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The theory of linear dispersive equations predicts that waves should spread out and disperse over time. However, it is a remarkable phenomenon, observed both in theory and practice, that once nonlinear effects are taken into account, solitary wave or soliton solutions can be created, which can be stable enough to persist indefinitely. The construction of such solutions can be relatively straightforward, but the fact that they are stable requires some significant amounts of analysis todoi:10.1090/s0273-0979-08-01228-7 fatcat:dmlt2nrix5fk7ofxcozjeonvsa