Global Bifurcation of Anti-plane Shear Fronts

Chen, Robin Ming; Walsh, Samuel; Wheeler, Miles H.

doi:10.1007/s00332-021-09684-7

Cited by 5 publications

(2 citation statements)

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“…One substantial advantage of this choice is that we are able to prove monotonicity properties of the solutions relatively easily, laying the groundwork for subsequent global bifurcation theoretic analysis. Indeed, the authors [8,10] and Hogancamp [24] use exactly this strategy to construct large solutions for two of the three applications considered in this paper; these works all rely in an essential way on the qualitative information garnered from the local theory.…”

Section: Introductionmentioning

confidence: 99%

Center manifolds without a phase space for quasilinear problems in elasticity, biology, and hydrodynamics

2022

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In this paper, we present a center manifold reduction theorem for quasilinear elliptic equations posed on infinite cylinders that is done without a phase space in the sense that we avoid explicitly reformulating the PDE as an evolution problem. Under suitable hypotheses, the resulting center manifold is finite dimensional and captures all sufficiently small bounded solutions. Compared with classical methods, the reduced ODE on the manifold is more directly related to the original physical problem and also easier to compute. The analysis is conducted directly in Hölder spaces, which is often desirable for elliptic equations. We then use this machinery to construct small bounded solutions to a variety of systems. These include heteroclinic and homoclinic solutions of the anti-plane shear problem from nonlinear elasticity; exact slow moving invasion fronts in a two-dimensional Fisher–KPP equation; and hydrodynamic bores with vorticity in a channel. The last example is particularly interesting in that we find solutions with critical layers and distinctive ‘half cat’s eye’ streamline patterns.

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Section: Introductionmentioning

confidence: 99%

Center manifolds without a phase space for quasilinear problems in elasticity, biology, and hydrodynamics

2022

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“…Regarding the ruling-out/realization of the loss of compactness alternative in the global theory, as was studied in [19,20], the established monotonicity property is strong enough to assert a "compactness or front" result stating that this possibility must manifest as a broadening phenomenon, leading to a monotone front type of solution at the end of the bifurcation curve. When the underlying system possesses a Hamiltonian structure, a so-called conjugate flow analysis can be casted utilizing the conserved quantities to rule out the broadening alternative [2,19,20,21,25,36]. Moreover, for some particular problems such a Hamiltonian structure may also allow one to obtain uniform bounds on solutions that can account for the realization of broadening [25].…”

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confidence: 99%

Global bifurcation of solitary waves to the Boussinesq $abcd$ system

Chen¹,

Jin²

2021

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The Boussinesq abcd system arises in the modeling of long wave small amplitude water waves in a channel, where the four parameters (a, b, c, d) satisfy one constraint. In this paper we focus on the solitary wave solutions to such a system. In particular we work in two parameter regimes where the system does not admit a Hamiltonian structure (corresponding to b = d). We prove via analytic global bifurcation techniques the existence of solitary waves in such parameter regimes. Some qualitative properties of the solutions are also derived, from which sharp results can be obtained for the global solution curves.Specifically, we first construct solutions bifurcating from the stationary waves, and obtain a global continuous curve of solutions that exhibits a loss of ellipticity in the limit. The second family of solutions bifurcate from the classical Boussinesq supercritical waves. We show that the curve associated to the second class either undergoes a loss of ellipticity in the limit or becomes arbitrarily close to having a stagnation point.

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