Nonlinear Partial Differential Equations for Scientists and Engineers

Partial differential equations arise frequently in the formulation of fundamental laws of nature and in the mathematical analysis of a wide variety of problems in applied mathematics, mathematical physics, and engineering science. This subject plays a central role in modern mathematical sciences, especially in physics, geometry, and analysis. Many problems of physical interest are described by partial differential equations with appropriate initial and/or boundary conditions. These problems are usually formulated as initial-value problems, boundary-value problems, or initial boundary-value problems. In order to prepare the reader for study and research in nonlinear partial differential equations, a broad coverage of the essential standard material on linear partial differential equations and their applications is required.

This chapter deals with the basic ideas and some important nonlinear model equations which arise in a wide variety of physical problems. This is followed by variational principles and the Euler-Lagrange equations. Also included are Plateau’s problem, Hamilton’s principle, Lagrange’s equations, Hamilton’s equations, the variational principle for nonlinear Klein-Gordon equations, and the variational principle for nonlinear water waves. Special attention is given to the Euler equation of motion, the continuity equation, linear water wave problems and their solutions, nonlinear finite amplitude waves (the Stokes waves), and linear and nonlinear dispersion relations. Finally, the modern theory of nonlinear water waves is formulated.

Many problems in mathematical, physical, and engineering sciences deal with the formulation and the solution of first-order partial differential equations. From a mathematical point of view, first-order equations have the advantage of providing a conceptual basis that can be utilized for second-, third-, and higher-order equations.

First-order, nonlinear, partial differential equations arise in various areas of physical sciences which include geometrical optics, fluid dynamics, and analytical dynamics. An important example of such equations is the Hamilton-Jacobi equation used to describe dynamical systems. Another famous example of the first-order nonlinear equations is the eikonal equation which arises in nonlinear optics and also describes the propagation of wave fronts and discontinuities for acoustic wave equations, Maxwell’s equations, and equations of elastic wave propagation. Evidently, first-order, nonlinear equations play an important role in the development of these diverse areas.

This chapter deals with the theory and applications of kinematic waves to several real-world problems, which include traffic flow on highways, flood waves in rivers, glacier flow, roll waves in an inclined channel, chromatographic models, and sediment transport in rivers. The general ideas and essential features of these problems are of wide applicability. Other applications of conservation laws include various chromatographic models in chemistry and the movement of pollutants in waterways. The propagation of traffic jams is almost similar to the shock waves that cause noise pollution near airports and spaceports. Kinematic wave phenomena also play an important role in traveling detonation and combustion fronts, the wetting water fronts observed in soils after rainfall, and the clanking of shunting trains. All of these problems are essentially based on the theory of kinematic waves developed by Lighthill and Whitham (1955). Many basic ideas and important features of hyperbolic waves and kinematic shock waves are found to originate from gas dynamics, so specific nonlinear models which describe Riemann’s simple waves with Riemann’s invariants and shock waves in gas dynamics are discussed. Considerable attention is also given to nonlinear hyperbolic systems and Riemann’s invariants, generalized simple waves, and generalized Riemann’s invariants.

Historically, the study of nonlinear dispersive waves started with the pioneering work of Stokes in 1847 on water waves. Stokes first proved the existence of periodic wavetrains which are possible in nonlinear dispersive wave systems. He also determined that the dispersion relation on the amplitude produces significant qualitative changes in the behavior of nonlinear waves. It also introduces many new phenomena in the theory of dispersive waves, not merely the correction of linear results. These fundamental ideas and the results of Stokes have provided a tremendous impact on the subject of nonlinear water waves, in particular, and on nonlinear dispersive wave phenomena, in general. Stokes’ profound investigations on water waves can be considered as the starting point for the modern theory of nonlinear dispersive waves. In fact, most of the fundamental concepts and results on nonlinear dispersive waves originated in the investigation of water waves. The study of nonlinear dispersive waves has proceeded at a very rapid pace with remarkable developments over the past three decades.

Many physical phenomena are described by the interaction of convection and diffusion and also by the interaction of diffusion and reaction. From a physical point of view, the convection-diffusion process and the diffusion-reaction process are quite fundamental in describing a wide variety of problems in physical, chemical, biological, and engineering sciences. Some nonlinear, partial differential equations that model these processes provide many new insights into the question of interaction of nonlinearity and diffusion. It is well known that the Burgers equation is the simplest nonlinear model equation representing phenomena described by a balance between convection and diffusion. The Fisher equation is another simplest nonlinear model equation which arises in a wide variety of problems involving diffusion and reaction.

Dispersion and nonlinearity play a fundamental role in wave motions in nature. The nonlinear shallow water equations that neglect dispersion altogether lead to breaking phenomena of the typical hyperbolic kind with the development of a vertical profile. In particular, the linear dispersive term in the Korteweg-de Vries equation prevents this from ever happening in its solution. In general, breaking can be prevented by including dispersive effects in the shallow water theory. The nonlinear theory provides some insight into the question of how nonlinearity affects dispersive wave motions. Another interesting feature is the instability and subsequent modulation of an initially uniform wave profile.

It has already been indicated in Section 2.3 that the nonlinear Schrödinger (NLS) equation arises in a wide variety of physical problems in fluid mechanics, plasma physics, and nonlinear optics. The most common applications of the NLS equation include self-focusing of beams in nonlinear optics, modeling of propagation of electromagnetic pulses in nonlinear optical fibers which act as wave guides, and stability of Stokes waves in water. Some formal derivations of the NLS equation have been obtained by several methods which include the multiple scales expansions, the asymptotic method, Whitham’s (1965) averaged variational equations, and Phillips’ (1981) resonant interaction equations. Zakharov and Shabat (1972) developed an ingenious inverse scattering method to show that the NLS equation is completely integrable. The NLS equation is of great importance in adding to our fundamental knowledge of the general theory of nonlinear dispersive waves.

This chapter deals with the theory and applications of nonlinear Klein-Gordon (KG) and Sine-Gordon (SG) equations. Special emphasis is given to various methods of solutions of these equations. The Green function method combined with integral transforms is employed to solve the linear Klein-Gordon equation. The Whitham averaging procedure and the Whitham averaged Lagrangian principle are used to discuss solutions of the nonlinear Klein-Gordon equation. Included are different ways of finding general and particular solutions of the sine-Gordon equation. Special attention is given to solitons, antisolitons, breather solutions and the energy associated with them, interaction of solitons, Bäcklund transformations, similarity solutions, and the inverse scattering method. Significant features of these methods and solutions are described with other ramifications.

Many physical systems involving nonlinear wave propagation include the effects of dispersion, dissipation, and/or the inhomogeneous property of the medium. The governing equations are usually derived from conservation laws. In simple cases, these equations are hyperbolic. However, in general, the physical processes involved are so complex that the governing equations are very complicated and, hence, are not integrable by analytic methods. So, special attention is given to seeking mathematical methods which lead to a less complicated problem yet retain all of the important physical features. In recent years, several asymptotic methods have been developed for the derivation of the evolution equations, which describe how some dynamical variables evolve in time and space. So, we begin this chapter with one simple method of construction of the linear evolution equation from a given frequency-wavenumber dispersion relation of the form This relation is multiplied by −iU (k) exp [i (kx − ωt)] and integrated with respect to the wavenumber k from −∞ to ∞ to obtain the equation where L(u) represents an operator and U (k) is an arbitrary function which is related to the function u (x, t) by the inverse Fourier transform