Novel PT-invariant Kink and Pulse Solutions For a Large Number of Real Nonlinear Equations

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For a large number of real nonlinear equations, either continuous or discrete, integrable or nonintegrable, uncoupled or coupled, we show that whenever a real nonlinear equation admits kink solutions in terms of βx, where β is the inverse of the kink width, it also admits solutions in terms of the PT-invariant combinations 2βx ± i 2 βx, i.e. the kink width is reduced by half to that of the real kink solution. We show that both the kink and the PT-invariant kink are linearly stable and obtain expressions for the zero mode in the case of several PT-invariant kink solutions. Further, for a number of real nonlinear equations we show that whenever a nonlinear equation admits periodic kink solutions in terms of (x,m), it also admits periodic solutions in terms of the PT-invariant combinations (x,m) ± i (x,m) as well as (x,m)± i (x,m). Finally, for coupled equations we show that one cannot only have complex PT-invariant solutions with PT eigenvalue +1 or -1 in both the fields but one can also have solutions with PT eigenvalue +1 in one field and -1 in the other field.