Nonlinear Physics with Mathematica for Scientists and Engineers

In this text on nonlinear physics, we are primarily interested in the problem of how to deal with physical phenomena described by nonlinear ordinary or partial differential equations (ODEs or PDEs), i.e., by equations which are nonlinear functions of the dependent variables.

In physics and engineering, the most readily visualized examples are usually those in classical mechanics. The starting point at the elementary level is New-ton’s second law, while at the more advanced level one can form the Lagrangian L = T -V where T is the kinetic energy andV the potential energy. We shall apply both approaches to nonlinear mechanics where the force is a nonlinear function of the displacement and, perhaps, the velocity.

Patterns pervade the world of nature as well as the world of the intellect. In the biological realm we are quite familiar with the stripes on a zebra, the spots on a leopard, and the colorful markings of certain birds, fish, and butterflies. In the physical world we may have noticed the pretty fringe patterns which occur when thin films of oil spread on a road surface or the wonderful shapes that ice crystals can assume when trees are coated after an ice storm. If we go into a wallpaper shop, we can be overwhelmed by the wide variety of patterns available, the patterns created by someone’s artistic imagination. If we talk to a scientist we will soon find that his or her goal in life is usually to discover (impose?) some underlying pattern to the phenomena under investigation and then attempt to mathematically model that pattern. In this section, we shall look at some attempts to understand or create patterns through the use of nonlinear modeling and concepts. Our first example is from the world of chemistry.

Most nonlinear systems cannot be solved exactly so we must resort to a variety of approaches in order to obtain an approximate solution. Where applicable, the phase plane portrait can serve as a valuable tool for qualitatively determining the types of possible solutions before resorting to numerical or (usually approximate) analytical methods for specific initial conditions. In this chapter, the concept of phase plane analysis will be examined in some depth, not for a specific problem, but for a wide class of physical problems described by the following system of first order equations: 4.1$$\begin{array}{*{20}{c}} {\frac{{dx}}{{dt}} = P(x,y),} & {\frac{{dy}}{{dt}} = Q(x,y),} \\ \end{array}$$ where P, Q are, in general, nonlinear functions of x and y and the independent variable has been taken here to be time t. In the laser competition equations (2.32), t would, of course, be replaced with z, the spatial coordinate The mathematician would refer to this set of equations as being autonomous, meaning that P and Q do not explicitly depend on t. Why it is desirable to restrict the discussion for the moment to autonomous equations will become readily apparent.1

The topological approach allows us to easily determine the types of solutions that a given physically interesting nonlinear ODE system will permit. With its qualitative nature established, the next step is to ascertain whether a given solution can be described by some sort of analytic expression.

The combination of finite-difference approximations to the derivatives and the use of a high speed digital computer leads to a very powerful approach to solving the nonlinear ordinary and partial differential equations of physics. For many nonlinear systems, particularly those where the nonlinear terms are not small corrections to an otherwise linear behavior, the numerical route may be the best or only feasible way to travel. For the nonlinear ODEs encountered earlier in the text, the student has been allowed to use the Mathematica numerical ODE solver without any explanation provided of the principles on which it is based. In this chapter, we would like to partially fill that void by briefly describing how some of the common numerical schemes for solving nonlinear ODEs are derived. Our aim is to provide a simple conceptual framework that will make the reader more comfortable with the numerical approach while progressing through the rest of the topics that lie ahead. It should be emphasized that we are not attempting to explain the code which underlies Mathematica’s NDSolve command which is about 500 pages long.

For autonomous nonlinear and non-conservative systems described by the differential equations 7.1$$\begin{array}{*{20}{c}} {\frac{{dx}}{{dt}} = P(x,y),} & {\frac{{dy}}{{dt}} = Q(x,y)} \\ \end{array}$$ a new kind of trajectory, the limit cycle, has been briefly encountered at various points in the preceding chapters. The Van der Pol (VdP) electronic oscillator with P(x, y) = y and Q(x, y) = ∈(1-x2)y - x, for example, made its debut in Chapter 2. In this chapter we would like to explore some of the more important properties of limit cycles in greater depth.

In Mathematica File MF09, the student has already seen some of the exciting possible solutions that can occur for a forced oscillator depending on the amplitude F chosen for the forcing term. The nonlinear system in that file is the Duffing oscillator 8.1$$\ddot{x} + 2\gamma \dot{x} + \alpha x + \beta {{x}^{3}} = F \cos (\omega t)$$ with γ the damping coefficient and the driving frequency. In mechanical terms, the lhs of the Duffing equation can be thought of as a damped nonlinear spring. With the forcing term on the rhs included, the following special cases have been extensively studied in the literature: 1Hard spring Duffing oscillator: α > 0, β> 02Soft spring Duffing oscillator: α > 0, β < 03Inverted Duffing oscillator: α < 0, β > 04Nonharmonic Duffing oscillator: α = 0, β > 0

In the study of forced oscillator phenomena, we have avoided plunging into heavy analysis because generally the details can be gory and are probably soon forgotten by the student. The virtue of maps, and the logistic map in particular, is that they are amenable to relatively simple, easily understandable, analysis because they are governed by finite difference equations rather than nonlinear differential equations. Despite their relative simplicity, nonlinear maps can guide us along the road to understanding many of the features that are seen in forced nonlinear oscillator systems such as the period doubling route to chaos, the stretching and folding of strange attractors, and so on. The emphasis will be on understanding rather than trying to establish the direct connection of a given map with a particular nonlinear ODE which is a nontrivial task beyond the scope of this text. Some new concepts like bifurcation diagrams and Lyapunov exponents, which will be encountered in this chapter, could have been introduced in the last chapter but are more easily dealt with in the framework of nonlinear maps.

In Section 3.2.1 the reader encountered the Korteweg–deVries (KdV) equation which has been successfully used to describe the propagation of solitons in various physical contexts, the most historically famous being for shallow water waves in a rectangular canal. Using subscripts to denote partial derivatives with respect to the distance x traveled and the time t elapsed, the KdV equation for the (normalized) transverse displacement ψ of the water waves is 10.1$${\psi _t} + \alpha \psi {\psi _x} + {\psi _{xxx}} = 0$$.

One approach to investigating the collisional stability of solitary waves, as well as the evolution of other input shapes, for the nonlinear PDEs that have been encountered is through the use of numerical simulation. The PDEs can be solved numerically using either explicit or implicit schemes. In either case, the starting point is the representation of the partial derivatives by finite difference approximations. This may be easily accomplished by generalizing the treatment outlined in Chapter 6 for nonlinear ODEs.

The inverse scattering method (ISM) is important because it uses linear techniques to solve the initial value problem for a wide variety of nonlinear wave equations of physical interest and to obtain N-soliton (N = 1, 2, 3,…) solutions. The KdV two-soliton solution was the subject of Mathematica File MF08 where it was animated. The ISM was first discovered and developed by Gardner, Greene, Kruskal and Miura [GGKM67] for the KdV equation. A general formulation of the method by Peter Lax [Lax68] soon followed. This nontrivial formulation is the subject of the next few sections. It is presented to give the reader the flavor of a more advanced topic in nonlinear physics. As you will see, the inverse scattering method derives its name from its close mathematical connection for the KdV case to the quantum mechanical scattering of a particle by a localized potential or tunneling through a barrier.

As Richard Feynmann reminds us in his wonderful quote [FLS77], science de-mands that all theory be checked by experiment. It is because nonlinear physics can often be so profoundly counter-intuitive that these laboratory investigations are, in our opinion, so important. Understanding is enhanced when experiments are used to check theory, so please attempt as many of the activities as you can. As you perform them, we hope that you will be amazed and startled by strange behavior, intrigued and terrorized by new ideas, and be able to amaze your friends as you relate your strange sightings! Remember that imagination is as important as knowledge, so exercise yours whenever possible. But please be careful, as exposure to nonlinear activities can be addictive, can provide fond memories, and can awaken an interest that lasts a lifetime.