Convergence of Solitary - Wave Solutions in Aperturbed Bi - Hamiltonian Dynamical System

In this part, we prove that the solitary wave solutions investigated in part I are extended as analytic functions in the complex plane, except at most countably many branch points and branch lines. We describe in detail how the limiting behavior of the complex sin-gularities allows the creation of non-analytic solutions with corners and/or compact support. This is the second in a series of two papers investigating the solitary wave solutions of the integrable model wave equation u t + u xxt = u x + u xxx + 3 uu x + uu xxx + 2u x u xx : (3.6) (We adopt the notation and numbering of statements from part I.) The ordinary diierential equation for travelling wave solutions u(x; t) = (x ? ct) is Substituting = a + a, where a is the undisturbed uid depth for our solitary wave solutions, and integrating the resulting equation twice, leads to the rst order equation (a + + cc + a)(0 a) 2 = ? 2 a (a + 3a + (+ c)) (3.17) To understand why analytic solitary wave solutions converge to non-analytic functions, such as compactons and peakons, having singularities on the real axis, we shall extend the solitary wave solutions described in Theorems 3.1 and 3.2 in Part I to functions deened in the complex plane to study singularity distribution of these functions. This method