De Vries equation

More recently, Palais [17] showed that the generic cases of soliton—the Korteweg de Vries equation (KdV), the nonlinear Schrodinger equation (NLS), the sine-Gordon equation (SGE)—can be given an SU(2) formulation. In each of the three cases considered below, V is a one-dimensional space that is embedded in the space of off-diagonal complex matrices, ( ) and in each case L(u) at- I u, where u is a potential, i. is a complex parameter, and a is the constant, diagonal, trace zero matrix... 

This is known as the de Vries equation. The sign of the rotatory power reverses on crossing the reflexion band (A,). When A < A , (4.1.55) reduces to (4.1.21), and when Xp X, p tends asymptotically to 0. The behaviour on either side of the reflexion band has been confirmed experimentally (fig. 4.1.8). 

Only a few solvents are known to dissolve cellulose completely, and solid cellulose decomposes before melting. Therefore, it is difficult to study the mesophase behavior of cellulose. Chanzy et al. [32] reported lyotropic mesophases of cellulose in a mixture of jV-methyl-morpholine-Af-oxide and water (20-50%), but were unable to determine the nature of the mesophase. Lyotropic cholesteric mesophase formation in highly concentrated mixtures of cellulose in trifluoroa-cetic acid + chlorinated-alkane solvent [33] and in ammonia/ammonium thiocyanate solutions [34] has been studied, and although poor textures were obtained in the polarizing microscope, high optical rotatory power has been measured in an optical rotation (ORD) experiment, which could be fitted to the de Vries equation [Eq. (3)] for selective reflection beyond the visible wavelength region and was taken as proof of a lyotropic chiral nematic phase. 

Adler, M., and Moser J. On a class of polynomials connected with the Korteweg-de Vries equation. Comm, Math, Phys, (1978), 1-30. 

Airault, H., McKean, H. P., and Moser, J. Rational and elliptic solution of the Korteweg-de Vries equation and related many-body problem. Comm, Pure Appl, Math 30 (1977), 95-148. 

Berezin, F. A., Perelomov, A. M. "Group-theoretical interpretation of Korte-weg-de Vries equations. Funkts. Analiz x yego Prilozhen. 14 (1980), No. 2, 50-51. 

Gardner, C. S. "Korteweg-de Vries equation and generalization. IV "The Korteweg-de Vries equation as a Hamiltonian system. J. Math. Phys. 12... 

Zakharov, V. E., and Faddeev, L. D. "The Korteweg-de Vries equation, a completely integrable Hamiltonian system. Funkts. Analiz i yego Prilozhen. 5 (1971), No. 4, 18-27. 

Miles, J. W. (1976) Korteweg-de Vries equation modified by viscosity. Phys. Fluids 19 1063... 

Nekorkin, V. I., and Velarde, M. G. (1994) Solitary waves of a dissipative Korteweg-de Vries equation describing Marangoni-Benard convection and other thermoconveclivc instabilities. Int. J. Bifurc. Chaos 4 1135-1146. 

Rednikov, A. Ye., Velarde, M. G., Ryazantsev, Yu. S., Nepomnyashchy A. A., and Kurdyumov, V. (1995), Cnoidal wave trains and solitary waves in a dissipation-modified Korteweg-de Vries equation. Acta Appl. Math. 39 457-475. 

Contents Introduction. - The Korteweg-de Vries Equation (KdV-Equation). - The Inverse Scattering Transformation (1ST) as Illustrated with the KdV. - Inverse Scattering Theory for Other Evolution Equations. - The Classical Sine-Gordon Equation (SGE). - Statistical Mechanics of the Sine-Gordon System. - Difference Equations The Toda Lattice. - Appendix Mathematical Details. - References. -Subject Index. 

However the de Vries equation is not valid in the region of A , which corresponds to a total reflection of circularly polarized light having the same sense as the helical pitch. This is often referred to as Bragg reflection, by analogy with X-ray diffraction, but only first order reflections are allowed for nor-... 

A. JEFFREY T. KAKUTANI, "Weak nonlinear dispersive waves A Discussion centred around the Korteweg-de Vries equation", SIAM. Rev.,... 

C.S. GARDNER, J.M. GREEN, M.D. KRUSKAL R.M. MIURA, "Korteweg-de Vries equation and generalizations. Part VI Methods of exact... 

The dynamics of a lattice with a net of oscillators (Toda [5]) connected by elastic springs Is considered often In the modern theory of nonlinear waves. The analysis gives the Korteweg - de Vries equation, and viscosity is taken into account with the Introduction of the Burgers term. 

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