There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

This book describes the remarkable connections that exist between the classical differential geometry of surfaces and modern soliton theory. The authors also explore the extensive body of literature from the nineteenth and early twentieth centuries by such eminent geometers as Bianchi, Darboux, Backlund, and Eisenhart on transformations of privileged classes of surfaces which leave key geometric properties unchanged. Prominent amongst these are Backlund-Darboux transformations with their remarkable associated nonlinear superposition principles and importance in soliton theory. It is with these transformations and the links they afford between the classical differential geometry of surfaces and the nonlinear equations of soliton theory that the present text is concerned. In this geometric context, solitonic equations arise out of the Gaus-Mainardi-Codazzi equations for various types of surfaces that admit invariance under Backlund-Darboux transformations. This text is appropriate for use at a higher undergraduate or graduate level for applied mathematicians or mathematical physics.

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Bäcklund and Darboux transformations : geometry and modern applications in soliton theory

SOME NOTATIONS AND REMARKS PARABOLIC EQUATIONS WITH ONE SPACE VARIABLE Equations with Power-Law Nonlinearities Equations with Exponential Nonlinearities Equations with Hyperbolic Nonlinearities Equations with Logarithmic Nonlinearities Equations with Trigonometric Nonlinearities Equations Involving Arbitrary Functions Nonlinear Schrodinger Equations and Related Equations PARABOLIC EQUATIONS WITH TWO OR MORE SPACE VARIABLES Equations with Two Space Variables Involving Power-Law Nonlinearities Equations with Two Space Variables Involving Exponential Nonlinearities Other Equations with Two Space Variables Involving Arbitrary Parameters Equations Involving Arbitrary Functions Equations with Three or More Space Variables Nonlinear Schrodinger Equations HYPERBOLIC EQUATIONS WITH ONE SPACE VARIABLE Equations with Power-Law Nonlinearities Equations with Exponential Nonlinearities Other Equations Involving Arbitrary Parameters Equations Involving Arbitrary Functions Equations of the Form wxy=F(x,y,w, wx, wy ) HYPERBOLIC EQUATIONS WITH TWO OR THREE SPACE VARIABLES Equations with Two Space Variables Involving Power-Law Nonlinearities Equations with Two Space Variables Involving Exponential Nonlinearities Nonlinear Telegraph Equations with Two Space Variables Equations with Two Space Variables Involving Arbitrary Functions Equations with Three Space Variables Involving Arbitrary Parameters Equations with Three Space Variables Involving Arbitrary Functions ELLIPTIC EQUATIONS WITH TWO SPACE VARIABLES Equations with Power-Law Nonlinearities Equations with Exponential Nonlinearities Equations Involving Other Nonlinearities Equations Involving Arbitrary Functions ELLIPTIC EQUATIONS WITH THREE OR MORE SPACE VARIABLES Equations with Three Space Variables Involving Power-Law Nonlinearities Equations with Three Space Variables Involving Exponential Nonlinearities Three-Dimensional Equations Involving Arbitrary Functions Equations with n Independent Variables EQUATIONS INVOLVING MIXED DERIVATIVES AND SOME OTHER EQUATIONS Equations Linear in the Mixed Derivative Equations Quadratic in the Highest Derivatives Bellman Type Equations and Related Equations SECOND-ORDER EQUATIONS OF GENERAL FORM Equations Involving the First Derivative in t Equations Involving Two or More Second Derivatives THIRD-ORDER EQUATIONS Equations Involving the First Derivative in t Equations Involving the Second Derivative in t Hydrodynamic Boundary Layer Equations Equations of Motion of Ideal Fluid (Euler Equations) Other Third-Order Nonlinear Equations FOURTH-ORDER EQUATIONS Equations Involving the First Derivative in t Equations Involving the Second Derivative in t Equations Involving Mixed Derivatives EQUATIONS OF HIGHER ORDERS Equations Involving the First Derivative in t and Linear in the Highest Derivative General Form Equations Involving the First Derivative in t Equations Involving the Second Derivative in t Other Equations SUPPLEMENTS: EXACT METHODS FOR SOLVING NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS Classification of Second-Order Semilinear Partial Differential Equations in Two Independent Variables Transformations of Equations of Mathematical Physics Traveling-Wave Solutions and Self-Similar Solutions. Similarity Methods Method of Generalized Separation of Variables Method of Functional Separation of Variables Generalized Similarity Reductions of Nonlinear Equations Group Analysis Methods Differential Constraints Method Painleve Test for Nonlinear Equations of Mathematical Physics Inverse Scattering Method Conservation Laws Hyperbolic Systems of Quasilinear Equations REFERENCES INDEX

Handbook of Nonlinear Partial Differential Equations

The theory of pseudo differential operators, discussed in § 1, is well suited for investigating various problems connected with elliptic differential equations. However, this theory fails to be adequate for studying equations of hyperbolic type, and one is then forced to examine a wider class of operators, the so-called Fourier integral operators (Egorov [1975], Hormander [1968, 1971, 1983, 1985], Kumano-go [1982], Shubin [1978], Taylor [1981], Treves [1980]).

Fourier Integral Operators

Reviews - Eigenfunction Expansions Associated with Second-Order Differential Equations. By E. C. Titchmarsh Pp. 175. 20s. 1946. (Oxford University)

It is observed that the Camassa–Holm equation describes pseudo-spherical surfaces and that therefore, its integrability properties can be studied by geometrical means. An sl(2, R)-valued linear problem whose integrability condition is the Camassa–Holm equation is presented, a ‘Miura transform’ and a ‘modified Camassa–Holm equation’ are introduced, and conservation laws for the Camassa–Holm equation are then directly constructed. Finally, it is pointed out that this equation possesses a nonlocal symmetry, and its flow is explicitly computed.

Geometric Integrability of the Camassa–Holm Equation

Computational and geometric aspects of nonlocal (infinitesimal) symmetries of nonlinear partial differential equations are considered. In particular, the relation of nonlocal symmetries with classical, generalized and internal symmetries is briefly discussed. A nonlocal symmetry for the Kaup–Kupershmidt equation is introduced and studied in some detail. Some explicit particular solutions are found with its help, and a Darboux-like transformation is also obtained.

Nonlocal symmetries and the Kaup–Kupershmidt equation

We study the generalized bosonic string equation
 $$ \Delta e^{-c\,\Delta}\phi = U(x,\phi), \quad c > 0$$
 on Euclidean space $${\mathbb {R}^n}$$
 . First, we interpret the nonlocal operator $${\Delta e^{-c\,\Delta}}$$
 using entire vectors of Δ in $${L^2(\mathbb{R}^n)}$$
 , and we show that if $${U(x, \phi) = \phi(x) + f(x)}$$
 , in which $${f \in L^2(\mathbb {R}^n)}$$
 , then there exists a unique real-analytic solution to the Euclidean bosonic string in a Hilbert space $${{\mathcal H}^{c,\infty}(\mathbb {R}^n)}$$
 we define precisely below. Second, we consider the case in which the potential $${U(x, \phi)}$$
 in the generalized bosonic string equation depends nonlinearly on $${\phi}$$
 , and we show that this equation admits real-analytic solutions in $${{\mathcal H}^{c,\infty}(\mathbb R^n)}$$
 under some symmetry and growth assumptions on U. Finally, we show that the above given equation admits real-analytic solutions in $${{\mathcal H}^{c,\infty}(\mathbb {R}^n)}$$
 if $${U(x, \phi)}$$
 is suitably near $${U_0(x, \phi) = \phi}$$
 , even if no symmetry assumptions are imposed.

Nonlinear Equations with Infinitely many Derivatives

Some relationships between local differential geometry of surfaces and integrability of evolutionary partial differential equations are studied. It is proven that every second order formally integrable equation describes pseudo-spherical surfaces. A classification of integrable equations of Boussinesq type is presented, and it is shown that they can be interpreted geometrically as “equations describing hyperbolic affine surfaces”

On formal integrability of evolution equations and local geometry of surfaces

The Camassa–Holm (CH) equation describes pseudo-spherical surfaces and therefore its integrability properties can be studied by geometrical means [Reyes, “Geometric integrability of the Camassa–Holm equation.” Letters in Mathematical Physics 59 (2002): 117–31]. Using this fact, we introduce a “Miura transform” and a “modified” Camassa–Holm (mCH) equation, in analogy with the Korteweg–de Vries theory. We obtain an infinite number of local conservation laws for CH from these data and also an infinite number of (non)local symmetries. We then compute conservation laws for mCH and also show that it describes pseudo-spherical surfaces, so that, in particular, it is the integrability condition of an -valued linear problem. Finally, we investigate mCH analytically: we define weak solutions and prove their existence and uniqueness.

Enrique G. Reyes

Papers

Geometric Integrability of the Camassa–Holm Equation

Nonlocal symmetries and the Kaup–Kupershmidt equation

Nonlinear Equations with Infinitely many Derivatives

On formal integrability of evolution equations and local geometry of surfaces

The Modified Camassa–Holm Equation