Long time behavior of solutions to the generalized Boussinesq equation in Sobolev spaces

In this paper, we study the generalized Boussinesq equation to model the water wave problem with surface tension. First, we investigate the initial value problem in the Sobolev spaces. We derive some conditions under which the solutions of this equation are global or blow-up in time, and next, we extend our results to the Bessel potential spaces. The asymptotic behavior of the solutions is also determined. The non-existence of solitary waves for some parameters is proved using Pohozaev-type identities. We generate solitary wave solutions of generalized Boussinesq equation using the Petviashvili iteration method numerically. In order to investigate the time evolution of solutions to the generalized Boussinesq equation, we propose the Fourier pseudo-spectral numerical method. After studying the time evolution of the single solitary wave, we focus on the gap interval where neither a global existence nor a blow-up result has been established theoretically. Our numerical results successfully fill the gaps left by the theoretical ones

Notes on solitary-wave solutions of Rosenau-type equations

The present paper is concerned with the existence of solitary wave solutions of Rosenau-type equations. By using two standard theories, Normal Form Theory and Concentration-Compactness Theory, some results of existence of solitary waves of three different forms are derived. The results depend on some conditions on the speed of the waves with respect to the parameters of the equations. They are discussed for several families of Rosenau equations present in the literature. The analysis is illustrated with a numerical study of generation of approximate solitary-wave profiles from a numerical procedure based on the Petviashvili iteration

On the Kadomtsev-Petviashvili equation with double-power nonlinearities

In this paper, we delve into the study of the generalized KP equation, which incorporates double-power nonlinearities. Our investigation covers various aspects, including the existence of solitary waves, their nonlinear stability, and instability. Notably, we address a broader class of nonlinearities represented by $\mu_1|u|^{p_1-1}u+\mu_2|u|^{p_2-1}u$, with $p_1>p_2$, encompassing cases where $\mu_1>0$ and $\mu_1<0<\mu_2$. One of the distinct features of our work is the absence of scaling, which introduces several challenges in establishing the existence of ground states. To overcome these challenges, we employ two different minimization problems, offering novel approaches to address this issue. Furthermore, our study includes a nuanced analysis to ascertain the stability of these ground states. Intriguingly, we extend our stability analysis to encompass cases where the convexity of the Lyapunov function is not guaranteed. This expansion of stability criteria represents a significant contribution to the field. Moving beyond the analysis of solitary waves, we shift our focus to the associated Cauchy problem. Here, we derive criteria that determine whether solutions exhibit finite-time blow-up or remain uniformly bounded within the energy space. Remarkably, our study unveils a notable gap in the existing literature, characterized by the absence of both theoretical evidence of blow-up and uniform boundedness. To explore this intriguing scenario, we employ the integrating factor method, providing a numerical investigation of solution behavior. This method distinguishes itself by offering spectral-order accuracy in space and fourth-order accuracy in time. Lastly, we rigorously establish the strong instability of the ground states, adding another layer of understanding to the complex dynamics inherent in the generalized KP equation

SAPSIZ MEŞE’DE KESİŞ YÖNÜ VE SU BAZLI VERNİK TÜRÜNÜN SES GEÇİŞ KAYBINA ETKİSİ

Bu çalışmada iç dekorasyonda önemli bir kullanım alanına sahip olan sapsız meşe ağacının ses geçiş kayıpları incelenmiştir. Çalışmada kesiş yönü ve su bazlı verniklerin (tek ve çift kompenantlı) ses geçiş kaybına etkileri araştırılmıştır. Bu amaçla 18 mm kalınlığındaki ağaç malzeme, teğet ve radyal yönde kesilerek tek ve çift kompenantlı verniklerle verniklenmiş, empedans tüpü kullanılarak ses geçiş kayıpları belirlenmiştir. Araştırma sonuçlarına göre çift kompenantlı su bazlı verniğin ve teğet kesitin ses geçiş kaybını olumlu yönde etkilediği tespit edilmiştir

Relaxation-based viscosity mapping for magnetic particle imaging

Magnetic particle imaging (MPI) has been shown to provide remarkable contrast for imaging applications such as angiography, stem cell tracking, and cancer imaging. Recently, there is growing interest in the functional imaging capabilities of MPI, where 'color MPI' techniques have explored separating different nanoparticles, which could potentially be used to distinguish nanoparticles in different states or environments. Viscosity mapping is a promising functional imaging application for MPI, as increased viscosity levels in vivo have been associated with numerous diseases such as hypertension, atherosclerosis, and cancer. In this work, we propose a viscosity mapping technique for MPI through the estimation of the relaxation time constant of the nanoparticles. Importantly, the proposed time constant estimation scheme does not require any prior information regarding the nanoparticles. We validate this method with extensive experiments in an in-house magnetic particle spectroscopy (MPS) setup at four different frequencies (between 250 Hz and 10.8 kHz) and at three different field strengths (between 5 mT and 15 mT) for viscosities ranging between 0.89 mPa • s-15.33 mPa • s. Our results demonstrate the viscosity mapping ability of MPI in the biologically relevant viscosity range. © 2017 Institute of Physics and Engineering in Medicine

Relaxation-based viscosity mapping for magnetic particle imaging

Magnetic particle imaging (MPI) has been shown to provide remarkable contrast for imaging applications such as angiography, stem cell tracking, and cancer imaging. Recently, there is growing interest in the functional imaging capabilities of MPI, where 'color MPI' techniques have explored separating different nanoparticles, which could potentially be used to distinguish nanoparticles in different states or environments. Viscosity mapping is a promising functional imaging application for MPI, as increased viscosity levels in vivo have been associated with numerous diseases such as hypertension, atherosclerosis, and cancer. In this work, we propose a viscosity mapping technique for MPI through the estimation of the relaxation time constant of the nanoparticles. Importantly, the proposed time constant estimation scheme does not require any prior information regarding the nanoparticles. We validate this method with extensive experiments in an in-house magnetic particle spectroscopy (MPS) setup at four different frequencies (between 250 Hz and 10.8 kHz) and at three different field strengths (between 5 mT and 15 mT) for viscosities ranging between 0.89 mPa • s-15.33 mPa • s. Our results demonstrate the viscosity mapping ability of MPI in the biologically relevant viscosity range. © 2017 Institute of Physics and Engineering in Medicine