Korteweg-de Vries equation on R

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The local and global well-posedness theory for the Korteweg-de Vries equation on the real line is as follows.

The KdV equation can also be generalized to a 2x2 system


</math>b_1 \partial_t v + \partial_x^3 v + b_2 a_3 \partial_x^3 u + v \partial_x v + b_2 a_2 u \partial_x u + b_2 a_1 \partial_x (uv) + r \partial_x v</math>

where b_1,b_2 are positive constants and a_1,a_2,a_3,r are real constants. This system was introduced in references.html#GeaGr1984 GeaGr1984 to study strongly interacting pairs of weakly nonlinear long waves, and studied further in references.html#BnPoSauTm1992 BnPoSauTm1992. In references.html#AsCoeWgg1996 AsCoeWgg1996 it was shown that this system was also globally well-posed on L^2.
It is an interesting question as to whether these results can be pushed further to match the KdV theory; the apparent lack of complete integrability in this system (for generic choices of parameters b_i, a_i, r) suggests a possible difficulty.