WE ANALYZE THE EXISTENCE OF T ?PERIODIC SOLUTIONS TO THE SECOND-ORDER INDEFINITE SINGULAR EQUATION U?? = ? H(T) COS ² U WHICH DEPENDS ON A POSITIVE PARAMETER ? > 0. HERE, H IS A SIGN-CHANGING FUNCTION WITH H = 0 AND WHERE THE NONLINEAR TERM OF THE EQUATION HAS TWO SINGULARITIES. FOR THE FIRST TIME, THE DEGENERATE CASE IS STUDIED, DISPLAYING AN UNEXPECTED FEATURE WHICH CONTRASTS WITH THE RESULTS KNOWN IN THE LITERATURE FOR INDEFINITE SINGULAR EQUATIONS. © 2019 AMERICAN INSTITUTE OF MATHEMATICAL SCIENCES. ALL RIGHTS RESERVED.

IN THIS PAPER WE GIVE A DETAILED ANALYSIS ON THE EXISTENCE OF TRAVELING WAVE (PERIODIC) SOLUTIONS TO A GENERALIZED KLEIN?GORDON EQUATION WITH A SINGULAR NONLINEARITY GENERATED BY TWO HORIZONTAL LINES. IN PARTICULAR, WE STUDY SOME SOLUTIONS THAT PASS THROUGH (OR MEET) THESE SINGULAR LINES. IN THIS SETTING, WE DISPLAY SOME EXAMPLES OF NEW SOLITON STRUCTURES.