# Solitons in mathematics and physics by Alan C. Newell

By Alan C. Newell

The soliton is a dramatic thought in nonlinear technological know-how. What makes this ebook special within the remedy of this topic is its specialise in the houses that make the soliton bodily ubiquitous and the soliton equation mathematically striking. right here, at the classical point, is the entity box theorists were postulating for years: an area touring wave pulse; a lump-like coherent constitution; the answer of a box equation with outstanding balance and particle-like homes. it's a basic mode of propagation in gravity- pushed floor and inner waves; in atmospheric waves; in ion acoustic and Langmuir waves in plasmas; in a few laser waves in nonlinear media; and in lots of biologic contexts, reminiscent of alpha- helix proteins

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81-94) and Newell (S1AM J. Appl. , 35 (1978), pp. 650-664), for further discussion of the importance of this resonance. One now rewrites the equation for the envelope a(£ = X —

Therefore the part of the mean field induced by the slow gradients in the envelope of the fast field can be solved for: Note in particular the possibility of resonance, namely, when the group velocity of the fast field equals the phase velocity of the long wave or mean field. The reader should consult the papers by Benney (Stud. Appl. , 55 (1976), pp. ; 56 (1977), pp. 81-94) and Newell (S1AM J. Appl. , 35 (1978), pp. 650-664), for further discussion of the importance of this resonance. One now rewrites the equation for the envelope a(£ = X —

We will meet both of these cases in the exercises and I will also point you to three concrete examples in physics where these effects are important. For now let us assume that the mean flow is not in the null space of L(0) as, for example, would be the case if L = d 2 /df 2 -c 2 d 2 /dx 2 + o>2. 48) is L(1)w0 and so we must choose the dependence of a on X and T! 42), L2 = +o>'L1 and so if Lx ^ 0 (which we assume) We then solve for u1? which contains second harmonics and perhaps a mean term proportional to aa*.