The 2019 study of Shatnawi and Amundsen (Stud. Appl. Math. 143, 192-210) classified the near-resonant response of a canonical one-parameter family of forced nonlinear wave equations on a bounded interval, comparing flux-type nonlinearities unuxu^nu_x, potential-type powers unu^n, and hybrid mixtures (1−β)u2+βuux(1-\beta)u^2+\beta uu_x, with a boundary parameter α\alpha tuning the linear spectrum between commensurate and effectively incommensurate. The classification, the explicit amplitude-frequency relations, and the numerically observed sharp single-to-multiple-mode transition in the hybrid model rest on algebraic solvability conditions whose mathematical structure was not analysed. The open problem is to complete that theory: determine existence, multiplicity, stability, and bifurcations of the coupled algebraic systems governing multimodal resonant responses (including degenerate cases such as vanishing leading coefficients), characterize analytically the transition threshold β∗(m,α)\beta^*(m,\alpha) of the hybrid nonlinearity and its scaling, derive how weak dissipation modifies the conservative amplitude equations and selects attracting steady states (regularizing the unbounded multimodal growth seen at the commensurate spectrum α=0\alpha=0), and assess the robustness of the parity/locality rules and amplitude scalings beyond the canonical one-dimensional constant-coefficient models. No resolution of these open questions is known in the literature to date.

Background

When a wave system in a bounded domain is driven periodically at a frequency close to one of its natural frequencies, linear theory predicts an unboundedly growing response, and nonlinearity, however weak, enters the leading balance. The standard tool is a perturbation expansion in the small forcing amplitude mm: at some order the expansion generates secular terms (corrections that grow linearly in time), and removing them imposes a solvability condition, an instance of the Fredholm alternative requiring that the projection of the forcing onto the kernel of the adjoint linear operator vanish. The outcome is an algebraic amplitude-frequency relation linking the modal amplitude AA, the detuning δ\delta (the offset of the driving frequency from the resonant eigenfrequency, scaled to the order at which it balances the nonlinearity), and the forcing. For a quadratic nonlinearity this relation is a cubic in AA; the classical prototype is Chester's analysis of resonant gas oscillations in a closed tube, in which dissipation through shocks and boundary layers enters the leading-order balance, and its successors developed small-rate damping theory for open and closed tubes (Jimenez; Seymour and Mortell).

Whether the steady response involves one spatial mode or many is governed by the linear spectrum. If the eigenfrequencies are commensurate (integer multiples of the resonant one, as in a closed-closed or open-closed tube), every harmonic generated by the nonlinearity is itself resonant and can be fed in turn; if they are incommensurate, only the forced mode can be sustained and the response is effectively single-mode. Related boundary-excited standing-wave problems with quadratic and cubic nonlinearities were also analysed by Rudenko, Hedberg, and Enflo.

Shatnawi and Amundsen (2019), building on a companion study (Amundsen, 2017) and the first author's thesis, introduced a canonical family of model problems that isolates these mechanisms: the wave equation

utt−uxx+f(u,ux)=0,0≤x≤1,u_{tt}-u_{xx}+f(u,u_x)=0,\qquad 0\le x\le 1,

with harmonic boundary forcing u(0,t)=msin⁡(λt)u(0,t)=m\sin(\lambda t) at one end and the one-parameter family of boundary conditions (α−1)u(1,t)−αux(1,t)=0(\alpha-1)u(1,t)-\alpha u_x(1,t)=0 at the other. The boundary parameter α\alpha tunes the spectrum through the transcendental eigenvalue equation tan⁡(ω0)=αω0/(α−1)\tan(\omega_0)=\alpha\omega_0/(\alpha-1): α=0\alpha=0 (Dirichlet end) gives eigenfrequencies nπn\pi and α=1\alpha=1 (Neumann end) gives (2n+1)π/2(2n+1)\pi/2, both commensurate, while generic values such as α=2\alpha=2 give effectively incommensurate spectra. They compared flux-type nonlinearities f=unuxf=u^nu_x (containing a spatial derivative) with potential-type powers f=unf=u^n and hybrid mixtures f=(1−β)u2+βuuxf=(1-\beta)u^2+\beta uu_x interpolating between them.

Their findings, obtained by perturbation analysis with Runge-Kutta finite-difference simulations: (i) a commensurate spectrum is necessary but not sufficient for a multimode response; odd-nn flux nonlinearities produce multiple-mode, formally unbounded responses at the commensurate value α=0\alpha=0, while α=1\alpha=1 (the other commensurate value) and incommensurate α\alpha are effectively single-mode, as are even-nn flux and potential-type nonlinearities across the tested α\alpha (a parity/locality rule); (ii) the response amplitude scales as A=O(m1/(2n+1))A=O(m^{1/(2n+1)}) for odd nn and A=O(m1/(n+1))A=O(m^{1/(n+1)}) for even nn; (iii) explicit single-mode amplitude-frequency relations of the form N1(α,ω0)A3+N2(α,ω0)Aδ=ω0πN_1(\alpha,\omega_0)A^3+N_2(\alpha,\omega_0)A\delta=\omega_0\pi (and a seventh-order analogue for u3uxu^3u_x, whose secular balance first appears at O(ϵ7)O(\epsilon^7)), with coefficients depending on α\alpha through the eigenfunction; (iv) in the hybrid model at α=0\alpha=0 the response changes sharply with the mixing parameter, from single-mode for β≤0.01\beta\le0.01 to multiple-mode for β≥0.5\beta\ge0.5 (numerically, at forcing m=10−4m=10^{-4} for β≥0.6\beta\ge0.6 and at m=10−2m=10^{-2} for the β=0.5\beta=0.5 transition point), but only numerically; and (v) since the models are conservative, the algebraic amplitude states correspond to neutrally stable fixed points (perturbations neither grow nor decay), so long-time simulations exhibit slow phase drift rather than convergence to a steady state. These results leave open the mathematical analysis of the coupled algebraic solvability systems that govern the multimodal cases, the analytic characterization of the single- to multiple-mode transition threshold in the hybrid nonlinearity, the effect of small dissipation on the amplitude equations and on attractor selection, and the extension of the classification beyond the canonical one-dimensional model forms.

Problem Statement

For the Shatnawi-Amundsen model family — the wave equation utt−uxx+f(u,ux)=0u_{tt}-u_{xx}+f(u,u_x)=0 on 0≤x≤10\le x\le1 with boundary forcing u(0,t)=msin⁡(λt)u(0,t)=m\sin(\lambda t) and boundary condition (α−1)u(1,t)−αux(1,t)=0(\alpha-1)u(1,t)-\alpha u_x(1,t)=0, for flux-type nonlinearities f=unuxf=u^nu_x, potential-type powers f=unf=u^n, and hybrid mixtures f=(1−β)u2+βuuxf=(1-\beta)u^2+\beta uu_x — complete the mathematical theory of the steady near-resonant response. The common scientific objective is to predict, for given nonlinearity, boundary parameter α\alpha, forcing amplitude mm, and dissipation, which steady resonant states exist, whether they are single- or multiple-mode, and which of them are selected as long-time attractors. Concretely: (1) derive and mathematically analyse the coupled algebraic solvability systems that govern the multimodal cases (flux-type nonlinearities at the commensurate boundary parameter α=0\alpha=0), determining existence, multiplicity, and stability of their real solution branches and their bifurcation structure, including degenerate cases in which leading coefficients vanish (for example N1=0N_1=0); (2) determine analytically the threshold β∗(m,α)\beta^*(m,\alpha), and its scaling, at which the hybrid nonlinearity transitions from single-mode to multiple-mode response, replacing the current purely numerical demarcation, which is available only at α=0\alpha=0, with the dependence on the forcing amplitude mm and the boundary parameter α\alpha made explicit; (3) incorporate weak dissipative effects into the amplitude equations of this model family and determine how they modify the conservative amplitude-frequency relations, whether the steady states become attracting, and whether unbounded multimodal growth at the commensurate spectrum α=0\alpha=0 is regularized, building on but going beyond the single-member damped analysis in the authors' thesis; and (4) assess whether the parity/locality rules (odd flux nonlinearities multimodal at α=0\alpha=0; even flux and potential-type nonlinearities single-mode) and the amplitude scalings A=O(m1/(2n+1))A=O(m^{1/(2n+1)}) (odd nn) and A=O(m1/(n+1))A=O(m^{1/(n+1)}) (even nn) persist beyond the canonical one-dimensional constant-coefficient setting — the target being this general extension, with the listed settings (inhomogeneous coefficients, other geometries, or higher spatial dimensions) illustrative of the broader class rather than a required enumeration. Answers may combine rigorous analysis with reproducible numerical computation. The four components are the required elements of this objective: each is independently meaningful, but a complete resolution addresses all four, and a result that resolves only a proper subset is a partial result.

References

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