Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: ,  investigate the spectral instability of periodic traveling wave solu- tions of the generalized Korteg-de Vries equation to long wavelength transverse perturbations in the generalized Kadomtsev-Petviashvili equation. By analyzing high and low frequency limits of the appropriate periodic Evans function,  derive an orientation index which yields sufficient conditions for such an instability to occur. This index is geometric in nature and applies to arbitrary periodic traveling waves with minor smoothness and convexity assumptions on the nonlinearity. Using the integrable structure of the ordinary differential equation governing the traveling wave profiles,  are then able to calculate the resulting orientation index for the elliptic function solutions of the Korteg-de Vries and modified Korteg-de Vries equations.
['Mathew A. Johnson', 'Kevin Zumbrun']
I am writing a research paper on a idea Transverse Instability of Periodic Traveling Waves in the Generalized Kadomtsev-Petviashvili Equation
[{"from": "human", "value": "I am writing a research paper on a idea Transverse Instability of Periodic Traveling Waves in the Generalized Kadomtsev-Petviashvili Equation"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: ,  investigate the spectral instability of periodic traveling wave solu- tions of the generalized Korteg-de Vries equation to long wavelength transverse perturbations in the generalized Kadomtsev-Petviashvili equation. By analyzing high and low frequency limits of the appropriate periodic Evans function,  derive an orientation index which yields sufficient conditions for such an instability to occur. This index is geometric in nature and applies to arbitrary periodic traveling waves with minor smoothness and convexity assumptions on the nonlinearity. Using the integrable structure of the ordinary differential equation governing the traveling wave profiles,  are then able to calculate the resulting orientation index for the elliptic function solutions of the Korteg-de Vries and modified Korteg-de Vries equations."}]