Non-unique weak solutions to periodic k-gKdV are constructed via convex integration in low-regularity spaces with zero initial data, establishing that nonlinearity integrability is necessary for unconditional uniqueness.
Nonuniqueness of weak solutions of the nonlinear Schroedinger equation
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abstract
Generalized solutions of the Cauchy problem for the one-dimensional periodic nonlinear Schr\"odinger equation, with certain nonlinearities, are not unique. For any $s<0$ there exist nonzero generalized solutions varying continuously in the Sobolev space $H^s$, with identically vanishing initial data.
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Non-unique solutions to the periodic gKdV equation
Non-unique weak solutions to periodic k-gKdV are constructed via convex integration in low-regularity spaces with zero initial data, establishing that nonlinearity integrability is necessary for unconditional uniqueness.