Well-posedness and dynamics of solutions to the generalized KdV with low power nonlinearity

Abstract

We consider two types of the generalized Korteweg -de Vries equation, where the nonlinearity is given with or without absolute values, and, in particular, including the low powers of nonlinearity, an example of which is the Schamel equation. We first prove the local well-posedness of both equations in a weighted subspace of H 1 that includes functions with polynomial decay, extending the result of [39] to fractional weights. We then investigate solutions numerically, confirming the well-posedness and extending it to a wider class of functions that includes exponential decay. We include a comparison of solutions to both types of equations, in particular, we investigate soliton resolution for the positive and negative data with different decay rates. Finally, we study the interaction of various solitary waves in both models, showing the formation of solitons, dispersive radiation and even breathers, all of which are easier to track in nonlinearities with lower power.

Stability and instability of solitary waves in fractional generalized KdV equation in all dimensions

“…(The one dimensional version of (1.15) with α = 2 was studied by the authors and collaborators in [34], where (1.15) was termed as GKdV to distinguish it from the standard gKdV equations without absolute value in the nonlinearity; there, a local well-posedness was obtained on some subset of a weighted Sobolev space). The following is a consequence of Theorem 1.3 for (1.15) with a non-integer m.…”

“…The assumption of the local well-posedness in H s (R d ) for the Cauchy problem associated to (1.15) is not always guaranteed, since the lack of regularity of the nonlinearity |u| m−1 for the fractional m > 1 yields extra difficulty when applying classical methods used for the well-posedness. Nevertheless, for results in this direction for (1.15) in 1d, refer to [34] and [57].…”