How to solve the following equation ?

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Discussion Overview

The discussion revolves around solving a nonlinear partial differential equation, with participants exploring methods and considerations for handling the equation, particularly in the context of complex solutions. The scope includes mathematical reasoning and technical explanations related to differential equations.

Discussion Character

  • Exploratory, Technical explanation, Mathematical reasoning

Main Points Raised

  • One participant notes the equation is nonlinear and invites others to share ideas on solving it.
  • Another participant suggests separating the cases based on the sign of the solution, proposing a standard approach.
  • A question is raised about how to separate the solution into positive and negative cases if the solution is complex.
  • It is proposed that if the function is complex, it can be expressed in terms of its real and imaginary parts, leading to two nonlinear differential equations for those parts.
  • There is an acknowledgment that the resulting equations may be complex and challenging to solve, but a claim is made that most nonlinear equations can be solved in closed form.

Areas of Agreement / Disagreement

Participants express differing views on how to approach the separation of cases for complex solutions, indicating that the discussion remains unresolved regarding the best method for handling complex-valued functions in this context.

Contextual Notes

The discussion does not clarify certain assumptions regarding the nature of the nonlinear partial differential equation or the specific conditions under which the proposed methods apply.

mark gazi
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It is a nonliear Partial differential equation . you can find it in the appendix.
If you have some good idea about this equation ,ok ,please reply to me .thanks!
 

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The first thing you would do, of course, is separate into "u> 0", "u< 0" cases. After that it is pretty much standard.
 
My question

First, thank you .but if the solution of the equation (U) is a complex number, how to separate the "u" into the two condition:(1) U>0; (2) U<0.
 
If u is a complex valued function, then separate it into real and imaginary parts: u(x)= p(x)+ i q(x). |u|2= p2+ q2 and you have two non-linear differential equations for p and q. They will still be rather nasty I am afraid. MOST non-linear equations can be solved in closed form.
 

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