New integrable PDEs of boomeronic type

Abstract

Novel integrable systems of coupled first-order autonomous PDEs in 1 + 1 dimensions (space x and time t) are presented. Integrable covariant 2-vector and 3-vector PDEs, as well as higher-order integrable PDEs for a single, or a couple, of scalar-dependent variables (including an extension of the sine-Gordon equation and a remarkably neat, highly nonlinear third-order PDE), are also obtained by appropriate reductions of the basic matrix equations. The Lax pairs that characterize the integrable character of the basic matrix PDEs are also exhibited, as well as their single-soliton solutions. These solitons generally exhibit the boomeronic (and, less generically, the trapponic) phenomenology, namely they do not move uniformly, but rather (in an appropriate reference system) come in from one end in the remote past and boomerang back to that same end in the remote future (boomerons), or are trapped to oscillate around a value fixed by the initial data (trappons).