Exact solutions for a generalized Higgs equation

Long time behavior of solutions of nonlinear classical field equations: the example of NLS defocusing

This paper describes a number of simple but quite effective methods for constructing exact solutions of nonlinear partial differential equations that involve a relatively small amount of intermediate calculations. The methods employ two main ideas: (i) simple exact solutions can serve to construct more complex solutions of the equations under consideration and (ii) exact solutions of some equations can serve to construct solutions of other, more complex equations. In particular, we propose a method for constructing complex solutions from simple solutions using translation and scaling. We show that in some cases, rather complex solutions can be obtained by adding one or more terms to simpler solutions. There are situations where nonlinear superposition allows us to construct a complex composite solution using similar simple solutions. We also propose a few methods for constructing complex exact solutions to linear and nonlinear PDEs by introducing complex-valued parameters into simpler solutions. The effectiveness of the methods is illustrated by a large number of specific examples (over 30 in total). These include nonlinear heat equations, reaction–diffusion equations, wave type equations, Klein–Gordon type equations, equations of motion through porous media, hydrodynamic boundary layer equations, equations of motion of a liquid film, equations of gas dynamics, Navier–Stokes equations, and some other PDEs. Apart from exact solutions to ‘ordinary’ partial differential equations, we also describe some exact solutions to more complex nonlinear delay PDEs. Along with the unknown function at the current time, u=u(x,t), these equations contain the same function at a past time, w=u(x,t−τ), where τ>0 is the delay time. Furthermore, we look at nonlinear partial functional-differential equations of the pantograph type, which, in addition to the unknown u=u(x,t), also contain the same functions with dilated or contracted arguments, w=u(px,qt), where p and q are scaling parameters. We propose an efficient approach to construct exact solutions to such functional-differential equations. Some new exact solutions of nonlinear pantograph-type PDEs are presented. The methods and examples in this paper are presented according to the principle “from simple to complex”.

An extended mapping method with a computerized symbolic computation is used for constructing a new exact travelling wave solutions for nonlinear evolution equations arising in physics, namely, generalized Zakharov Kuznetsov equation with variable coefficients. As a result, many exact travelling wave solutions are obtained which include new periodic wave solution, trigonometric function solutions and rational solutions. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations with variable coefficients arising in mathematical physics. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010

The Kudryashov and Sinelshchikov (KS) equation address pressure waves in liquid-gas bubble mixtures while considering heat transport and viscosity. This study mainly includes two types of generalized solutions: polynomial function traveling wave solutions and rational function traveling wave solutions. In this study, we constructed the KS equation’s exact traveling and solitary wave solutions with variable coefficients by the generalized unified method (GUM). These newly created solutions play a significant role in mathematical physics, optical fiber physics, plasma physics, and other applied science disciplines. We illustrated the dynamical behavior of the discovered solutions in three dimensions. We proposed the possibility of discussing wave interaction and other wave structures using bilinear form related to the Hirota method for the fractional solutions.

A new sub-equation method applied to obtain exact travelling wave solutions of some complex nonlinear equations

Our aim in this research work is to formulate the exact traveling and solitary wave solutions of nonlinear diffusion reaction (DR) equation with quadratic and cubic nonlinearities by implementing the new technique which is a modified mathematical method. We have investigated the density independent nonlinear diffusion equation with convective flux term. As a result, we have found a variety of new exact traveling and solitary wave solutions in the form of dark solitons, bright solitons, combined dark-bright solitons, traveling wave, periodic wave solutions and we also represent the physical structure of the obtained solutions by two- and three-dimensional graphics by using the Mathematica software. This work proves the power, reliability and fruitfulness of this new technique.

The nonlinear Schrodinger equation (NLSE) with the aid of three order dispersion terms is investigated to find the exact solutions via the extended $ (\frac{G'}{G^{2}})$ -expansion method and the first integral method. Many exact traveling wave solutions, such as trigonometric, hyperbolic, rational, soliton and complex function solutions, are characterized with some free parameters of the problem studied. It is corroborated that the proposed techniques are manageable, straightforward and powerful tools to find the exact solutions of nonlinear partial differential equations (PDEs). Some figures are plotted to describe the propagation of traveling wave solutions expressed by the hyperbolic functions, trigonometric functions and rational functions.

The sinh-Gordon equation is simply the classical wave equation with a nonlinear sinh source term. It arises in diverse scientific applications including differential geometry theory, integrable quantum field theory, fluid dynamics, kink dynamics, and statistical mechanics. It can be used to describe generic properties of string dynamics for strings and multi-strings in constant curvature space. In the present paper, we study a generalized sinh-Gordon equation with variable coefficients with the goal of obtaining analytical traveling wave solutions. Our results show that the traveling waves of the variable coefficient sinh-Gordon equation can be derived from the known solutions of the standard sinh-Gordon equation under a specific selection of a choice of the variable coefficients. These solutions include some real single and multi-solitons, periodic waves, breaking kink waves, singular waves, periodic singular waves, and compactons. These solutions might be valuable when scientists model some real-life phenomena using the sinh-Gordon equation where the balance between dispersion and nonlinearity is perturbed.

Using a traveling wave reduction technique, we have shown that Maccari equation, (2?1)-dimensional nonlinear Schrodinger equation, medium equal width equation, (3?1)-dimensional modified KdV-Zakharov- Kuznetsev equation, (2?1)-dimensional long wave-short wave resonance interaction equation, perturbed nonlinear Schrodinger equation can be reduced to the same family of auxiliary elliptic-like equations. Then using extended F-expansion and projective Riccati equation methods, many types of exact traveling wave solutions are obtained. With the aid of solutions of the elliptic-like equation, more explicit traveling wave solutions expressed by the hyper- bolic functions, trigonometric functions and rational func- tions are found out. It is shown that these methods provide a powerful mathematical tool for solving nonlinear evolu- tion equations in mathematical physics. A variety of structures of the exact solutions of the elliptic-like equation are illustrated.

In the development of quantum mechanics in the 1920s, both matrix mechanics (developed by Born, Heisenberg and Jordon) and wave mechanics (developed by Schrödinger) prevailed. These early attempts corresponded to the quantum mechanics of particles. Matrix mechanics was found to lead directly to the Schrödinger equation, and the Schrödinger equation could be used to derive the alternative problem for matrix mechanics. Later emphasis lay on the development of the dynamics of fields, where the classical field equations were quantized (see, for example, Weinberg). Today, quantum field theory is one of the most successful physical theories ever developed. The symmetry between particle and wave mechanics is exploited herein. One of the important properties of quantum mechanics is that it is linear, leading to some confusion about how to treat the problem of nonlinear classical field equations. In the present paper we address the case of classical nonlinear soliton equations which are exactly integrable in terms of the periodic/quasiperiodic inverse scattering transform. This means that all physical spectral solutions of the soliton equations can be computed exactly for these specific boundary conditions. Unfortunately, such solutions are highly nonlinear, leading to difficulties in solving the associated quantum mechanical problems. Here we find a strategy for developing the quantum mechanical solutions for soliton dynamics. To address this difficulty, we apply a recently derived result for soliton equations, i.e., that all solutions can be written as quasiperiodic Fourier series. This means that soliton equations, in spite of their nonlinear solutions, are perfectly linearizable with quasiperiodic boundary conditions, the topic of finite gap theory, i.e., the inverse scattering transform with periodic/quasiperiodic boundary conditions. We then invoke the result that soliton equations are Hamiltonian, and we are able to show that the generalized coordinates and momenta also have quasiperiodic Fourier series, a generalized linear superposition law, which is valid in the case of nonlinear, integrable classical dynamics and is here extended to quantum mechanics. Hamiltonian dynamics with the quasiperiodicity of inverse scattering theory thus leads to matrix mechanics. This completes the main theme of our paper, i.e., that classical, nonlinear soliton field equations, linearizable with quasiperiodic Fourier series, can always be quantized in terms of matrix mechanics. Thus, the solitons and their nonlinear interactions are given an explicit description in quantum mechanics. Future work will be formulated in terms of the associated Schrödinger equation.

Construction of exact solutions to nonlinear PDEs with delay using solutions of simpler PDEs without delay

The exact parametric representations of the traveling wave solutions for a nonlinear elastic rod equation are considered. By using the method of planar dynamical systems, in different parameter regions, the phase portraits of the corresponding traveling wave system are given. Exact explicit kink wave solutions, periodic wave solutions and some unbounded wave solutions are obtained.

Based on the invariant subspace method, a symbolic computation scheme and its corresponding MAPLE package are developed to construct exact solutions for nonlinear evolution equations. In the symbolic computation scheme, a crucial step is constructing the linear differential equations as invariant subspaces that systems of evolution equations admin and taking their solutions as subspaces to construct exact solutions. The MAPLE package is proved to provide an easy way for constructing exact solutions of evolution equations automatically by only inputting several necessary parameters. Three different types of examples are given to illustrate the scope and demonstrate the validity of our package, especially for wave equation. The results of the examples reveal that there are polynomial subspaces, trigonometric subspaces, exponential subspaces and other complex subspaces as invariant subspaces that evolutions equations admit. In addition, our MAPLE software package provides a helpful and easy-to-use tool in science and engineering to deal with a wide variety of (1+1) dimensional nonlinear evolution equations.

The main goal of this Focus Point volume is to explore new advances on theoretical and computational techniques for uncertainty quantification (UQ) of modeling and simulation in relevant problems appearing in physics sciences. The papers submitted to this volume were received by invitation. As a result we have collected a number of high-quality contributions dealing with a rich and update picture of UQ analysis that includes its interplay with laws in physics problems, mainly, formulated via differential equations. The following text highlights some of the primary aspects of the topics covered as well as the contributions of the published papers. The consideration of randomness in the analysis of mathematical models for analyzing relevant problems related to health is currently a main research topic. The paper "The role of noise in the tumor dynamics under chemotherapy treatment" proposes a two-dimensional stochastic nonlinear differential system, taking into account the impact of chemotherapy on both the tumor and the healthy cells. The randomness is driven by two uncorrelated Gaussian noises with different levels of intensity. Authors analyze this interesting problem combining a number of techniques that include bifurcation diagrams, phase portraits, time series and probability density plots. The contribution is aimed at offering a trustworthy explanation of tumor growth under medical treatment. The Focus Point volume continues with the study of a second model also belonging to Medicine, specifically, to Epidemiology. In "Random resampling numerical simulations applied to a SEIR compartmental model," authors revisit the SEIR model addressed to study the dynamics of COVID-19 in Spain during its first pandemic-wave in 2020. Authors concentrate on estimating model parameters taking into account data uncertainties by means of bootstrap sampling. The validation of the stochastic analysis is performed by different goodness-of-fit measures. They show that the point estimates obtained by the bootstrap samples improve the ones of the original data. The use of fractional and stochastic calculus has emerged, mainly, during the last decades with important applications in Physics. In "Hölder continuity of mild solutions of space-time fractional stochastic heat equation driven by colored noise," authors combine the Caputo fractional derivative with the fractional Laplace operator to analyze the heat equation with noise driven by a colored noise. The paper establishes Hölder continuity of mild random field solutions of the aforementioned PDE by, first, taking advantage of key properties associated to the space-time fractional Green function. In "Asymptotic controllability of nonlinear Fokker-Planck equations," the author studies a problem related with stabilization of the control for a Fokker-Plank PDE. This equation often appears in kinetic theory of statistical mechanics and, contains, as a particular case, the classical Einstein-Smoluchowski equation, which describes the time evolution of the probability density of Brownian particles in a field of forms with a given potential. In the paper, one identifies a class of final states (reachable states) for which the problem has solution. The results can have interesting applications in control theory and physics. The stochastic of multiphysics systems is often a challenging problem because different physical scales and physical effects must be properly combined to accurately conduct the corresponding analysis. When the study is done in the practical setting, it usually involves the rigorous treatment of large sets of data. In the paper "Random stick-slip oscillations in a multiphysics system" (Fig. 12 ), one combines stick-slip oscillations, electromechanical coupling and uncertainties to characterize and quantify the stochastic response of a multiphysics system in terms of two physical quantities, duration and position, which are dependent random variables that must be jointly studied.

A technique which is known as Sumudu Transform Method (STM) is studied for the construction of solutions of a most general form of delay differential equations of pantograph type. This is a pioneer study on using the STM to construct the solutions of delay differential equations of pantograph type with variable coefficients. We obtain the exact and approximate solutions of nonlinear problems with multiproportional delays and variable coefficients. The strength of STM is illustrated in reducing the complex computational work as compared to the well-known methods. This paper shows how to succinctly identify the Lagrange multipliers for nonlinear delay differential equations with variable coefficients, using the STM. The potency and suitability of the STM are exhibited by giving expository examples. The solutions of nonlinear Volterra integro-differential equations of pantograph type are also obtained.