Convergence of Solitary-wave Solutions in a Perturbed Bi-hamiltonian Dynamical System. Ii. Complex Analytic Behavior and Convergence to Non-analytic Solutions

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 singularities 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 ut + νuxxt = αux + βuxxx + 3 ν uux + uuxxx + 2uxuxx. (3.6) (We adopt the notation and numbering of statements from part I.) The ordinary differential equation for travelling wave solutions u(x, t) = φ(x− ct) is (α+ c)φ′ + (β + cν + φ)φ′′′ + 3 ν φφ′ + 2φ′φ′′ = 0. (3.7) Substituting φ = φa + a, where a is the undisturbed fluid depth for our solitary wave solutions, and integrating the resulting equation twice, leads to the first order equation ν(φa + β + cν + a)(φ ′ a) 2 = −φa(φa + 3a+ ν(α+ 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 defined 1 School of Mathematics, University of Minnesota, Minneapolis, MN 55455. 2 Research supported in part by NSF Grant DMS 95–00931 and BSF Grant 94–00283. AMS subject classifications: 34A20, 34C35, 35B65, 58F05, 76B25 Typeset by AMS-TEX 1 in the complex plane to study singularity distribution of these functions. This method not only provides another way to prove the last two theorems, but also makes it clear that singularities of solitary wave solutions are approaching the real axis in the process of convergence. Thus, roughly speaking, the singularities of compactons or peakons come from those complex singularities of analytic solitary wave solutions, which are close to the real axis. The explicit form (3.3) of solitary wave solutions of the KdV equation shows that they are restriction to the real axis of meromorphic functions with countably many poles in the complex plane so that their analytic extension is unique. In contrast to these functions, extensions of solitary wave solutions under our consideration do not have poles but branch points. These branch points play an important role in the formation of singularities of compactons and peakons. The analytic extension of these solitary wave solutions enables us to understand the loss in analyticity of their limiting compacton or peakon solutions. For any complex number z ∈ C, the real part and the imaginary part of z are denoted by Rz and Iz, respectively. The real part u(x, y) and imaginary part v(x, y) of an analytic function w = F (z) = F (x + iy) = u(x, y) + iv(x, y) will be called the velocity potential and stream function respectively. The level sets of the velocity potential, u(x, y) = u0, and the stream function, v(x, y) = v0, are called the equipotentials and streamlines of F , respectively. Finally, logw is the single-valued branch of the natural logarithmic function Logw, defined as logw = log |w|+ iargw with −π < argw ≤ π. 4. Analytic extensions of solitary wave solutions for ν > 0. We shall consider the solitary wave solutions in Case I , when ν > 0, and Case II , when ν < 0, separately because of their different structures as functions defined on the complex plane. We begin with the compacton case where ν > 0. Under the assumption of Theorem 3.1, Equation (3.17) has an orbitally unique and analytic solitary wave solution φa. Rescaling (3.17) by φa(x) = −ν(α+ c+ 3a ν )φ(x), we reduce it to the equation (δφ+ ǫ)(φ′)2 = φ(1− φ), (4.1) where δ = ν and ǫ = −(β + cν + a)/(α+ c+ 3a ν ). The phase plane portrait of (4.1) indicates that its solitary wave solution φ is a positive, even function with unit amplitude and decaying to zero at infinity. Therefore, as x > 0, the solution φ satisfies the integral equation x = ∫ 1