ASYMPTOTIC STABILITY OF SOLITARY WAVES OF THE 3D QUADRATIC ZAKHAROV-KUZNETSOV EQUATION

We consider the quadratic Zakharov-Kuznetsov equation partial differential tu+ partial differential x increment u+ partial differential xu2 = 0 on R3. A solitary wave solution is given by Q(x - t, y, z), where Q is the ground state solution to -Q + increment Q + Q(2) = 0. We prove the asymptotic stability of these solitary wave solutions. Specifically, we show that initial data close to Q in the energy space, evolves to a solution that, as t -> infinity, converges to a rescaling and shift of Q(x - t, y, z) in L2 in a rightward shifting region x > delta t - tan theta y2 + z2 for 0 < theta <pi 3 - delta.