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Question: How do I find Lax pair for this equation?

Question:How do I find Lax pair for this equation?

alaloush 35 Maple 2018
pde detools pdetools assume
January 25 2020
0

Hello
    In this example, we have the KdV equation    
         u[t] - 6 uu[x] + u[xxx] = 0                
    I would like to find the Lax pair for the KdV equation, which are    
               Lψ=λψ                
               ψ[t] = Mψ                
    where    
              L[t]+ML-LM = 0  called a compatibility condition               
    So, I will start from this purpose    
    Then we will assume M in the form   
    we will assume M in the form   
              M := a3*Dx^3+a2*Dx^2+a1*Dx+a0              
    then by using M and L in the form   L[t]+ML-LM = 0 we can find   
     ( ) Dx^5+( ) Dx^4+( ) Dx^3+( ) Dx^2+( ) Dx+( )=0              
    then we can find a_i ;i=0,1,2,3   
    In the following maple code to do that   
    my question is    
   . ?How I can continue the solution to get a_i ;i=0,1,2,3 using maple code -  
      .?is thir any maple packge to find Lax pair for PDE -  


 

> 

restart; with(DEtools); with(PDEtools)

     in this exampile we have KdV equation

      u[t]-6*uu[x]+u[xxx] = 0

    I would like to find the Lax pair for the KdV equation, which are

       Lψ=λψ

    psi[t] = M*psi

   where``

    L[t]+ML-LM = 0    called  a compatibility  condition

    So, I  will start from this purpose

     L:=-Dx^2+u;

    then we will assume M in the form

    M := a3*Dx^3+a2*Dx^2+a1*Dx+a0

    then by using M and L in the form L[t]+ML-LM = 0 we can find

    ( ) Dx^5+( ) Dx^4+( ) Dx^3+( ) Dx^2+( ) Dx+( )=0

    then we can find a_i ;i=0,1,2,3

    

  in the fllowing maple code to do that

 my question is ;

  1) How I can continue the solution  to get a_i ;i=0,1,2,3 using maple code  ?

  2) is thir any maple packge to find  Lax pair for PDE ?

 

> 

alias(u = u(x, t)); declare(u(x, t)); alias(a3 = a3(x, t)); declare(a3(x, t)); alias(a2 = a2(x, t)); declare(a2(x, t)); alias(a1 = a1(x, t)); declare(a1(x, t)); alias(a0 = a0(x, t)); declare(a0(x, t))

u

  

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

  

u, a3

  

` a3`(x, t)*`will now be displayed as`*a3

  

u, a3, a2

  

` a2`(x, t)*`will now be displayed as`*a2

  

u, a3, a2, a1

  

` a1`(x, t)*`will now be displayed as`*a1

  

u, a3, a2, a1, a0

  

` a0`(x, t)*`will now be displayed as`*a0

 (1)
> 

_Envdiffopdomain := [Dx, x]

[Dx, x]

 (2)
> 

L := -Dx^2+u

-Dx^2+u

 (3)
> 

M := Dx^3*a3+Dx^2*a2+Dx*a1+a0

a3*Dx^3+a2*Dx^2+a1*Dx+a0

 (4)
> 

 

 

 
> 

LM := expand(mult(L, M))

-a3*Dx^5-2*Dx^4*(diff(a3, x))-a2*Dx^4+Dx^3*u*a3-Dx^3*(diff(diff(a3, x), x))-2*Dx^3*(diff(a2, x))-Dx^3*a1+Dx^2*u*a2-Dx^2*(diff(diff(a2, x), x))-2*Dx^2*(diff(a1, x))-Dx^2*a0+Dx*u*a1-Dx*(diff(diff(a1, x), x))-2*Dx*(diff(a0, x))+u*a0-(diff(diff(a0, x), x))

 (5)
> 

ML := expand(mult(M, L))

-a3*Dx^5-a2*Dx^4+Dx^3*u*a3-Dx^3*a1+3*Dx^2*a3*(diff(u, x))+Dx^2*u*a2-Dx^2*a0+3*Dx*a3*(diff(diff(u, x), x))+2*Dx*a2*(diff(u, x))+Dx*u*a1+a3*(diff(diff(diff(u, x), x), x))+a2*(diff(diff(u, x), x))+a1*(diff(u, x))+u*a0

 (6)
> 

Commutator := simplify(ML-LM)

a3*(diff(diff(diff(u, x), x), x))+(3*Dx*a3+a2)*(diff(diff(u, x), x))+diff(diff(a0, x), x)+Dx*(diff(diff(a1, x), x))+Dx^2*(diff(diff(a2, x), x))+Dx^3*(diff(diff(a3, x), x))+(3*Dx^2*a3+2*Dx*a2+a1)*(diff(u, x))+2*Dx^4*(diff(a3, x))+2*Dx^3*(diff(a2, x))+2*Dx^2*(diff(a1, x))+2*Dx*(diff(a0, x))

 (7)
> 

sol := diff(L, t)-Commutator = 0

diff(u, t)-a3*(diff(diff(diff(u, x), x), x))-(3*Dx*a3+a2)*(diff(diff(u, x), x))-(diff(diff(a0, x), x))-Dx*(diff(diff(a1, x), x))-Dx^2*(diff(diff(a2, x), x))-Dx^3*(diff(diff(a3, x), x))-(3*Dx^2*a3+2*Dx*a2+a1)*(diff(u, x))-2*Dx^4*(diff(a3, x))-2*Dx^3*(diff(a2, x))-2*Dx^2*(diff(a1, x))-2*Dx*(diff(a0, x)) = 0

 (8)
> 

collect(sol, [Dx, x])

-2*Dx^4*(diff(a3, x))+(-(diff(diff(a3, x), x))-2*(diff(a2, x)))*Dx^3+(-3*a3*(diff(u, x))-(diff(diff(a2, x), x))-2*(diff(a1, x)))*Dx^2+(-2*a2*(diff(u, x))-3*a3*(diff(diff(u, x), x))-(diff(diff(a1, x), x))-2*(diff(a0, x)))*Dx-a1*(diff(u, x))-a2*(diff(diff(u, x), x))-a3*(diff(diff(diff(u, x), x), x))-(diff(diff(a0, x), x))+diff(u, t) = 0

 (9)

 

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``
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NULL


 

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