Seven common errors in finding exact solutions of nonlinear differential equations

We analyze the common errors of the recent papers in which the solitary wave solutions of nonlinear differential equations are presented. Seven common errors are formulated and classified. These errors are illustrated by using multiple examples of the common errors from the recent publications. We show that many popular methods in finding of the exact solutions are equivalent each other. We demonstrate that some authors look for the solitary wave solutions of nonlinear ordinary differential equations and do not take into account the well - known general solutions of these equations. We illustrate several cases when authors present some functions for describing solutions but do not use arbitrary constants. As this fact takes place the redundant solutions of differential equations are found. A few examples of incorrect solutions by some authors are presented. Several other errors in finding the exact solutions of nonlinear differential equations are also discussed.Comment: 42 page

A transformed rational function method and exact solutions to the 3+1 dimensional Jimbo-Miwa equation

A direct approach to exact solutions of nonlinear partial differential equations is proposed, by using rational function transformations. The new method provides a more systematical and convenient handling of the solution process of nonlinear equations, unifying the tanh-function type methods, the homogeneous balance method, the exp-function method, the mapping method, and the F-expansion type methods. Its key point is to search for rational solutions to variable-coefficient ordinary differential equations transformed from given partial differential equations. As an application, the construction problem of exact solutions to the 3+1 dimensional Jimbo-Miwa equation is treated, together with a B\"acklund transformation.Comment: 13 page

Multiscale characterisation of chimneys/pipes: Fluid escape structures within sedimentary basins

Evaluation of seismic reflection data has identified the presence of fluid escape structures cross-cutting overburden stratigraphy within sedimentary basins globally. Seismically-imaged chimneys/pipes are considered to be possible pathways for fluid flow, which may hydraulically connect deeper strata to the seabed. The properties of fluid migration pathways through the overburden must be constrained to enable secure, long-term subsurface carbon dioxide (CO2) storage. We have investigated a site of natural active fluid escape in the North Sea, the Scanner pockmark complex, to determine the physical characteristics of focused fluid conduits, and how they control fluid flow. Here we show that a multi-scale, multi-disciplinary experimental approach is required for complete characterisation of fluid escape structures. Geophysical techniques are necessary to resolve fracture geometry and subsurface structure (e.g., multi-frequency seismics) and physical parameters of sediments (e.g., controlled source electromagnetics) across a wide range of length scales (m to km). At smaller (mm to cm) scales, sediment cores were sampled directly and their physical and chemical properties assessed using laboratory-based methods. Numerical modelling approaches bridge the resolution gap, though their validity is dependent on calibration and constraint from field and laboratory experimental data. Further, time-lapse seismic and acoustic methods capable of resolving temporal changes are key for determining fluid flux. Future optimisation of experiment resource use may be facilitated by the installation of permanent seabed infrastructure, and replacement of manual data processing with automated workflows. This study can be used to inform measurement, monitoring and verification workflows that will assist policymaking, regulation, and best practice for CO2 subsurface storage operations

The study of soliton fission and fusion in (2+1)-dimensional nonlinear system

By means of a variable separation approach and an extended homogeneous balance method, a general variable separation excitation of a (2+1)-dimensional nonlinear system is derived. Based on the derived solution with arbitrary functions, we reveal soliton fission and fusion phenomena in the (2+1)-dimensional soliton system