Question: How to get solution Soll11 for the reduced Liouville non linear pde ?

The modified Liouville equation

How to solve this pde for a general solution ?

The general solution in this form exist.

restart;

with(PDEtools): declare(u(x,t)); U:=diff_table(u(x,t));
PDE1:=U[t,t]=a^2*U[x,x]+b*exp(beta*U[]);
Sol11:=u(x,t)=1/beta*ln(2*(B^2-a^2*A^2)/(b*beta*(A*x+B*t+C)^2));
Sol12:=S->u(x,t)=1/beta*ln(8*a^2*C/(b*beta))
-2/beta*ln(S*(x+A)^2-S*a^2*(t+B)^2+S*C);
Test11:=pdetest(Sol11,PDE1);
Test12:=pdetest(Sol12(1),PDE1);
Test13:=pdetest(Sol12(-1),PDE1);

u(x, t)*`will now be displayed as`*u

 

table( [(  ) = u(x, t) ] )

 

diff(diff(u(x, t), t), t) = a^2*(diff(diff(u(x, t), x), x))+b*exp(beta*u(x, t))

 

u(x, t) = ln(2*(-A^2*a^2+B^2)/(b*beta*(A*x+B*t+C)^2))/beta

 

proc (S) options operator, arrow; u(x, t) = ln(8*a^2*C/(b*beta))/beta-2*ln(S*(x+A)^2-S*a^2*(t+B)^2+S*C)/beta end proc

 

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(1)

The Soll11 can be plotted with a Explore plot in this form of soll11 with th eparameters , but suppose i try to get the general solution in Maple ?

infolevel[pdsolve] := 3

pdsolve(PDE1, generalsolution)

ans := pdsolve(PDE1);

What solvin gstrategy to follow ? : the pde is a non-linear wave eqation  with a exponentiel sourceterm
It seems that the pde can reduced to a ode? :

 

with(PDEtools):
declare(u(x,t));

# Stap 1: Definieer de PDE
PDE := diff(u(x,t), t,t) = a^2 * diff(u(x,t), x,x) + b * exp(beta * u(x,t));

# Stap 2: Definieer de transformatie naar karakteristieke variabelen
# Nieuw: x en t uitgedrukt in ξ en η
tr := {
    x = (xi + eta)/2,
    t = (eta - xi)/(2*a)
};

# Pas de transformatie toe op de PDE
simplified_PDE := dchange(tr, PDE, [xi, eta], params = [a, b, beta], simplify);

# Stap 3: Definieer de algemene oplossing
solution := u(x,t) = (1/beta) * ln(
    (-8*a^2/(b*beta)) *
    diff(_F1(x - a*t), x) * diff(_F2(x + a*t), x) /
    (_F1(x - a*t) + _F2(x + a*t))^2
);

# Stap 4: Controleer de oplossing (optioneel)
pdetest(solution, PDE);  # Moet 0 teruggeven als correct

u(x, t)*`will now be displayed as`*u

 

diff(diff(u(x, t), t), t) = a^2*(diff(diff(u(x, t), x), x))+b*exp(beta*u(x, t))

 

{t = (1/2)*(eta-xi)/a, x = (1/2)*xi+(1/2)*eta}

 

a^2*(diff(diff(u(xi, eta), xi), xi)-2*(diff(diff(u(xi, eta), eta), xi))+diff(diff(u(xi, eta), eta), eta)) = a^2*(diff(diff(u(xi, eta), xi), xi))+2*a^2*(diff(diff(u(xi, eta), eta), xi))+a^2*(diff(diff(u(xi, eta), eta), eta))+b*exp(beta*u(xi, eta))

 

u(x, t) = ln(-8*a^2*(D(_F1))(-a*t+x)*(D(_F2))(a*t+x)/(b*beta*(_F1(-a*t+x)+_F2(a*t+x))^2))/beta

 

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(2)

missing some steps here : solution u  without  the pde reduced ?
there is a ode ?

# Definieer de ODE # vorige stappen ontbreken van de reduktie
ode := (v^2 - a^2) * diff(f(xi), xi, xi) = b * exp(beta * f(xi));

# Algemene oplossing zoeken
sol := dsolve(ode, f(xi));

(-a^2+v^2)*(diff(diff(f(xi), xi), xi)) = b*exp(beta*f(xi))

 

f(xi) = ln((1/2)*c__1*(tan((1/2)*(-c__1*a^2*beta+c__1*beta*v^2)^(1/2)*(c__2+xi)/(a^2-v^2))^2+1)/b)/beta

(3)

 

, ,

Question : how do i arrive on Soll11   in Maple  ?

 

Download liouville_reduced_2-2-2025_mprimes_vraag.mw

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