Interfacial Nonlinear Love Waves

• Interfacial nonlinear Love waves are guided shear–horizontal acoustic modes localized near the interface between dissimilar elastic media.
• They are modeled by coupled nonlinear Schrödinger equations that capture resonant, finite-amplitude dynamics and dispersion effects in layered structures.
• Analytical and numerical studies reveal tunable pulse properties, supporting applications in surface-wave logic, delay lines, and acoustic sensing.

Interfacial nonlinear Love waves are finite-amplitude, shear–horizontal surface-guided waves localized near the interface between dissimilar elastic media. In one formulation, they are surface acoustic pulses propagating along the interface between a thin guiding layer and an elastic half-space in the presence of an atomically thin resonant transition layer; under acoustic self-induced transparency, the resulting envelope dynamics admit a two-component vector soliton (Adamashvili, 2016). In another formulation, they are Love-type interface waves in incompressible hyperelastic or hyper-viscoelastic layered solids, where finite strain, constitutive nonlinearity, and viscoelasticity modify the classical dispersion and long-time phase-speed behavior (McAdam et al., 18 Mar 2026). The topic therefore spans both resonantly driven interfacial nonlinear optics analogies in acoustics and continuum-mechanical generalizations of the classical Love-wave problem.

1. Geometric setting and defining characteristics

The defining feature of a Love wave is interfacial trapping of a shear–horizontal mode in a layered medium. In the resonant surface-acoustic configuration, the elastic half-space occupies $z \le 0$, the guiding layer occupies $0<z\le h$, and the layer thickness satisfies $h \ll \lambda$, where $\lambda$ is the Love-wave wavelength. At $z=0$, an ultrathin transition layer of thickness $d \ll \lambda$ contains a dilute ensemble of two-level centers with density $n_0$; these centers may be paramagnetic impurity atoms or semiconductor quantum dots with spin $S=\tfrac12$. A static magnetic field $H_0 \hat z$ produces the Zeeman splitting $\omega_0=\beta_0 H_0/\hbar$ (Adamashvili, 2016).

In the hyperelastic formulation, the geometry is a two-layered medium in the $0<z\le h$0–$0<z\le h$1 plane, with $0<z\le h$2 corresponding to layer 1 and $0<z\le h$3 to layer 2. The motion is antiplane: the only nonzero displacement is $0<z\le h$4 in the $0<z\le h$5-direction, $0<z\le h$6. The assumptions are incompressibility, isotropy, and homogeneity, with the strain-energy density $0<z\le h$7 depending on the first invariant $0<z\le h$8 (McAdam et al., 18 Mar 2026).

These two descriptions use different constitutive mechanisms and coordinate conventions, but both treat Love waves as SH-polarized interfacial modes whose existence depends on the contrast between a guiding layer and a substrate. A plausible implication is that “interfacial nonlinear Love wave” is best understood as a family of guided SH waves for which either resonant interfacial physics or bulk material nonlinearity becomes dynamically relevant.

2. Boundary-value structure and resonant interface coupling

For the resonant interfacial model, the free surface at $0<z\le h$9 satisfies the vanishing-shear-stress condition

$h \ll \lambda$0

At the interface $h \ll \lambda$1, displacement is continuous and the shear stress has a jump generated by the resonant transition layer: $h \ll \lambda$2 where

$h \ll \lambda$3

Introducing the interfacial shear strain

$h \ll \lambda$4

and a Fourier decomposition in $h \ll \lambda$5, the field reduces to a scalar wave-operator equation

$h \ll \lambda$6

where $h \ll \lambda$7 encodes the Love-mode dispersion and $h \ll \lambda$8 are the transverse sound speeds in the guiding layer and substrate (Adamashvili, 2016).

The spin–phonon interaction is introduced through

$h \ll \lambda$9

With $\lambda$0, the Bloch equations become

$\lambda$1

Under acoustic self-induced transparency, with pulse duration $\lambda$2 and area conditions, slowly varying envelopes may be introduced for $\lambda$3, $\lambda$4, and $\lambda$5. A multiple-scale expansion then reduces the interfacial problem to coupled nonlinear Schrödinger equations for the envelope components $\lambda$6 and $\lambda$7: $\lambda$8

$\lambda$9

The coefficients $z=0$0, $z=0$1, $z=0$2, and $z=0$3 represent, respectively, the group velocities, dispersion coefficients, and self- and cross-phase nonlinearities. This places the resonant Love-wave problem in the standard vector-envelope soliton class, but with coefficients fixed by interfacial acoustics and the resonant layer rather than by a bulk optical medium.

3. Explicit vector soliton and parameter scaling

Seeking a steady-state pulse propagating with velocity $z=0$4 via

$z=0$5

gives a two-component sech-profile,

$z=0$6

with common width $z=0$7. The amplitudes and width are constrained by the algebraic relations

$z=0$8

and

$z=0$9

In terms of the physical strain, the solution has a two-frequency structure,

$d \ll \lambda$0

where $d \ll \lambda$1 (Adamashvili, 2016).

The material dependence enters through the Love-mode dispersion function $d \ll \lambda$2 and its derivatives, which depend on layer thickness $d \ll \lambda$3, densities $d \ll \lambda$4, and transverse sound speeds $d \ll \lambda$5; through the spin–phonon coupling $d \ll \lambda$6 and impurity density $d \ll \lambda$7, which enter $d \ll \lambda$8; and through the inhomogeneous linewidth $d \ll \lambda$9 via the broadening function $n_0$0.

The scaling laws summarized for this solution are: $n_0$1

$n_0$2

and $n_0$3 lies between the two group speeds $n_0$4 and is sensitive to $n_0$5 and $n_0$6 through $n_0$7. Accordingly, the properties of the nonlinear Love wave depend on the parameters of the transition resonance layer, the connected elastic media, and the transverse structure of the Love mode. This makes the interfacial vector soliton a coupled object: its localization is geometric, its nonlinearity is resonant, and its spectral composition is intrinsically two-component.

4. Hyperelastic and viscoelastic continuum formulations

A distinct route to interfacial nonlinear Love waves starts from incompressible hyperelasticity. For antiplane shear motion in an incompressible isotropic homogeneous material, the balance of linear momentum yields

$n_0$8

with

$n_0$9

In one spatial dimension, this reduces to

$S=\tfrac12$0

In the purely linear, neo-Hookean limit, the two-layer problem becomes

$S=\tfrac12$1

At the interface $S=\tfrac12$2, displacement continuity and traction continuity hold: $S=\tfrac12$3 At the free surface $S=\tfrac12$4, $S=\tfrac12$5, and as $S=\tfrac12$6, $S=\tfrac12$7. For plane waves of the form

$S=\tfrac12$8

the dispersion relation is

$S=\tfrac12$9

with the classical Love-wave existence condition

$H_0 \hat z$0

For the cubic Yeoh constitutive law,

$H_0 \hat z$1

the derivatives are

$H_0 \hat z$2

The corresponding two-dimensional PDE becomes

$H_0 \hat z$3

In one dimension this reduces to

$H_0 \hat z$4

The $H_0 \hat z$5 term is the linear restoring force, while the terms proportional to $H_0 \hat z$6 and $H_0 \hat z$7 are the cubic and quintic nonlinearities. Adding viscoelasticity through the pseudo-potential

$H_0 \hat z$8

produces additional damping–dispersion terms, including in one dimension

$H_0 \hat z$9

so that the mixed derivative $\omega_0=\beta_0 H_0/\hbar$0 appears (McAdam et al., 18 Mar 2026).

5. Trapping, phase-speed evolution, and numerical behavior

In the hyperelastic and viscoelastic framework, the relation between nonlinearity and the classical trapping condition is central. Full $\omega_0=\beta_0 H_0/\hbar$1-dimensional numerical simulations are performed in a rectangular $\omega_0=\beta_0 H_0/\hbar$2 domain of width $\omega_0=\beta_0 H_0/\hbar$3 and total depth $\omega_0=\beta_0 H_0/\hbar$4, with upper layer thickness $\omega_0=\beta_0 H_0/\hbar$5. The initial condition is a “Gaussian explosion” centered at $\omega_0=\beta_0 H_0/\hbar$6,

$\omega_0=\beta_0 H_0/\hbar$7

with $\omega_0=\beta_0 H_0/\hbar$8. The material parameters are piecewise constant, with $\omega_0=\beta_0 H_0/\hbar$9 for $0<z\le h$00 and $0<z\le h$01 for $0<z\le h$02, while the nonlinear coefficients $0<z\le h$03, $0<z\le h$04, and the viscosity $0<z\le h$05 are chosen globally. The numerical scheme uses the method of lines, second-order finite differences on a nonuniform mesh, Neumann conditions at $0<z\le h$06, Dirichlet far fields, and Matlab’s ode23 adaptive Runge–Kutta time stepping (McAdam et al., 18 Mar 2026).

Wave speed is tracked at the interface $0<z\le h$07 and free surface $0<z\le h$08 by following the crest or largest-amplitude point along $0<z\le h$09: $0<z\le h$10 with $0<z\le h$11 defined by $0<z\le h$12 and $0<z\le h$13 maximal. The simulations show that, after an initial transient, both interface and surface speeds settle to the same constant value, namely the larger of $0<z\le h$14. If $0<z\le h$15, then $0<z\le h$16 as $0<z\le h$17; if $0<z\le h$18, then $0<z\le h$19. The early-time deviation from the purely linear Love-wave phase speed is of order $0<z\le h$20 and typically decays exponentially: $0<z\le h$21

Several numerical observations clarify the nonlinear Love-wave picture. In the linear case $0<z\le h$22, one recovers the classical two-layer refraction/reflection pattern, with a trapped Love-wave front moving along $0<z\le h$23 only when $0<z\le h$24. With modest nonlinearity $0<z\le h$25, the interface and surface wavefronts develop small amplitude-dependent speed shifts and higher-harmonic ripples, but still satisfy $0<z\le h$26. Adding viscosity $0<z\le h$27 damps high-frequency ripples, broadens the front, and causes the early-time speed to evolve more slowly, but in all cases $0<z\le h$28. If $0<z\le h$29, no sustained trapped mode appears and most energy leaks into the half-space, agreeing with linear theory.

A common misconception is that finite-amplitude nonlinearity necessarily invalidates the classical Love-wave existence condition. The simulations do not support that conclusion in moderate-time dynamics: in the fully nonlinear case, the variable wave speed of interface and surface waves generally satisfies $0<z\le h$30. The long-time limit, however, tends to the larger material wave speed. This suggests a distinction between transient interfacial trapping and asymptotic propagation speed.

6. Material realization, observables, and applications

For the resonant vector-soliton model, a numerical case study is given for a ZnO/LiNbO$0<z\le h$31 layered structure with Fe$0<z\le h$32 parameters. Using

$0<z\le h$33

$0<z\le h$34

$0<z\le h$35

the numerical estimates are

$0<z\le h$36

$0<z\le h$37

These parameters are stated to lie within reach of current surface-acoustic-wave experiment setups (Adamashvili, 2016).

The observable quantities differ across the two model classes. In the resonant model, the defining observables are the two envelope components $0<z\le h$38, the common pulse width $0<z\le h$39, the soliton velocity $0<z\le h$40, and the physical interfacial strain $0<z\le h$41 with its two-frequency structure. In the hyperelastic and viscoelastic model, the principal observables are the interface and free-surface wavefronts, their speed histories $0<z\le h$42, and the transition from trapped or partially trapped dynamics to long-time propagation at $0<z\le h$43.

Potential applications identified for the resonant interfacial vector soliton include surface-wave logic and delay lines, acoustic signal processing, and high-precision sensing, benefiting from the enhanced nonlinearity and tight modal confinement of Love-mode vector solitons at the interface. A plausible implication is that the broader significance of interfacial nonlinear Love waves lies in combining the modal selectivity of guided SH acoustics with mechanisms—resonant, hyperelastic, or viscoelastic—that make phase speed, pulse width, spectral content, and localization tunable by material design and operating regime.