Stability analysis of multi-spatial Riesz and multi-fractional non-linear damped wave equations involving Caputo-Fabrizio derivative

Global existence and nonexistence of solutions for a system of nonlinear damped wave equations

All low‐order conservation laws are found for a general class of nonlinear wave equations in one dimension with linear damping which is allowed to be time‐dependent. Such equations arise in numerous physical applications and have attracted much attention in analysis. The conservation laws describe generalized momentum and boost momentum, conformal momentum, generalized energy, dilational energy, and light cone energies. Both the conformal momentum and dilational energy have no counterparts for nonlinear undamped wave equations in one dimension. All of the conservation laws are obtainable through Noether's theorem, which is applicable because the damping term can be transformed into a time‐dependent self‐interaction term by a change of dependent variable. For several of the conservation laws, the corresponding variational symmetries have a novel form which is different than any of the well‐known variational symmetries admitted by nonlinear undamped wave equations in one dimension.

Nonlinear wave equation is a typical nonlinear evolution equation with important theoretical significance and application value. The physical meaning of damping is the attenuation of force, or the energy dissipation of objects in motion. It is to stop the object from continuing to move. When an object vibrates under the action of an external force, a reaction force will be generated to attenuate the external force, which is called damping force. This paper studies the conservation laws for one dimensional damped nonlinear wave equation with constant and non-constant damping coefficient. Multipliers and corresponding conserved quantities are derived for wave equation with simple self-interaction term and constant damping coefficient using method of multipliers. The results are generalized in presence of more complicated source term and non-constant damping coefficient. Certain type of equations conserving all energy-momentum related quantities is found. For each individual conserved quantity, the corresponding multiplier and the corresponding type of equations are found.

We study low regularity behavior of the nonlinear wave equation in $\mathbb {R}^2$ augmented by the viscous dissipative effects described by the Dirichlet-Neumann operator. Problems of this type arise in fluid-structure interaction where the Dirichlet-Neumann operator models the coupling between a viscous, incompressible fluid and an elastic structure. We show that despite the viscous regularization, the Cauchy problem with initial data $(u,u_t)$ in $H^s(\mathbb {R}^2)\times H^{s-1}(\mathbb {R}^2)$ is ill-posed whenever $0 < s < s_{cr}$, where the critical exponent $s_{cr}$ depends on the degree of nonlinearity. In particular, for the quintic nonlinearity $u^5$, the critical exponent in $\mathbb {R}^2$ is $s_{cr} = 1/2$, which is the same as the critical exponent for the associated nonlinear wave equation without the viscous term. We then show that if the initial data is perturbed using a Wiener randomization, which perturbs initial data in the frequency space, then the Cauchy problem for the quintic nonlinear viscous wave equation is well-posed almost surely for the supercritical exponents $s$ such that $-1/6 < s \le s_{cr} = 1/2$. To the best of our knowledge, this is the first result showing ill-posedness and probabilistic well-posedness for the nonlinear viscous wave equation arising in fluid-structure interaction.

The goal of this paper is to bring new mathematical and physical methods to the problem of shallow water wave motions, for which there are many important engineering applications. There are four issues we address in this paper: (1) Nonlinear integrable wave equations have (quasiperiodic) Fourier series solutions. This means that, even though the wave dynamics are nonlinear, they can nevertheless be described as a linear superposition of sine waves, a surprising result. Nonlinearity arises from particular phase locking of the Fourier modes. (2) We discuss how these Fourier series solutions can always be computed to high accuracy for a particular nonlinear wave equation. While quasiperiodic Fourier series are more complicated than ordinary periodic Fourier series (a standard tool of ocean engineering as the Fast Fourier transform, or FFT), the advantage of dealing with a linear superposition law for nonlinear equations is quite useful. (3) We apply these Fourier series to the modelling of water waves. Amazingly, we use the linearity property of the quasiperiodic Fourier series, and we find a new wave model for solving nonlinear wave motion. Our model, which we call quasilinear, has properties similar to the well-known linear model already applied to ocean engineering problems for over 90 years [Paley and Weiner, 1935] [Longuet-Higgins, 1957]. An additional particular transformation (due to Baker [1907], Its &amp; Matveev [1976], Mumford [1982]) to the quasilinear model allows us to simply simulate typical oceanic nonlinear wave motions and to analyze data. (4) We discuss determination of the important probability distributions of water waves for amplitudes, heights and crests from knowledge of the quasiperiodic Fourier series solutions. The results given herein are based upon a simply stated Theorem: Given an integrable wave equation and/or its Hamiltonian perturbations we can write the full spectral solutions of the equations as quasiperiodic Fourier series and we can also analytically determine the probability distributions of various properties of the wave field (including, amplitudes, crest heights and wave heights). We are interested in shallow water wave motion in the present paper and our nonlinear wave equation of choice is the Korteweg-deVries equation (KdV). Our simple discussion has therefore introduced quasiperiodic Fourier methods and probability distributions to the solutions of the KdV equation. Furthermore, the Kadomtsev-Petviashvili equation is a two-dimensional shallow water wave equation (a generalization of the KdV equation, also with Hamiltonian perturbations) and it too can be treated in a similar manner. In this context this paper is dedicated to a discussion of important physical and engineering tools for the shallow water domain.

Chapter 12 The nonlinear Schrödinger equation as both a PDE and a dynamical system

Nonlinear damped wave equation: Existence and blow-up

The last twenty years has seen the birth and subsequent evolution of a fundamental new idea in nonlinear wave research: Rogue waves, freak waves or extreme events in the wave field dynamics can often be classified as coherent structure solutions of the requisite nonlinear partial differential wave equations (PDEs). Since a large number of generic nonlinear PDEs occur across many branches of physics, the approach is widely applicable to many fields including the dynamics of ocean surface waves, internal waves, plasma waves, acoustic waves, nonlinear optics, solid state physics, geophysical fluid dynamics and turbulence (vortex dynamics and nonlinear waves), just to name a few. The first goal of this paper is to give a classification scheme for solutions of this type using the inverse scattering transform (IST) with periodic boundary conditions. In this context the methods of algebraic geometry give the solutions of particular PDEs in terms of Riemann theta functions. In the classification scheme the Riemann spectrum fully defines the coherent structure solutions and their mutual nonlinear interactions. I discuss three methods for determining the Riemann spectrum: (1) algebraic-geometric loop integrals, (2) Schottky uniformization and (3) the Nakamura-Boyd approach. I give an overview of several nonlinear wave equations and graph some of their coherent structure solutions using theta functions. The second goal is to discuss how theta functions can be used for developing data analysis (nonlinear Fourier) algorithms; nonlinear filtering techniques allow for the extraction of coherent structures from time series. The third goal is to address hyperfast numerical models of nonlinear wave equations (which are thousands of times faster than traditional spectral methods).

In this paper we present some results on the global existence of weak solutions to a nonlinear variational wave equation and some related problems. We first introduce the main tools, the L p Young measure theory and related compactness results, in the first section. Then we use the L p Young measure theory to prove the global existence of dissipative weak solutions to the asymptotic equation of the nonlinear wave equation, and comment on its relation to Camassa-Holm equations in the second section. In the third section, we prove the global existence of weak solutions to the original nonlinear wave equation under some restrictions on the wave speed. In the last section, we present global existence of renormalized solutions to two-dimensional model equations of the asymptotic equation, which is also the so-called vortex density equation arising from sup-conductivity.

PurposeThis paper aims to solve linear and non-linear shallow water wave equations using homotopy perturbation method (HPM). HPM is a straightforward method to handle linear and non-linear differential equations. As such here, one-dimensional shallow water wave equations have been considered to solve those by HPM. Interesting results are reported when the solutions of linear and non-linear equations are compared.Design/methodology/approachHPM was used in this study.FindingsSolution of one-dimensional linear and non-linear shallow water wave equations and comparison of linear and non-linear coupled shallow water waves from the results obtained using present method.Originality/valueCoupled non-linear shallow water wave equations are solved.

Let $G$ be a compact Lie group. In this article, we investigate the Cauchy problem for a nonlinear wave equation with the viscoelastic damping on $G$. More precisely, we investigate some $L^2$-estimates for the solution to the homogeneous nonlinear viscoelastic damped wave equation on $G$ utilizing the group Fourier transform on $G$. We also prove that there is no improvement of any decay rate for the norm $\|u(t,\,\cdot )\|_{L^2(G)}$ by further assuming the $L^1(G)$-regularity of initial data. Finally, using the noncommutative Fourier analysis on compact Lie groups, we prove a local in time existence result in the energy space $\mathcal {C}^1([0,\,T],\,H^1_{\mathcal {L}}(G)).$

Invariant subspaces, exact solutions and stability analysis of nonlinear water wave equations

In this work, we proposed a new method called Laplace–Padé–Caputo fractional reduced differential transform method (LPCFRDTM) for solving a two-dimensional nonlinear time-fractional damped wave equation subject to the appropriate initial conditions arising in various physical models. LPCFRDTM is the amalgamation of the Laplace transform method (LTM), Padé approximant, and the well-known reduced differential transform method (RDTM) in the Caputo fractional derivative senses. First, the solution to the problem is gained in the convergent power series form with the help of the Caputo fractional-reduced differential transform method. Then, the Laplace–Padé approximant is applied to enlarge the domain of convergence. The advantage of this method is that it solves equations simply and directly without requiring enormous amounts of computational work, perturbations, or linearization, and it expands the convergence domain, leading to the exact answer. To confirm the effectiveness, accuracy, and convergence of the proposed method, four test-modeling problems from mathematical physics nonlinear wave equations are considered. The findings and results showed that the proposed approach may be utilized to solve comparable wave equations with nonlinear damping and source components and to forecast and enrich the internal mechanism of nonlinearity in nonlinear dynamic events.

Chapter 7 On the Global Weak Solutions to a Variational Wave Equation

Large time behavior of solutions for a system of nonlinear damped wave equations