First-iteration adjoint-state sensitivity maps are examined for qualitative ultrasonic imaging of defects in orthotropic plates under a single pitch–catch illumination. The five signed kernels of the classical least-squares misfit are displayed as absolute-value normalised images, not as material updates of an iterative FWI reconstruction.We form a reflection–transmission residual-wave interaction indicator, correlating two fields back-propagated from residuals on opposed apertures. It is bilinear in the two residual families, where every single-residual construction is linear in one; since the degree of homogeneity in the data is invariant under linear data-space operators, no reweighting of the misfit produces it. It is not the propagator term of the first variation of any standard single-state constrained functional, both fields sharing the same time orientation. It does, however, admit an exact variational status in the perturbation variable, and a gradient in the discrete space where the maps are computed.The set-up being mirror-symmetric about the axis through the inclusion, all four C66 constructions vanish quadratically along it; reversing the source polarisation moves that zero to C11, C22 and C12, as predicted: a causal test. For the density and the three normal-block coefficients the classical maps localise the inclusion to within two tenths of a defect radius, which the indicator does not improve. Its benefit is specific to C66, whose multipolar map keeps a fitted centre of symmetry within half a defect radius of the inclusion under both apertures, against classical displacements of 0.83 and at least 1.5 radii.
We derive and classify a family of exact longitudinal travelling solitons in a periodic binary array of coupled nonlinear waveguides. The two sublattices are allowed to have different dispersion laws and Kerr nonlinearities, while the transverse field distribution is described by a Bloch mode. We consider bright and dark states with common longitudinal profiles proportional to sech(z/d) and tanh (z/d), respectively. Their substitution into the coupled propagation equations yields a dispersion-slope synchronism condition, a nonlinear amplitude balance, and a parameter-dependent 2 × 2 eigenvalue problem. Eliminating the soliton amplitude, width, and velocity reduces the existence problem to a single scalar nonlinear equation for an auxiliary parameter Z, whose positive and negative admissible roots correspond to bright and dark states. For opposite-sign Kerr nonlinearities, NA > 0 and NB < 0, we obtain an explicit analytical classification of the lower spectral branch. The sign of the dispersion contrast selects the soliton type: RA < RB supports bright states, whereas RA > RB supports dark states. The coupling strength and Bloch wavenumber enter only through the effective coupling Teff=T|cos(q/2)|. We derive a closed-form critical boundary Teff,c beyond which no admissible mixed state exists. At this boundary, the algebraic root continues through Z=0, but the physical soliton delocalizes according to G∝(Teff,c−Teff)1/2 and d∝(Teff,c−Teff)−1/2. The analytical classification and critical laws are confirmed by direct numerical root searches and continuation of the exact solution branches.
In this work, we investigate higher-order rational and semi-rational waves and their interactions in the defocusing reverse-space nonlocal coupled Lakshmanan–Porsezian–Daniel system. An N-fold Darboux transformation, combined with spectral degeneration and a Taylor expansion, yields a unified determinant representation for both kinds of solutions. Degeneration of the plane-wave spectral data generates higher-order rational waves, whereas the nondegenerate branch produces semi-rational waves formed by rational solitons interacting with breather-like periodic waves. Furthermore, we graphically illustate the effects of the degeneration order, shift function, background polarization, and nondegenerate-branch parameter on localization, peak splitting, component redistribution, periodic modulation, and transitions between rational and semi-rational wave structures. Moreover, these solutions provide exact reference profiles for testing numerical solvers and future stability analyses of nonlocal higher-order vector models.
In this paper, we investigate the N-soliton solutions and develop a reverse space-time symmetry-informed physics-informed neural network (RST-PINN) for the reverse space-time nonlocal complex three-component mKdV (NCTC-mKdV) equation. Based on the associated Lax pair, the analytic properties of the Jost solutions are analyzed and the corresponding matrix Riemann-Hilbert problem is established. By solving the reflectionless Riemann-Hilbert problem under the symmetry constraints of the discrete spectrum, the general N-soliton solutions are derived. The dynamical behaviors of several representative one- and two-soliton solutions are further examined, revealing the effects of the spectral and polarization parameters on the amplitude, localization, internal oscillation, propagation direction, and interaction patterns. Furthermore, RST-PINN is proposed to accurately approximate representative nonsingular solutions of the NCTC-mKdV equation. By explicitly embedding the reverse space-time nonlocal symmetry into the loss function, the proposed framework enhances the symmetry preservation and prediction accuracy of the learned solutions. Numerical experiments show that the RST-PINN predictions agree well with the exact solutions, demonstrating the effectiveness of the proposed framework. The combination of exact soliton theory and symmetry-informed neural learning provides an effective framework for studying nonlocal integrable systems.
