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For the following differential equation

$\displaystyle-\frac{\partial ^2\phi2 (x)}{\partial x^2}+\lambda ~[\phi2 (x)]^3-\mu ^2~\phi2 (x)=\phi2(x)~\delta(x)$

-D[\[Phi]2[x], {x, 2}] - \[Mu]^2*\[Phi]2[x] + \[Lambda]*\[Phi]2[x]^3 == \[Phi]2[x]*DiracDelta[x]

One possible solution for $\phi2[x]$ is given by $\text{$\phi $2}(\text{x})\text{:=}\displaystyle\frac{\mu \tanh \left(\frac{\mu \sqrt{x^2}}{\sqrt{2}}\right)}{\sqrt{\lambda }}$

\[Phi]2[x_] := (\[Mu]/Sqrt[\[Lambda]])*Tanh[(Sqrt[x^2]*\[Mu])/Sqrt[2]]

The following command (hopefully?) validates the solution:

Simplify[-D[\[Phi]2[x], {x, 2}] - \[Mu]^2*\[Phi]2[x] + \[Lambda]*\[Phi]2[x]^3 == \[Phi]2[x]*DiracDelta[x]]

(*True*)

Apparently, another identity/solution is when the RHS is replaced by $a~constant~~\times~ \delta(x)$

When trying to validate this in MMa, the following doesn't seem to work:

Simplify[-D[\[Phi][x], {x, 2}] - \[Mu]^2*\[Phi][x] + \[Lambda]*\[Phi][x]^3 == constant1*DiracDelta[x]]

(*0 == constant1*DiracDelta[x]*)

Is there a trick here, or is the earlier "validation" of $\phi2(x)~\delta(x)$ as a solution itself misleading?

Note that $\sqrt{x^2}$ is being used for $|x|$, since MMa's Abs[x] didn't work for the validation.

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  • $\begingroup$ Up to Wiki en.wikipedia.org/wiki/Distribution_(mathematics), the product $ \phi2(x)~\delta(x)$ is defined if $ \phi2(x)$ is a smooth function. Therefore, $ \text{$\phi $2}(\text{x})\text{:=}\displaystyle\frac{\mu \tanh \left(\frac{\mu \sqrt{x^2}}{\sqrt{2}}\right)}{\sqrt{\lambda }}$, not being smooth at the origin, is not any solution of the ODE under consideration. $\endgroup$ – user64494 Jun 19 at 17:57
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I don't think, that it is allowed to check an ode with DiracDelta pointwise. Though the first True of your simplification must be wrong.

The general solution fullfills -D[\[Phi]2[x], {x, 2}] - \[Mu]^2*\[Phi]2[x] + \[Lambda]*\[Phi]2[x]^3==0 , x>0 . Dirac only forces a sudden change of the initial conditions.

Integrate your ode in a small range Element[x, {-\[CurlyEpsilon], \[CurlyEpsilon]}]

-\[Phi]2'[\[CurlyEpsilon]]+\[Phi]2'[-\[CurlyEpsilon]]+ 2\[CurlyEpsilon](-\[Mu]^2*\[Phi]2[0] + \[Lambda]*\[Phi]2[0]^3) ==\[Phi]2[0]

Limit \[CurlyEpsilon] -> 0 gives the new initial condition

\[Phi]2'[SuperPlus[0]]==\[Phi]2'[0]-\[Phi]2[0] (*RHS \[Phi][x] DiracDelta[x]*)

\[Phi]2'[SuperPlus[0]]==\[Phi]2'[0]-\[Phi]2[0] (*RHS 1  DiracDelta[x]*)
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  • $\begingroup$ Thank you Ulrich. I am trying to trace the exact commands. What would be the command in MMa to perform the following (or the equivalent you suggested), to get from $\int_{-b}^{b} \phi2^{\prime \prime}(x) -a^{2}\int_{-b}^{b} \phi2(x) = k~\int_{-b}^{b} \phi2(x)~\delta(x)$ So that it leads to (LHS) = $\lim_{b\to 0}\,\phi^{\prime}(b) - \phi2^{\prime}(- b)=[\phi2^{\prime}(0)]$ and (RHS) = $k~\int_{-b}^{b} \phi(x)~\delta(x) = k~\phi2(0)$ = $[\phi2^{\prime}(0)]=k\,\phi2(0)$ = $\phi2(x)~=~A~e^{\mp~a~|x|}$ $\endgroup$ – user58293 Jan 19 at 13:59
  • $\begingroup$ @user58293 Is this a new question/new ode?The initialvalue problem \[Phi]''[x] - a^2 \[Phi][x] == k \[Phi][x] \[Delta][x], \[Phi][0] == \[Phi]0, \[Phi]'[0] == \[Phi]p0 is "equivalent" to \[Phi]''[x] - a^2 \[Phi][x] == 0, \[Phi][0] == \[Phi]0, \[Phi]'[0] == \[Phi]p0+k \[Phi][0] The new initial conditions follows from integrating the ode. $\endgroup$ – Ulrich Neumann Jan 19 at 14:11
  • $\begingroup$ What's the command to go from integration with limits, and get the "new" initial condition that contains "Superplus"? The modified question in the comment is to try and capture (perhaps) a slightly more common/general case and ensure that my MMa command inputs are exactly as you intended. Thanks again Ulrich! $\endgroup$ – user58293 Jan 19 at 14:35
  • $\begingroup$ The command is Integrate["ode" ,{x,-b,b},Assumption->b>0 ] with following Limit[...,b->0]. Thereby you have to help MMA a little bit to integrate the LHS. $\endgroup$ – Ulrich Neumann Jan 19 at 14:37
  • $\begingroup$ My apologies, the steps/commands are still not obvious to me. $\endgroup$ – user58293 Jan 20 at 0:21
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For the second question(see comment) proceed as follows:

initial value problem

 {Derivative[2][\[Phi]][x] - a^2*\[Phi][x] ==  k*\[Phi][x]*DiracDelta[x],  
 \[Phi][0]==\[Phi]0,\[Phi]'[0]==\[Phi]p0}

integrating the ode in the range -b

 Derivative[1][\[Phi]][b]-Derivative[1][\[Phi]][-b]+  2 b a^2*\[Phi][2]
 ==k*\[Phi][0]

Limit b->0

Derivative[1][\[Phi]][b]-Derivative[1][\[Phi]][-b] ==k*\[Phi][0]

Here you can see the sudden change of Derivative[1][[Phi]][x] due to the Dirac-impuls!

The equivalent initial value problem is

{Derivative[2][\[Phi]][x] - a^2*\[Phi][x] ==0 ,  
 \[Phi][0]==\[Phi]0,\[Phi]'[0]==\[Phi]p0 +k *\[Phi][0]}

can be solved easily

DSolve[{Derivative[2][\[Phi]][x] - a^2*\[Phi][x] ==0, \[Phi][0] == \[Phi]0, \[Phi]'[0] == \[Phi]p0 +k*\[Phi][0]}, \[Phi], x][[1]]
(*{\[Phi] ->Function[{x}, (E^(-a x) (a \[Phi]0 + a E^(2 a x) \[Phi]0 - k \[Phi]0 +E^(2 a x) k \[Phi]0 - \[Phi]p0 + E^(2 a x) \[Phi]p0))/(2 a)]}*)
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