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Korteweg-de Vries equation.

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"Solitons are nonlinear waves which behave like interacting particles. When first proposed in the 19th century, leading mathematical physicists denied that such a thing could exist. Now they are regularly observed in nature, shedding light on phenomena like rogue waves and DNA transcription. Solitons of light are even used by engineers for data transmission and optical switches. Furthermore, unlike most nonlinear partial differential equations, soliton equations have the remarkable property of being exactly solvable. Explicit solutions to those equations provide a rare window into what is possible in the realm of nonlinearity. Glimpses of Soliton Theory reveals the hidden connections discovered over the last half-century that explain the existence of these mysterious mathematical objects. It aims to convince the reader that, like the mirrors and hidden pockets used by magicians, the underlying algebro-geometric structure of soliton equations provides an elegant explanation of something seemingly miraculous. Assuming only multivariable calculus and linear algebra, the book introduces the reader to the KdV Equation and its multisoliton solutions, elliptic curves and Weierstrass wp-functions, the algebra of differential operators, Lax Pairs and their use in discovering other soliton equations, wedge products and decomposability, the KP Hierarchy, and Sato's theory relating the Bilinear KP Equation to the geometry of Grassmannians. Notable features of the book include: careful selection of topics and detailed explanations to make the subject accessible to undergraduates, numerous worked examples and thought-provoking exercises, footnotes and lists of suggested readings to guide the interested reader to more information, and use of Mathematica® to facilitate computation and animate solutions. The second edition refines the exposition in every chapter, adds more homework exercises and projects, updates references, and includes new examples involving non-commutative integrable systems. Moreover, the chapter on KdV multisolitons has been greatly expanded with new theorems providing a thorough analysis of their behavior and decomposition."--Provided by publisher.

Korteweg-de Vries equation. --- Differential equations, Partial. --- Geometry, Algebraic.

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Mathematical physics --- Solitons --- Korteweg-de Vries equation --- Solitons. --- Korteweg-de Vries, Équation de. --- Pulses, Solitary wave --- Solitary wave pulses --- Wave pulses, Solitary --- Connections (Mathematics) --- Nonlinear theories --- Wave-motion, Theory of --- KdV equation --- Differential equations, Nonlinear

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Differential equations, Partial --- Hamiltonian systems --- Mathematical physics --- Solitons --- Equations aux dérivées partielles --- Systèmes hamiltoniens --- Physique mathématique --- Equations aux dérivées partielles --- Systèmes hamiltoniens --- Physique mathématique --- Solitons. --- Korteweg-de Vries equation --- Korteweg-de Vries, Équation de --- Differential equations, Nonlinear --- Équations différentielles non linéaires --- Lie groups --- Lie, Groupes de --- Lie algebras --- Lie, Algèbres de

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Mathematical physics --- Differential equations, Partial --- Solitons --- Scattering (Mathematics) --- Equations aux dérivées partielles --- Dispersion (Mathématiques) --- Numerical solutions --- Solutions numériques --- 517.93 --- -Scattering (Mathematics) --- Pulses, Solitary wave --- Solitary wave pulses --- Wave pulses, Solitary --- Connections (Mathematics) --- Nonlinear theories --- Wave-motion, Theory of --- Scattering theory (Mathematics) --- Boundary value problems --- Scattering operator --- Partial differential equations --- Special differential equations. Systems of analytic mechanics, automatic control, operators. Dynamic systems --- Solitons. --- Numerical solutions. --- Scattering (Mathematics). --- 517.93 Special differential equations. Systems of analytic mechanics, automatic control, operators. Dynamic systems --- Equations aux dérivées partielles --- Dispersion (Mathématiques) --- Solutions numériques --- Numerical analysis --- Korteweg-de Vries equation --- Korteweg-de Vries, Équation de --- Korteweg-de Vries equation. --- Korteweg-de Vries, Équation de --- Dynamique différentiable --- Differential equations, Partial - Numerical solutions --- Equations aux derivees partielles --- Scattering