Robert M. Miura

Korteweg‐de Vries Equation and Generalizations. II. Existence of Conservation Laws and Constants of Motion

With extensive use of the nonlinear transformations presented in Paper I of the series, a variety of conservation laws and constants of motion are derived for the Korteweg‐de Vries and related equations. A striking connection with the Sturm‐Liouville eigenvalue problem is exploited.

The Korteweg–deVries Equation: A Survey of Results

The Korteweg–de Vries equation \[ u_t + uu_x + u_{xxx} = 0\] is a nonlinear partial differential equation arising in the study of a number of different physical systems, e.g., water waves, plasma physics, anharmonic lattices, and elastic rods. It describes the long time evolution of small-but-finite amplitude dispersive waves. From detailed studies of properties of the equation and its solutions, the concept of solitons was introduced and the method for exact solution of the initial-value problem using inverse scattering theory was developed. A survey of these and other results for the Korteweg–deVries equation are given, including conservation laws, an alternate method for exact solution, soliton solutions, asymptotic behavior of solutions, Backlund transformation, and a nonlinear WKB method. The recent literature contains many extensions of these ideas to a number of other nonlinear evolution equations of physical interest and to other classes of equations. Some of these equations and results are indica...

Korteweg‐deVries Equation and Generalizations. V. Uniqueness and Nonexistence of Polynomial Conservation Laws

The conservation laws derived in an earlier paper for the Korteweg‐deVries equation are proved to be the only ones of polynomial form. An algebraic operator formalism is developed to obtain explicit formulas for them.

A mathematical model for spreading cortical depression

A mathematical model is derived from physiological considerations for slow potential waves (called spreading depression) in cortical neuronal structures. The variables taken into account are the intra- and extracellular concentrations of Na+, Cl-, K+, and Ca++, together with excitatory and inhibitor transmitter substances. The general model includes conductance changes for these various ions, which may occur at nonsynaptic and synaptic membrane together with active transport mechanisms (pumps). A detailed consideration of only the conductance changes due to transmitter release leads to a system of nonlinear diffusion equations coupled with a system or ordinary differential equations. We obtain numerical solutions of a set of simplified model equations involving only K+ and Ca++ concentrations. The solutions agree qualitatively with experimentally obtained time-courses of these two ionic concentrations during spreading depression. The numerical solutions exhibit the observed phenomena of solitary waves and annihilation of colliding waves.

SINGULAR PERTURBATION ANALYSIS OF BOUNDARY-VALUE PROBLEMS FOR DIFFERENTIAL-DIFFERENCE EQUATIONS*

A study is made of a class of boundary-value problems for linear second-order differential-difference equations in which the highest-order derivative is multiplied by a small parameter. Consideration of the exact solution of simple model equations provides insight into the appropriate use of singular perturbation techniques for more general problems. The resulting analysis leads to several novel features which are not present in problems without differences.

Singular perturbation analysis of boundary-value problems for differential-difference equations. VI.: small shifts with rapid oscillations

This paper continues the authors’ study of boundary-value problems for singularly perturbed linear second-order differential-difference equations with small shifts. This study was initiated in the companion paper [SIAM J. Appl. Math., 54 (1994), pp. 249–272]. In this paper, the study is extended to problems that have solutions that exhibit rapid oscillations. Restrictions on the sizes of the shifts in terms of the small parameter are found such that, generally, the shifted terms cannot be replaced with truncated Taylor series. In particular, it is shown that, even when the shifts are small relative to the width of an oscillation, they can affect the solution to leading order. The conclusion is that oscillatory solutions are more sensitive to small delays than are layer solutions. It is shown that a suitably modified version of the standard WKB method can be used to obtain leading-order oscillatory solutions of these differential-difference equations. These preliminary studies of differential-difference eq...

Nonlinear phenomena in physics and biology

Computation and Innovation in the Nonlinear Sciences.- to Nonlinear Waves.- Remarks on Nonlinear Evolution Equations and the Inverse Scattering Transform.- The Linearity of Nonlinear Soliton Equations and the Three Wave Interaction.- Contour Dynamics: A Boundary Integral Evolutionary Method for Inviscid Incompressible Flows.- Numerical Computation of Nonlinear Waves.- Bifurcations, Fluctuations and Dissipative Structures.- Chemical Oscillations.- Models in Neurobiology.- Nonlinear Waves in Neuronal Cortical Structures.- Bifurcations in Insect Morphogenesis I.- Bifurcations in Insect Morphogenesis II.- Selection and Evolution in Molecular Systems.- Escape from Domains of Attraction for Systems Perturbed by Noise.- Seminars.- Error Propagation in Translation and its Relevance to the Nucleation of Life.- Pseudopotentials and Symmetries for Generalized Nonlinear Schrodinger Equations.- Superposition Principles for Nonlinear Differential Equations.- On Some Nonlinear Schrodinger Equations.- Asymptotic Evaluation Methods of Nonlinear Differential Equations Near the Instability Point.- Nonlinear Superposition of Simple Waves in Nonhomogeneous Systems.- A Method of Solving Nonlinear Differential Equations.- Lecturers.- Participants.

Accurate computation of the stable solitary wave for the FitzHugh-Nagumo equations

Comparisons are made between three different methods for computing the stable solitary wave solution for the FitzHugh-Nagumo equations which consist of a nonlinear diffusion equation coupled to an ordinary differential equation in time. They model the Hodgkin-Huxley equations which describe the propagation of the nerve impulse down the axon. Two of the methods involve the travelling wave equations. Previous accurate numerical computations of these equations as an initial-value problem using a shooting method lead to inaccurate values for the wave speed; however, nonlinear corrections to the initial values are shown to yield accurate values. A boundary-value method applies “asymptotic boundary conditions” and uses a spline-collocation code called COLSYS for numerical solution of boundary-value problems which leads to accurate wave profiles and speeds. The third method is to solve an initial-boundary-value problem with an adaptive outgoing wave condition for the partial differential equations where the solitary wave emerges as the stable long time solution. The concept of a “wave integral” is introduced and they are derived to determine the wave speed used in the adaptive boundary condition and to measure the closeness of the computed solutions to the exact solitary wave solution.

Singular Perturbation Analysis of Boundary Value Problems for Differential-Difference Equations III. Turning Point Problems

This paper continues a study of a class of boundary-value problems for linear second-order differential-difference equations in which the second-order derivative is multiplied by a small parameter (SIAM J. Appl. Math., 42 (1982), pp. 502–531; 45 (1985), pp. 687–707). The previous papers focused on problems involving boundary and interior layer phenomena, rapid oscillations, and resonance behavior. The problems studied here have solutions which exhibit turning point behavior, i.e., transition regions between rapid oscillations and exponential behavior. The presence of the shift terms can induce large amplitudes and multiphase behavior over parts of the interval. A combination of exact solutions, singular perturbation methods, and numerical computations are used in these studies.

Application of a Nonlinear WKB Method to the Korteweg–DeVries Equation

The WKB method used in quantum mechanics for solving linear second order ordinary differential equations is generalized to apply to nonlinear partial differential equations. In particular, this nonlinear WKB method, which is similar to the averaging method due to Whitham, is used to study nearly-periodic solutions of the Korteweg–deVries equation when the dispersion parameter is small. The emphasis of this paper is on a detailed analysis of the leading-order problem arising from the application of the nonlinear WKB method. An explicit representation of the leading-order solution is obtained in terms of unknown functions whose qualitative properties are studied. These unknown functions are governed by a first order system of nonlinear partial differential equations which is of hyperbolic type.