nonlinear equations
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1: Mark J. Ablowitz
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►Ablowitz is an applied mathematician who is interested in solutions of nonlinear wave equations.
Certain nonlinear equations are special; e.
…Some of the relationships between IST and Painlevé equations are discussed in two books: Solitons and the Inverse Scattering Transform and Solitons, Nonlinear Evolution Equations and Inverse Scattering.
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2: 3.8 Nonlinear Equations
§3.8 Nonlinear Equations
… ► … ►Corresponding numerical factors in this example for other zeros and other values of are obtained in Gautschi (1984, §4). ►§3.8(vii) Systems of Nonlinear Equations
►For fixed-point iterations and Newton’s method for solving systems of nonlinear equations, see Gautschi (1997a, Chapter 4, §9) and Ortega and Rheinboldt (1970). …3: 23.21 Physical Applications
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§23.21(ii) Nonlinear Evolution Equations
►Airault et al. (1977) applies the function to an integrable classical many-body problem, and relates the solutions to nonlinear partial differential equations. …4: Sidebar 22.SB1: Decay of a Soliton in a Bose–Einstein Condensate
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►Jacobian elliptic functions arise as solutions to certain nonlinear Schrödinger equations, which model many types of wave propagation phenomena.
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5: 21.9 Integrable Equations
6: Peter A. Clarkson
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►His well-known book Solitons, Nonlinear Evolution Equations and Inverse Scattering (with M.
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7: Bernard Deconinck
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►He has worked on integrable systems, algorithms for computations with Riemann surfaces, Bose-Einstein condensates, and methods to investigate the stability of solutions of nonlinear wave equations.
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8: 29.19 Physical Applications
9: 22.19 Physical Applications
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§22.19(iii) Nonlinear ODEs and PDEs
►Many nonlinear ordinary and partial differential equations have solutions that may be expressed in terms of Jacobian elliptic functions. These include the time dependent, and time independent, nonlinear Schrödinger equations (NLSE) (Drazin and Johnson (1993, Chapter 2), Ablowitz and Clarkson (1991, pp. 42, 99)), the Korteweg–de Vries (KdV) equation (Kruskal (1974), Li and Olver (2000)), the sine-Gordon equation, and others; see Drazin and Johnson (1993, Chapter 2) for an overview. …10: 9.16 Physical Applications
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►Airy functions play a prominent role in problems defined by nonlinear wave equations.
These first appeared in connection with the equation governing the evolution of long shallow water waves of permanent form, generally called solitons, and are predicted by the Korteweg–de Vries (KdV) equation (a third-order nonlinear partial differential equation).
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