The role of asymmetry and volume of thrombotic masses in the formation of local deformation of the aneurysmal-altered vascular wall: An in vivo study and mathematical modeling.

Currently, the primary factor indicating the necessity of an operation for an abdominal aortic aneurysm (AAA) is the diameter at its widest part. However, in practice, a large number of aneurysm ruptures occur before reaching a critical size. This means that the mechanics of aneurysm growth and remodeling have not been fully elucidated. This study presents a novel method for assessing the elastic properties of an aneurysm using an ultrasound technique based on tracking the oscillations of the vascular wall as well as the inner border of the thrombus. Twenty nine patients with AAA and eighteen healthy volunteers were considered. The study presents the stratification of a group of patients according to the elastic properties of the aneurysm, depending on the relative volume of intraluminal thrombus masses. Additionally, the neural network analysis of CT angiography images of these patients shows direct (r = 0.664271) correlation with thrombus volume according to ultrasound data, the reliability of the Spearman correlation is p = 0.000215. The use of finite element numerical analysis made it possible to reveal the mechanism of the negative impact on the AAA integrity of an asymmetrically located intraluminal thrombus. The aneurysm itself is considered as a complex structure consisting of a wall, intraluminal thrombus masses, and areas of calcification. When the thrombus occupies > 70% of the lumen of the aneurysm, the deformations of the outer and inner surfaces of the thrombus have different rates, leading to tensile stresses in the thrombus. This poses a risk of its detachment and subsequent thromboembolism or the rupture of the aneurysm wall. This study is the first to provide a mechanistic explanation for the effects of an asymmetrical intraluminal thrombus in an abdominal aortic aneurysm. The obtained results will help develop more accurate risk criteria for AAA rupture using non-invasive conventional diagnostic methods.

Bi-Solitons on the Surface of a Deep Fluid: An Inverse Scattering Transform Perspective Based on Perturbation Theory.

We investigate theoretically and numerically the dynamics of long-living oscillating coherent structures-bi-solitons-in the exact and approximate models for waves on the free surface of deep water. We generate numerically the bi-solitons of the approximate Dyachenko-Zakharov equation and fully nonlinear equations propagating without significant loss of energy for hundreds of the structure oscillation periods, which is hundreds of thousands of characteristic periods of the surface waves. To elucidate the long-living bi-soliton complex nature we apply an analytical-numerical approach based on the perturbation theory and the inverse scattering transform (IST) for the one-dimensional focusing nonlinear Schrödinger equation model. We observe a periodic energy and momentum exchange between solitons and continuous spectrum radiation resulting in repetitive oscillations of the coherent structure. We find that soliton eigenvalues oscillate on stable trajectories experiencing a slight drift on a scale of hundreds of the structure oscillation periods so that the eigenvalue dynamics is in good agreement with predictions of the IST perturbation theory. Based on the obtained results, we conclude that the IST perturbation theory justifies the existence of the long-living bi-solitons on the surface of deep water that emerge as a result of a balance between their dominant solitonic part and a portion of continuous spectrum radiation.

Solitons in a Box-Shaped Wave Field with Noise: Perturbation Theory and Statistics.

We investigate the fundamental problem of the nonlinear wave field scattering data corrections in response to a perturbation of initial condition using inverse scattering transform theory. We present a complete theoretical linear perturbation framework to evaluate first-order corrections of the full set of the scattering data within the integrable one-dimensional focusing nonlinear Schrödinger equation (NLSE). The general scattering data portrait reveals nonlinear coherent structures-solitons-playing the key role in the wave field evolution. Applying the developed theory to a classic box-shaped wave field, we solve the derived equations analytically for a single Fourier mode acting as a perturbation to the initial condition, thus, leading to the sensitivity closed-form expressions for basic soliton characteristics, i.e., the amplitude, velocity, phase, and its position. With the appropriate statistical averaging, we model the soliton noise-induced effects resulting in compact relations for standard deviations of soliton parameters. Relying on a concept of a virtual soliton eigenvalue, we derive the probability of a soliton emergence or the opposite due to noise and illustrate these theoretical predictions with direct numerical simulations of the NLSE evolution. The presented framework can be generalized to other integrable systems and wave field patterns.

Anomalous errors of direct scattering transform.

Theory of direct scattering transform for nonlinear wave fields containing solitons is revisited to overcome fundamental difficulties hindering its stable numerical implementation. With the focusing one-dimensional nonlinear Schrödinger equation serving as a model, we study a crucial fundamental property of the scattering problem for multisoliton potentials demonstrating that in many cases phase and space position parameters of solitons cannot be identified with standard machine precision arithmetics making solitons in some sense "uncatchable." Using the dressing method we find the landscape of soliton scattering coefficients in the plane of the complex spectral parameter for multisoliton wave fields truncated within a finite domain, allowing us to capture the nature of such anomalous numerical errors. They depend on the size of the computational domain L leading to a counterintuitive exponential divergence when increasing L in the presence of a small uncertainty in soliton eigenvalues. Then we demonstrate how one of the scattering coefficients loses its analytical properties due to the lack of the wave-field compact support in case of L→∞. Finally, we show that despite this inherent direct scattering transform feature, the wave fields of arbitrary complexity can be reliably analysed using high-precision arithmetics even in the presence of noise opening broad perspectives in nonlinear physics.

Direct scattering transform of large wave packets.

We apply the Magnus expansion to the Zakharov-Shabat system, providing a basis for a systematic construction of high-order numerical schemes to solve the direct scattering problem of the integrable one-dimensional nonlinear Schrödinger equation. The presented numerical simulations of previously unreachable wave fields with up to 128 solitons employing second-, fourth-, and sixth-order schemes stresses the need for delicate numerics to identify the eigenvalues and especially phase coefficients. This approach lays the foundation for the study of large optical wave packets, providing fundamental information about their scattering data content and origin of various nonlinear effects.