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janhardo

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

janhardo 700
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 700
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 700
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 700
May 03 2025
0 0

@nm
i am using maple 2024...

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janhardo 700
May 03 2025
0 0




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a example , but M is not exact calculate...

janhardo 700
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)
> 

 


 

Download berekening_coeiff_M_is_onjuist_en_niet_berekend_3-5-2025.mw

@mmcdara Thanks for the nice proof ...

janhardo 700
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 700
May 03 2025
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janhardo 700
May 03 2025
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janhardo 700
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)

 (1)
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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)
> 


 

Download clarifying_solving_new_ode_33-5-2025mprimes.mw

  ...

janhardo 700
May 03 2025
0 0


 

> 

ode := diff(h(phi), phi) = (omega*f^(-h(phi))+mu+nu*f^h(phi))/ln(f); subs_rule := h(phi) = ln(y(phi))/ln(f); new_ode := eval(ode, subs_rule); riccati_ode := diff(y(phi), phi) = nu*y(phi)^2+mu*y(phi)+omega; y_solution := dsolve(riccati_ode, y(phi)); h_solution := ln(y(phi))/ln(f); specific_ode := eval(riccati_ode, {mu = -1, nu = 1, omega = 0}); y_solution_example := dsolve(specific_ode, y(phi)); h_solution_example := ln(rhs(y_solution_example))/ln(exp(1)); simplify(h_solution_example); -b

diff(h(phi), phi) = (omega*f^(-h(phi))+mu+nu*f^h(phi))/ln(f)

  

h(phi) = ln(y(phi))/ln(f)

  

(diff(y(phi), phi))/(y(phi)*ln(f)) = (omega*f^(-ln(y(phi))/ln(f))+mu+nu*f^(ln(y(phi))/ln(f)))/ln(f)

  

diff(y(phi), phi) = nu*y(phi)^2+mu*y(phi)+omega

  

y(phi) = -(1/2)*(mu-tan((1/2)*(-mu^2+4*nu*omega)^(1/2)*(c__1+phi))*(-mu^2+4*nu*omega)^(1/2))/nu

  

ln(y(phi))/ln(f)

  

diff(y(phi), phi) = y(phi)^2-y(phi)

  

y(phi) = 1/(1+exp(phi)*c__1)

  

ln(1/(1+exp(phi)*c__1))

  

ln(1/(1+exp(phi)*c__1))

 (1)
> 

 


 

Download ode_naar_riccati_type_ode3-5-2025_mprimes.mw

@vv I only raised A and B each to t...

janhardo 700
May 03 2025
0 0

@vv 
I only raised A and B each to the second power, has nothing to do with : A^2 = B^2  does NOT imply A = B.

@mmcdara  Is A=B ? ...

janhardo 700
May 03 2025
0 0

@mmcdara 
Is A=B ? 

@salim-barzani ...

janhardo 700
May 02 2025
0 0

@salim-barzani 
Glimpses of Soliton Theory (Second Edition) by Alex Kasman

  > ....

janhardo 700
April 30 2025
0 0
> 

diff(G(xi), xi) = A+B*G(xi)+C*G(xi)^2; Delta := 4*A*C-B^2; assume(C <> 0); assume(Delta > 0); G1 := (sqrt(Delta)*tan((1/2)*sqrt(Delta)*(d[0]+xi))-B)/(2*C); check1 := simplify(diff(G1, xi)-C*G1^2-B*G1-A); G2 := -(sqrt(Delta)*cot((1/2)*sqrt(Delta)*(d[0]+xi))+B)/(2*C); check2 := simplify(diff(G2, xi)-C*G2^2-B*G2-A); forget(Delta); assume(Delta < 0); G3 := -(sqrt(-Delta)*tanh((1/2)*sqrt(-Delta)*(d[0]+xi))+B)/(2*C); check3 := simplify(diff(G3, xi)-C*G3^2-B*G3-A)

diff(G(xi), xi) = A+B*G(xi)+C*G(xi)^2

  

4*A*C-B^2

  

(1/2)*((4*A*C-B^2)^(1/2)*tan((1/2)*(4*A*C-B^2)^(1/2)*(d[0]+xi))-B)/C

  

0

  

-(1/2)*((4*A*C-B^2)^(1/2)*cot((1/2)*(4*A*C-B^2)^(1/2)*(d[0]+xi))+B)/C

  

0

  

-(1/2)*((-4*A*C+B^2)^(1/2)*tanh((1/2)*(-4*A*C+B^2)^(1/2)*(d[0]+xi))+B)/C

  

0

 (1)
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restart; with(PDEtools); E := diff(G(xi), xi) = A+B*G(xi)+C*G(xi)^2; Delta := 4*A*C-B^2; assume(C <> 0); assume(Delta > 0); G1 := (sqrt(Delta)*tan((1/2)*sqrt(Delta)*(d[0]+xi))-B)/(2*C); odetest(G(xi) = G1, E); G2 := -(sqrt(Delta)*cot((1/2)*sqrt(Delta)*(d[0]+xi))+B)/(2*C); odetest(G(xi) = G2, E); forget(Delta); assume(Delta < 0); G3 := -(sqrt(-Delta)*tanh((1/2)*sqrt(-Delta)*(d[0]+xi))+B)/(2*C); odetest(G(xi) = G3, E)

(1/2)*((4*A*C-B^2)^(1/2)*tan((1/2)*(4*A*C-B^2)^(1/2)*(d[0]+xi))-B)/C

  

0

  

-(1/2)*((4*A*C-B^2)^(1/2)*cot((1/2)*(4*A*C-B^2)^(1/2)*(d[0]+xi))+B)/C

  

0

  

-(1/2)*((-4*A*C+B^2)^(1/2)*tanh((1/2)*(-4*A*C+B^2)^(1/2)*(d[0]+xi))+B)/C

  

0

 (2)
 

Download ricatti_ode_3_oplossingen_mprimes_30-4-2025.mw

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