Solving Painleve Analysis for KdV & NLS: Saravanan's Research

[saravanan13]

Hai to everyone...
I am Saravanan, Research scholar from theoretical physics.
I am attempting to solve the painleve analysis for KdV and cubic NLS(nonlinear Schrodinger equation) equation.
After arriving at the arbitrary functions by substituting the Laurent's series in given equation I couldn able to conclude what makes the given equation an Integrable or Non integrable?
There is compatibility condition that arises from recurrence relation, what is meant by compatibility condition?
thanks in well advance...

 

[jvicens]

Hello Saravanan,

Thank you for sharing your research interests with us. It's always exciting to hear about new developments in theoretical physics.

To answer your question, the concept of integrability in mathematics and physics refers to the ability of a system to be solved using certain methods or techniques. In the context of differential equations, integrable equations are those that can be solved explicitly, meaning that their solutions can be written down in a closed form. Non-integrable equations, on the other hand, cannot be solved explicitly and may require numerical or approximate methods for finding solutions.

In the case of KdV and cubic NLS equations, their integrability is related to the existence of soliton solutions. Solitons are special solutions that retain their shape and speed after colliding with each other, making them stable and robust. The presence of soliton solutions in a differential equation is a strong indication of integrability.

The compatibility condition you mentioned arises from the fact that the Laurent series expansion used in the Painleve analysis must satisfy certain conditions in order to be valid. This condition ensures that the solutions obtained from the analysis are consistent and can be used to solve the original equation.

I hope this helps to clarify your doubts. Keep up the good work in your research and best of luck with your analysis. If you have any further questions, please don't hesitate to ask. Best regards.