Instability and Nonuniqueness for the b-Novikov Equation

The b-Novikov equation is a one-parameter family of Camassa-Holm-type equations with cubic nonlinearities that possess multipeakon traveling wave solutions and for b = 3 gives the well known Novikov equation, which is integrable. Here, using appropriate two-peakon solutions, instability and nonuniqueness for the initial value problem of the b-Novikov equation is studied when the initial data belong in Sobolev spaces H-s, S < 3/2, on both the line and the circle. The rectangular region of the bs-plane defined by b > 2 and s < 3/2 is divided into three subregions. The subregion that is below the line segment s = 2 - b/4, 2 < b < 4, is characterized by the phenomenon of nonuniqueness. Then, to the right of this subregion the phenomenon of norm inflation occurs, which leads to instability and discontinuity of the solution map. However, on the segment s = 2 - b/4, 2 < b < 4, either nonuniqueness or discontinuity may occur. All these are proved by constructing appropriate two-peakon solutions with arbitrary small initial size data that collide in arbitrary small time T. These solutions may become arbitrarily large near T. For b <= 2, the two-peakon solutions do not work since there is no collision. Finally, it is well known that for s > 3/2 there is well-posedness no matter what is the value of b.