Coupled Boussinesq-type long-wave models for imperfectly bonded layered waveguides: continuum-level derivation within dynamic nonlinear elasticity

Long nonlinear longitudinal waves in a two-layer elastic waveguide joined by a soft adhesive layer (an imperfect interface) were shown in 2009 by Khusnutdinova, Samsonov and Zakharov to be governed, in two regimes of adhesive-stiffness scaling, by coupled Boussinesq-type systems derived from a nonlinear composite lattice, with all coefficients explicit algebraic functions of the microscopic lattice and adhesive parameters. The same work showed numerically that breaking the symmetry between the layers turns classical solitary waves into generalized solitary waves carrying small co-propagating radiative ripples, and closed with the conjecture that analogous coupled systems can be derived directly within dynamic nonlinear elasticity. The later literature has developed the accompanying analysis - asymptotic constructions of radiating solitary waves with exponentially small tails in the weakly-broken-symmetry regime, initial-value theory and Ostrovsky reductions, qualitative experimental observation of radiating bulk strain solitons in layered bars, and nondestructive-testing applications of the coupled systems - but has continued to take the lattice-derived systems as given. The conjectured continuum-level derivation within dynamic nonlinear elasticity remains unestablished. The problem is recorded as open.

Background

Layered elastic structures - bars, plates and films bonded by thin adhesive layers - are ubiquitous in engineering, and their dynamic response depends not only on the bulk materials but on the compliance of the bond. When the adhesive is much softer than the layers, the interface is called imperfect: the two layers are coupled only through the compliant interlayer, and an effective macroscopic model must retain this coupling. The classical description of long (wavelength large compared with the layer thickness), weakly nonlinear longitudinal waves in a single elastic waveguide is a Boussinesq-type equation, a two-way nonlinear dispersive partial differential equation of the schematic form $u_{tt}-c_0^2u_{xx}=(\text{quadratic nonlinearity})+(\text{dispersion})$; in the doubly-dispersive form the dispersion enters through the mixed derivative $u_{ttxx}$, and in the regularized (or improved) Boussinesq equation the fourth-order spatial term $u_{xxxx}$ is asymptotically traded for $u_{ttxx}$, which suppresses a short-wave instability and is numerically preferable. Such equations also arise as continuum limits of Fermi-Pasta-Ulam (FPU) lattices - chains of masses joined by weakly nonlinear springs - in which case the equation's coefficients are explicit functions of the microscopic spring constants.

Khusnutdinova, Samsonov and Zakharov (Theoretical and Mathematical Physics 159, 819 (2009); companion paper Physical Review E 79, 056606 (2009)) constructed a composite lattice for a two-layer waveguide with an adhesive bond: each layer is an anharmonic chain of oscillating dipoles (particles carrying longitudinal, transverse, rotational and internal-separation degrees of freedom), and the layers are joined by adhesive springs of stiffness $\beta_g$. Under a long-wave amplitude-wavelength asymptotic ordering, and for two scalings of the adhesive stiffness, they derived coupled systems of doubly-dispersive Boussinesq equations for the longitudinal displacements $u(x,t)$ and $w(x,t)$ of the two layers, with all coefficients explicit algebraic combinations of lattice and adhesive constants. In nondimensional variables the regularized canonical form of one system reads

$$\begin{aligned} u_{tt}-u_{xx}&=u_xu_{xx}+u_{ttxx}-\delta(u-w),\\ w_{tt}-c^2w_{xx}&=\alpha w_xw_{xx}+\beta w_{ttxx}+\gamma(u-w), \end{aligned}$$

where $c,\alpha,\beta$ measure the contrast between the layers and $\delta,\gamma$ are proportional to the adhesive stiffness. For identical layers ($c=\alpha=\beta=1$) the system admits the symmetric reduction $w=u$ to a single regularized Boussinesq equation with classical localized solitary waves ($u_x$ of $\operatorname{sech}^2$ shape). When the symmetry is broken, the linear dispersion relation of the coupled system has two branches and the speed of a would-be solitary wave can resonate with the upper branch at a finite wavenumber; the numerical experiments of the 2009 work showed that a pure solitary-wave profile then evolves into a generalized solitary wave - a localized core trailed by small-amplitude co-propagating oscillatory ripples (the steady two-sided form is called a generalized solitary wave; the one-sided, slowly decaying form seen dynamically is called a radiating solitary wave).

The lattice derivation also showed that the single-layer doubly-dispersive equation coincides with a Boussinesq-type equation previously derived within nonlinear elasticity for macroscopic waveguides. On this basis the authors conjectured - this is the closing open problem of the paper - that the coupled systems admit an analogous direct derivation within dynamic nonlinear elasticity (the full equations of motion with a nonlinear constitutive law, such as a Murnaghan-type law, for the layers and a compliant adhesive interface); their closing remarks encourage the study of the newly derived model systems and the derivation of analogous systems within the framework of nonlinear elasticity theory.

Problem Statement

The problem is to establish the closing conjecture of Khusnutdinova, Samsonov and Zakharov (2009) for the coupled Boussinesq-type long-wave description of imperfectly bonded two-layer elastic waveguides derived at the lattice level: to derive the coupled systems directly within dynamic nonlinear elasticity.

Within dynamic nonlinear elasticity - the full equations of motion for two elastic layers with a nonlinear constitutive law, together with asymptotically consistent conditions on a thin, soft (compliant) adhesive interface - derive the coupled Boussinesq-type systems for the two longitudinal displacements, in both regimes of adhesive-stiffness scaling treated in the lattice derivation: the regime where the interlayer coupling enters at the same asymptotic order as the leading nonlinearity and dispersion, and the regime where it enters at leading order with modified effective wave speeds and additional derivative couplings. The derivation must reproduce the coupling structure of the lattice-derived systems (the linear interlayer coupling terms, the effective-speed corrections, and the nonlinear interlayer terms of the stronger-scaling system) and provide coefficient formulas in terms of continuum elastic constants and interface parameters that correspond, under the natural identification of lattice and continuum constants, to the lattice-derived coefficients.

References

1. K. R. Khusnutdinova, A. M. Samsonov, A. S. Zakharov, Nonlinear long-wave models for imperfectly bonded layered waveguides, Theoretical and Mathematical Physics 159(3), 819-832 (2009). DOI: 10.1007/s11232-009-0070-y. https://doi.org/10.1007/s11232-009-0070-y
2. K. R. Khusnutdinova, A. M. Samsonov, A. S. Zakharov, Nonlinear layered lattice model and generalized solitary waves in imperfectly bonded structures, Physical Review E 79, 056606 (2009). DOI: 10.1103/PhysRevE.79.056606. https://doi.org/10.1103/PhysRevE.79.056606
3. K. R. Khusnutdinova, K. R. Moore, Initial-value problem for coupled Boussinesq equations and a hierarchy of Ostrovsky equations, Wave Motion 48(8), 738-752 (2011). DOI: 10.1016/j.wavemoti.2011.04.003; arXiv:1104.3432. https://arxiv.org/abs/1104.3432
4. R. H. J. Grimshaw, K. R. Khusnutdinova, K. R. Moore, Radiating solitary waves in coupled Boussinesq equations, IMA Journal of Applied Mathematics 82(4), 802-820 (2017). DOI: 10.1093/imamat/hxx014; arXiv:1802.02473. https://doi.org/10.1093/imamat/hxx014
5. K. R. Khusnutdinova, M. R. Tranter, On radiating solitary waves in bi-layers with delamination and coupled Ostrovsky equations, Chaos 27, 013112 (2017). DOI: 10.1063/1.4973854; arXiv:1702.07575. https://doi.org/10.1063/1.4973854
6. K. R. Khusnutdinova, M. R. Tranter, Weakly-Nonlinear Solution of Coupled Boussinesq Equations and Radiating Solitary Waves, in: Dynamical Processes in Generalized Continua and Structures (Advanced Structured Materials), 321-343 (2019). DOI: 10.1007/978-3-030-11665-1_18; arXiv:1810.13358. https://doi.org/10.1007/978-3-030-11665-1_18
7. K. R. Khusnutdinova, M. R. Tranter, Periodic solutions of coupled Boussinesq equations and Ostrovsky-type models free from zero-mass contradiction, Chaos 32, 113132 (2022). DOI: 10.1063/5.0112982; arXiv:2102.03090. https://doi.org/10.1063/5.0112982
8. K. R. Khusnutdinova, M. R. Tranter, Nonlinear Longitudinal Bulk Strain Waves in Layered Elastic Waveguides, in: Applied Wave Mathematics II (Mathematics of Planet Earth), 125-150 (2019). DOI: 10.1007/978-3-030-29951-4_6; arXiv:1910.00524. https://arxiv.org/abs/1910.00524
9. N. Duruk, H. A. Erbay, A. Erkip, Blow-up and global existence for a general class of nonlocal nonlinear coupled wave equations, Journal of Differential Equations 250(3), 1448-1459 (2011). DOI: 10.1016/j.jde.2010.09.002; arXiv:1009.1263. https://doi.org/10.1016/j.jde.2010.09.002
10. Ş. Pasinlioğlu, Long-time behavior of solutions to the general class of coupled nonlocal nonlinear wave equations, Zeitschrift für angewandte Mathematik und Physik 75, 209 (2024). DOI: 10.1007/s00033-024-02342-4. https://doi.org/10.1007/s00033-024-02342-4
11. G. V. Dreiden, A. M. Samsonov, I. V. Semenova, K. R. Khusnutdinova, Observation of a radiating bulk strain soliton in a solid-state waveguide, Technical Physics 56, 889-892 (2011). DOI: 10.1134/S1063784211060041. https://doi.org/10.1134/S1063784211060041
12. G. V. Dreiden, K. R. Khusnutdinova, A. M. Samsonov, I. V. Semenova, Bulk strain solitary waves in bonded layered polymeric bars with delamination, Journal of Applied Physics 112, 063516 (2012). DOI: 10.1063/1.4752713. https://doi.org/10.1063/1.4752713
13. F. E. Garbuzov, K. R. Khusnutdinova, I. V. Semenova, On Boussinesq-type models for long longitudinal waves in elastic rods, Wave Motion 88, 129-143 (2019). DOI: 10.1016/j.wavemoti.2019.02.004; arXiv:1810.07684. https://doi.org/10.1016/j.wavemoti.2019.02.004
14. J. S. Tamber, D. J. Chappell, M. R. Tranter, Delamination detection in layered waveguides using Ostrovsky wave packets, Proceedings of the Royal Society A 481, 20240574 (2025). DOI: 10.1098/rspa.2024.0574; arXiv:2405.00388. https://doi.org/10.1098/rspa.2024.0574
15. H. J. Hupkes, J. A. McGinnis, R. W. S. Westdorp, J. D. Wright, Radiating Solitary Waves in an FPUT Lattice with Random Coefficients, preprint (2025). arXiv:2509.20165. https://arxiv.org/abs/2509.20165
16. Ş. Pasinlioğlu, Kuple Düzgünleştirilmiş Boussinesq Denklemlerinin Yalnız Dalga Çözümlerinin Kararlılığı Üzerine Sayısal Bir Çalışma [A numerical study of the stability of solitary wave solutions of coupled regularized Boussinesq equations], Iğdır Üniversitesi Fen Bilimleri Enstitüsü Dergisi 15(3), 1089-1099 (2025). DOI: 10.21597/jist.1640298. https://doi.org/10.21597/jist.1640298