a matrix

experiment with procedure parameters 

@mmcdara  "I always wonder how...

janhardo 745
May 03 2025
0 0

@mmcdara 
"I always wonder how to interpret a question like "How to show these two expressions are the same?"
we are working in Maple and trying to get an answer in Maple, it's as simple as that

@nm i am using maple 2024......

janhardo 745
May 03 2025
0 0

@nm
i am using maple 2024...

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janhardo 745
May 03 2025
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a example , but M is not exact calculate...

janhardo 745
May 03 2025
0 0

a example , but M is not exact calculated , so the solution is not fitting


 

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restart; with(DEtools); original_ODE := diff(U(xi), xi, xi)+U(xi)^2-2*U(xi) = 0; M := 2; ansatz := y^2*a[2]+y*a[1]+a[0]; L := 2; varkappa := 1; eta := 1; rho_prime := (xi/y+varkappa+eta*y)/ln(L); y_prime := y*rho_prime*ln(L); y_double_prime := diff(y_prime, xi); U_prime := 2*y*y_prime*a[2]+y_prime*a[1]; U_double_prime := a[1]*y_double_prime+2*a[2]*(y*y_double_prime+y_prime^2); substituted_ODE := eval(original_ODE, [U(xi) = ansatz, diff(U(xi), xi) = U_prime, diff(U(xi), xi, xi) = U_double_prime]); substituted_ODE := expand(simplify(substituted_ODE)); coefficient_eqs := [coeff(coeff(lhs(substituted_ODE), y, 0), xi, 0) = 0, coeff(coeff(lhs(substituted_ODE), y, 1), xi, 0) = 0, coeff(coeff(lhs(substituted_ODE), y, 2), xi, 0) = 0]; solution := solve(coefficient_eqs, {a[0], a[1], a[2]}); print(solution)

diff(diff(U(xi), xi), xi)+U(xi)^2-2*U(xi) = 0

  

2

  

y^2*a[2]+y*a[1]+a[0]

  

2

  

1

  

1

  

(xi/y+1+y)/ln(2)

  

y*(xi/y+1+y)

  

1

  

2*y^2*(xi/y+1+y)*a[2]+y*(xi/y+1+y)*a[1]

  

a[1]+2*a[2]*(y+y^2*(xi/y+1+y)^2)

  

a[1]+2*a[2]*(y+y^2*(xi/y+1+y)^2)+(y^2*a[2]+y*a[1]+a[0])^2-2*a[2]*y^2-2*a[1]*y-2*a[0] = 0

  

y^4*a[2]^2+2*y^4*a[2]+2*y^3*a[1]*a[2]+4*xi*y^2*a[2]+4*y^3*a[2]+2*y^2*a[0]*a[2]+y^2*a[1]^2+2*xi^2*a[2]+4*xi*y*a[2]+2*y*a[0]*a[1]-2*y*a[1]+2*y*a[2]+a[0]^2-2*a[0]+a[1] = 0

  

[a[0]^2-2*a[0]+a[1] = 0, 2*a[0]*a[1]-2*a[1]+2*a[2] = 0, 2*a[0]*a[2]+a[1]^2 = 0]

  

{a[0] = 0, a[1] = 0, a[2] = 0}, {a[0] = 2, a[1] = 0, a[2] = 0}, {a[0] = 4/3, a[1] = 8/9, a[2] = -8/27}

  

{a[0] = 0, a[1] = 0, a[2] = 0}, {a[0] = 2, a[1] = 0, a[2] = 0}, {a[0] = 4/3, a[1] = 8/9, a[2] = -8/27}

 (1)
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solution_U := 4/3+(8/9)*L^rho(xi)-(8/27)*(L^rho(xi))^2; rho_prime := (xi/L^rho(xi)+varkappa+eta*L^rho(xi))/ln(L); U_prime := diff(solution_U, xi); U_double_prime := diff(U_prime, xi); ODE_check := solution_U^2+U_double_prime-2*solution_U; simplify(ODE_check)

4/3+(8/9)*2^rho(xi)-(8/27)*(2^rho(xi))^2

  

(xi/2^rho(xi)+1+2^rho(xi))/ln(2)

  

(8/9)*2^rho(xi)*(diff(rho(xi), xi))*ln(2)-(16/27)*(2^rho(xi))^2*(diff(rho(xi), xi))*ln(2)

  

(8/9)*2^rho(xi)*(diff(rho(xi), xi))^2*ln(2)^2+(8/9)*2^rho(xi)*(diff(diff(rho(xi), xi), xi))*ln(2)-(32/27)*(2^rho(xi))^2*(diff(rho(xi), xi))^2*ln(2)^2-(16/27)*(2^rho(xi))^2*(diff(diff(rho(xi), xi), xi))*ln(2)

  

(4/3+(8/9)*2^rho(xi)-(8/27)*(2^rho(xi))^2)^2+(8/9)*2^rho(xi)*(diff(rho(xi), xi))^2*ln(2)^2+(8/9)*2^rho(xi)*(diff(diff(rho(xi), xi), xi))*ln(2)-(32/27)*(2^rho(xi))^2*(diff(rho(xi), xi))^2*ln(2)^2-(16/27)*(2^rho(xi))^2*(diff(diff(rho(xi), xi), xi))*ln(2)-8/3-(16/9)*2^rho(xi)+(16/27)*(2^rho(xi))^2

  

(16/9)*(1+(2/3)*2^rho(xi)-(2/9)*4^rho(xi))^2-8/3+(8/9)*ln(2)*(2^rho(xi)-(2/3)*4^rho(xi))*(diff(diff(rho(xi), xi), xi))+(8/9)*((diff(rho(xi), xi))^2*ln(2)^2-2)*2^rho(xi)-(32/27)*4^rho(xi)*(diff(rho(xi), xi))^2*ln(2)^2+(16/27)*4^rho(xi)

 (2)
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Download berekening_coeiff_M_is_onjuist_en_niet_berekend_3-5-2025.mw

@mmcdara Thanks for the nice proof ...

janhardo 745
May 03 2025
0 0

@mmcdara 

Thanks for the nice proof , it is an ingenious proof though ..could follow it with some help

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janhardo 745
May 03 2025
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janhardo 745
May 03 2025
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janhardo 745
May 03 2025
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restart; with(DEtools); ode := diff(rho(xi), xi) = (xi*L^(-rho(xi))+varkappa+eta*L^rho(xi))/ln(L)

diff(rho(xi), xi) = (xi*L^(-rho(xi))+varkappa+eta*L^rho(xi))/ln(L)

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Gamma := -4*eta*zeta+varkappa^2; sol1 := L^rho(xi) = -varkappa/(2*eta)+sqrt(-Gamma)*tan((1/2)*sqrt(-Gamma)*xi)/(2*eta); sol2 := L^rho(xi) = -varkappa/(2*eta)-sqrt(-Gamma)*cot((1/2)*sqrt(-Gamma)*xi)/(2*eta); rho_sol1 := solve(sol1, rho(xi)); rho_sol2 := solve(sol2, rho(xi))

-4*eta*zeta+varkappa^2

  

L^rho(xi) = -(1/2)*varkappa/eta+(1/2)*(4*eta*zeta-varkappa^2)^(1/2)*tan((1/2)*(4*eta*zeta-varkappa^2)^(1/2)*xi)/eta

  

L^rho(xi) = -(1/2)*varkappa/eta-(1/2)*(4*eta*zeta-varkappa^2)^(1/2)*cot((1/2)*(4*eta*zeta-varkappa^2)^(1/2)*xi)/eta

  

ln((1/2)*((4*eta*zeta-varkappa^2)^(1/2)*tan((1/2)*(4*eta*zeta-varkappa^2)^(1/2)*xi)-varkappa)/eta)/ln(L)

  

ln(-(1/2)*((4*eta*zeta-varkappa^2)^(1/2)+varkappa*tan((1/2)*(4*eta*zeta-varkappa^2)^(1/2)*xi))/(eta*tan((1/2)*(4*eta*zeta-varkappa^2)^(1/2)*xi)))/ln(L)

 (2)
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sol3 := L^rho(xi) = -varkappa/(2*eta)+sqrt(Gamma)*tanh((1/2)*sqrt(Gamma)*xi)/(2*eta); sol4 := L^rho(xi) = -varkappa/(2*eta)-sqrt(Gamma)*coth((1/2)*sqrt(Gamma)*xi)/(2*eta); rho_sol3 := solve(sol3, rho(xi)); rho_sol4 := solve(sol4, rho(xi))

L^rho(xi) = -(1/2)*varkappa/eta+(1/2)*(-4*eta*zeta+varkappa^2)^(1/2)*tanh((1/2)*(-4*eta*zeta+varkappa^2)^(1/2)*xi)/eta

  

L^rho(xi) = -(1/2)*varkappa/eta-(1/2)*(-4*eta*zeta+varkappa^2)^(1/2)*coth((1/2)*(-4*eta*zeta+varkappa^2)^(1/2)*xi)/eta

  

ln((1/2)*((-4*eta*zeta+varkappa^2)^(1/2)*(exp((1/2)*(-4*eta*zeta+varkappa^2)^(1/2)*xi))^2-(exp((1/2)*(-4*eta*zeta+varkappa^2)^(1/2)*xi))^2*varkappa-(-4*eta*zeta+varkappa^2)^(1/2)-varkappa)/(eta*((exp((1/2)*(-4*eta*zeta+varkappa^2)^(1/2)*xi))^2+1)))/ln(L)

  

ln(-(1/2)*((-4*eta*zeta+varkappa^2)^(1/2)*(exp((1/2)*(-4*eta*zeta+varkappa^2)^(1/2)*xi))^2+(exp((1/2)*(-4*eta*zeta+varkappa^2)^(1/2)*xi))^2*varkappa+(-4*eta*zeta+varkappa^2)^(1/2)-varkappa)/(((exp((1/2)*(-4*eta*zeta+varkappa^2)^(1/2)*xi))^2-1)*eta))/ln(L)

 (3)
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ode1 := eval(ode, rho(xi) = rho_sol1); `assuming`([simplify(ode1)], [L > 0, L <> 1, eta <> 0, -4*eta*zeta+varkappa^2 < 0]); ode2 := eval(ode, rho(xi) = rho_sol2); `assuming`([simplify(ode2)], [L > 0, L <> 1, eta <> 0, -4*eta*zeta+varkappa^2 < 0])

(1/2)*(4*eta*zeta-varkappa^2)*(1+tan((1/2)*(4*eta*zeta-varkappa^2)^(1/2)*xi)^2)/(((4*eta*zeta-varkappa^2)^(1/2)*tan((1/2)*(4*eta*zeta-varkappa^2)^(1/2)*xi)-varkappa)*ln(L)) = (xi*L^(-ln((1/2)*((4*eta*zeta-varkappa^2)^(1/2)*tan((1/2)*(4*eta*zeta-varkappa^2)^(1/2)*xi)-varkappa)/eta)/ln(L))+varkappa+eta*L^(ln((1/2)*((4*eta*zeta-varkappa^2)^(1/2)*tan((1/2)*(4*eta*zeta-varkappa^2)^(1/2)*xi)-varkappa)/eta)/ln(L)))/ln(L)

  

-2*(1+tan((1/2)*(4*eta*zeta-varkappa^2)^(1/2)*xi)^2)*(eta*zeta-(1/4)*varkappa^2)/((-(4*eta*zeta-varkappa^2)^(1/2)*tan((1/2)*(4*eta*zeta-varkappa^2)^(1/2)*xi)+varkappa)*ln(L)) = (1/2)*((-4*eta*zeta+varkappa^2)*tan((1/2)*(4*eta*zeta-varkappa^2)^(1/2)*xi)^2-4*xi*eta+varkappa^2)/((-(4*eta*zeta-varkappa^2)^(1/2)*tan((1/2)*(4*eta*zeta-varkappa^2)^(1/2)*xi)+varkappa)*ln(L))

  

-2*(-(1/4)*varkappa*(4*eta*zeta-varkappa^2)^(1/2)*(1+tan((1/2)*(4*eta*zeta-varkappa^2)^(1/2)*xi)^2)/(eta*tan((1/2)*(4*eta*zeta-varkappa^2)^(1/2)*xi))+(1/4)*((4*eta*zeta-varkappa^2)^(1/2)+varkappa*tan((1/2)*(4*eta*zeta-varkappa^2)^(1/2)*xi))*(4*eta*zeta-varkappa^2)^(1/2)*(1+tan((1/2)*(4*eta*zeta-varkappa^2)^(1/2)*xi)^2)/(eta*tan((1/2)*(4*eta*zeta-varkappa^2)^(1/2)*xi)^2))*eta*tan((1/2)*(4*eta*zeta-varkappa^2)^(1/2)*xi)/(((4*eta*zeta-varkappa^2)^(1/2)+varkappa*tan((1/2)*(4*eta*zeta-varkappa^2)^(1/2)*xi))*ln(L)) = (xi*L^(-ln(-(1/2)*((4*eta*zeta-varkappa^2)^(1/2)+varkappa*tan((1/2)*(4*eta*zeta-varkappa^2)^(1/2)*xi))/(eta*tan((1/2)*(4*eta*zeta-varkappa^2)^(1/2)*xi)))/ln(L))+varkappa+eta*L^(ln(-(1/2)*((4*eta*zeta-varkappa^2)^(1/2)+varkappa*tan((1/2)*(4*eta*zeta-varkappa^2)^(1/2)*xi))/(eta*tan((1/2)*(4*eta*zeta-varkappa^2)^(1/2)*xi)))/ln(L)))/ln(L)

  

-2*csc((1/2)*(4*eta*zeta-varkappa^2)^(1/2)*xi)*(eta*zeta-(1/4)*varkappa^2)/(ln(L)*(varkappa*sin((1/2)*(4*eta*zeta-varkappa^2)^(1/2)*xi)+(4*eta*zeta-varkappa^2)^(1/2)*cos((1/2)*(4*eta*zeta-varkappa^2)^(1/2)*xi))) = (1/2)*(cot((1/2)*(4*eta*zeta-varkappa^2)^(1/2)*xi)^2*(-4*eta*zeta+varkappa^2)-4*xi*eta+varkappa^2)/(ln(L)*(varkappa+(4*eta*zeta-varkappa^2)^(1/2)*cot((1/2)*(4*eta*zeta-varkappa^2)^(1/2)*xi)))

 (4)
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Download clarifying_solving_new_ode_33-5-2025mprimes.mw

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