Abstract
We obtain real-valued, time-periodic and radially symmetric solutions of the cubic Klein–Gordon equation which are weakly localized in space. Various families of such ‘breather’ solutions are shown to bifurcate from any given nontrivial stationary solution. The construction of weakly localized breathers in three space dimensions is, to the author’s knowledge, a new concept and based on the reformulation of the cubic Klein–Gordon equation as a system of coupled nonlinear Helmholtz equations involving suitable conditions on the far field behavior.
Highlights
Introduction and main resultsWe construct real-valued solutions U(t, x) of the cubic Klein–Gordon equation∂t2U − ΔU + m2U = Γ(x) U3 on R × R3 (1)where1 Γ ∈ L∞ rad(R3) ∩ Cl1oc(R3) and m > 0 is a parameter
We obtain real-valued, time-periodic and radially symmetric solutions of the cubic Klein–Gordon equation ∂t2U − ΔU + m2U = Γ(x)U3 on R × R3, which are weakly localized in space
Further, to ask for the global bifurcation picture given some trivial family T = {(w0, λ) | λ ∈ R}. (Here λ ∈ R denotes a bifurcation parameter which in our case is not visible in the differential equation and will be properly introduced later.) Typically, global bifurcation theorems state that a maximal bifurcating continuum of solutions (U, λ) emanating from T at (w0, λ0) is unbounded unless it returns to T at some point (w0, λ0), λ0 = λ0. In the former case, a satisfactory characterization of global bifurcation structures should provide a criterion whether or not unboundedness results from another stationary solution w1 = w0 with {(w1, λ) | λ ∈ R} belonging to the maximal continuum. Since it is not obvious at all whether and how such a criterion might be derived within our framework, we focus on the local result, which already adds new aspects to the state of knowledge about the existence of breather solutions summarized
Summary
Where Γ ∈ L∞ rad(R3) ∩ Cl1oc(R3) and m > 0 is a (mass) parameter. Here we restrict ourselves to the case of three space dimensions which is the most relevant one for applications in physics and which allows to use the tools established in [23]. The approach in [16] incorporates more general potentials and nonlinearities and is based on variational techniques It provides ground state solutions, which are possibly ‘large’—in contrast to our local bifurcation methods, which only yield solutions close to a given stationary one as described in theorem 1, i.e. with a typically ‘small’. Klainerman [18, 19] and Shatah [26, 27] independently developed new techniques leading to significant improvements in the study of uniqueness questions and of the asymptotic behavior of solutions as t → ∞ These results work in settings with high regularity and admit more general nonlinearities with growth assumptions for small arguments, which includes the cubic one as a special case. Our methods instead focus on several global properties of the solutions Uα(t, x) such as periodicity in time and localization as well as decay rates in space, i.e. the defining properties of breathers
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