Nonlinear fiber optics concerns with the nonlinear optical phenomena occurring inside optical fibers. Although the field ofnonlinear optics traces its beginning to 1961, when a ruby laser was first used to generate the second-harmonic radiation inside a crystal [1], the use ofoptical fibers as a nonlinear medium became feasible only after 1970 when fiber losses were reduced to below 20 dB/km [2]. Stimulated Raman and Brillouin scatterings in single-mode fibers were studied as early as 1972 [3] and were soon followed by the study of other nonlinear effects such as self- and crossphase modulation and four-wave mixing [4]. By 1989, the field ofnonlinear fiber optics has advanced enough that a whole book was devoted to it [5]. This book or its second edition has been translated into Chinese, Japanese, and Russian languages, attesting to the worldwide activity in the field of nonlinear fiber optics.

/pdf/nonlinear-fiber-optics-4oa9a2uvam.pdf

Nonlinear Fiber Optics

This essential reference for graduate students and researchers provides a unified treatment of earthquakes and faulting as two aspects of brittle tectonics at different timescales. The intimate connection between the two is manifested in their scaling laws and populations, which evolve from fracture growth and interactions between fractures. The connection between faults and the seismicity generated is governed by the rate and state dependent friction laws - producing distinctive seismic styles of faulting and a gamut of earthquake phenomena including aftershocks, afterslip, earthquake triggering, and slow slip events. The third edition of this classic treatise presents a wealth of new topics and new observations. These include slow earthquake phenomena; friction of phyllosilicates, and at high sliding velocities; fault structures; relative roles of strong and seismogenic versus weak and creeping faults; dynamic triggering of earthquakes; oceanic earthquakes; megathrust earthquakes in subduction zones; deep earthquakes; and new observations of earthquake precursory phenomena.

The Mechanics of Earthquakes and Faulting

Symplectic structures, their Bäcklund transformations and hereditary symmetries

WAVE BREAKING FOR NONLINEAR NONLOCAL SHALLOW WATER EQUATIONS 233 THEOREM 2.1. Let T>O and vE C 1 ([0, T); H 2(R)). Then for every t~ [0, T) there exists at least one point ~(t)ER with ,~(t) := in~ Ivy(t, x)] = ~ ( t , ~(t)), and the function m is almost everywhere differentiable on (0, T) with dm dt (t)=vt~(t,~(t)) a.e. on (O,T). Proof. Let c>0 stand for a generic constant. Fix te[0, T) and define m(t):=infxcR[v~(t,x)]. If m(t))O we have that v(t , . ) is nondecreasing on R and therefore v(t,. ) 0 (recall v(t,. )cL2(R)) , so that we may assume re(t)<0. Since vx(t,. ) c H I ( R ) we see that limlxl~ ~ Vx(t, x)=0 so that there exists at least a ~(t) e R with re(t) =v~(t, ~(t)). Let now s, tC [0, T) be fixed. If re(t) <~rn(s) we have 0 < re(s) .~(t) = i n f [~x (~, x ) ] ~ ( t , ~(t)) <. ~=(~, ~(t)) -~x(t, ~(t)), and by the Sobolev embedding HI(R) C L ~ ( R ) we conclude that Im(8)-.~(t)l ~< Ivx(t)-v~(s)lL~(~) < c Iv~(t)-v~(8)l.l(R). Hence the mean-value theorem for functions with values in Banach spaces-Hi(R) in the present case--yields (see [12]) jm(t)-m(s)l<~clt-s j m a x [IVt~(T)JHI(R)], t, se[O,T). O~T~max{s,t} Since vt~cC([O,T), Hi(R)) , we see that m is locally Lipschitz on [0, T) and therefore Rademacher's theorem (cf. [14]) implies that m is almost everywhere differentiable on (0,T). Fix tC(0, T). We have that v~(t+h)-vx(t)h vt~(t) Hl(R) ---~0 as h--*O, and therefore vx( t+h ,y ) -vx ( t , y ) sup vtx(t,y) --~0 as h---~O, (2.1) ycl~ h in view of the continuous embedding H 1 ( R ) c L ~ (R). 234 A. C O N S T A N T I N AND J. E S C H E R By the definition of m, m(t+h) = v~(t+h, ((t+h)) <. v~(t+h, ((t)). Consequently, given h>0 , we obtain m( t+h) -m( t ) <<. h Letting h--~O + and using (2.1), we find lim sup m(t+h) -m( t ) h~_~0 + h

/pdf/wave-breaking-for-nonlinear-nonlocal-shallow-water-equations-2y25ekrd18.pdf

Wave breaking for nonlinear nonlocal shallow water equations

The survey contains a description of the connection between the infinite-dimensional Lie algebras of Kats-Moody and systems of differential equations generalizing the Korteweg-de Vries and sine-Gordon equations and integrable by the method of the inverse scattering problem. A survey of the theory of Kats-Moody algebras is also given.

Lie algebras and equations of Korteweg-de Vries type

A discussion of the theory and applications of classical solitons is presented with a brief treatment of quantum mechanical effects which occur in particle physics and quantum field theory. The subjects addressed include: solitary waves and solitons, scattering transforms, the Schroedinger equation and the Korteweg-de Vries equation, and the inverse method for the isospectral Schroedinger equation and the general solution of the solvable nonlinear equations. Also considered are: isolation of the Korteweg-de Vries equation in some physical examples, the Zakharov-Shabat/AKNS inverse method, kinks and the sine-Gordon equation, the nonlinear Schroedinger equation and wave resonance interactions, amplitude equations in unstable systems, and numerical studies of solitons. 45 references.

Solitons and Nonlinear Wave Equations

We have found new hierarchies of Korteweg–de Vries and Boussinesq equations which have multiple soliton solutions. In contrast to the stan­dard hierarchy of K. de V. equations found by Lax, these equations do not appear to fit the present inverse formalism or possess the various pro­perties associated with it such as Backlund transformations. The most interesting of the new K. de V. equations is ( u nx ≡ ∂ n u /∂ x n ) ( u 4 x + 30 uu 2 x + 60 u 3 ) x + u t = 0. We have proved that this equation has N -soliton solutions but we have been able to find only two soliton solutions for the rest of this hierarchy. The above equation has higher conservation laws of rank 3, 4, 6 and 7 but none of rank 2, 5 and 8 and hence it would seem that an unusual series of conservation laws exists with every third one missing. Apart from the Boussinesq equation itself, which has N -soliton solutions, ( u xx + 6 u 2 ) xx + u xx – u tt = 0 we have found only two-soliton solutions to the rest of this second class. The new equations have bounded oscillating solutions which do not occur for the K. de V. equation itself.

A New Hierarchy of Korteweg-De Vries Equations

Like a number of other nonlinear dispersive wave equations the sine–Gordonequation z , xt = sin z has both multi-soliton solutions and an infinity of conserved densities which are polynomials in z , x , z , xx , etc. We prove that the generalized sine–Gordon equation z , xt = F ( z ) has an infinity of such polynomial conserved densities if, and only if, F ( z ) = A e αz + B e – αz for complex valued A, B and α ≠ 0. If F ( z ) does not take the form A e αz + B e βz there is no p. c. d. of rank greater than two. If α ≠ – β there is only a finite number of p. c. ds. If α = – β then if A and B are non-zero all p. c. ds are of even rank; if either A or B vanishes the p. c. ds are of both even and odd ranks. We exhibit the first eleven p. c. ds in each case when α = – β and the first eight when α ≠ – β . Neither the odd rank p. c. ds in the case α = – β , nor the particular limited set of p. c. ds in the case when α ≠ – β have been reported before. We connect the existence of an infinity of p. c. ds with solutions of the equations through an inverse scattering method, with Backlund transformations and, via Noether’s theorem, with infinitesimal Backlund transformations. All equations with Backlund transformations have an infinity of p. c. ds but not all such p. c. ds can be generated from the Backlund transformations. We deduce that multiple sine–Gordon equations like z , xt = sin z + ½ sin ½ z , which have applications in the theory of short optical pulse propagation, do not have an infinity of p. c. ds. For these equations we find essentially three conservation laws: one and only one of these is a p. c. d. and this is of rank two. We conclude that the multiple sine–Gordons will not be soluble by present formulations of the inverse scattering method despite numerical solutions which show soliton like behaviour. Results and conclusions are wholly consistent with the theorem that the generalized sine–Gordon equation has auto-Backlund transformations if, and only if Ḟ ( z ) – α 2 F ( z ) = 0.

Polynomial conserved densities for the sine-Gordon equations

We have used Wahlquist & Estabrook's prolongation method to construct a Lie algebra for the higher order Korteweg de Vries (K. de V.) equation ($q\_{4x}$ + $\alpha qq\_{2x}$ + $\beta q\_{x}^{2}$ + $\gamma q^{3})\_{x}$ + $q$$\_{t}$ = 0 with ($q$$\_{nx}$ $\equiv $ $\equiv \_{x}^{n}q$). This has a single soliton solution when $\beta $ = 30 - $\alpha $; $\gamma $ = 2$\alpha $. The only closure for the algebra we have found fixes $\alpha $ at $\alpha $ = 20 or $\alpha $ = 30. These two values are the only ones for which this equation has been found to have $N$-soliton solutions. The case $\alpha $ = 20 gives the standard Lax higher order K. de V. equation. The case $\alpha $ = 30 produces the equation of Caudrey, Dodd & Gibbon (1976): 0 = $q$$\_{t}$ + ($q$$\_{4x}$ + 30$qq\_{2x}$ + 60$q^{3}$)$\_{x}$, for which no known inverse method has been found until now. By finding a realization for the algebra we have constructed a 3 $\times $ 3 eigenvalue problem for this latter case which can be trasformed into the Lax formulation: $\tilde{L}$$\_{t}$ = [$\tilde{A}$, $\tilde{L}$].

The prolongation structure of a higher order Korteweg-de Vries equation

The generalized sine-Gordon equations $z\_{,xt}=F(z)$ in two independent variables $x$, mi include the sine-Gordon $z\_{,xt}=$sin z and the multiple sine-Gordon's like $z\_{,xt}$ = sin $z+\frac{1}{2}$ sin $\frac{1}{2}z$. Among other physical applications all these sine-Gordon's are significant to the theory of intense ultra-short optical pulse propagation. The sine-Gordon itself has analytical multisoliton solutions. It also has an infinity of polynomial conserved densities and has auto-Backlund transformations which generate a second solution of the sine-Gordon from a first solution - particularly from the solution $z\equiv 0$. We prove first that the generalized multi-dimensional sine-Gordon in two or more space variables $x^{1},x^{2}$... has no auto-Backlund transformations. Next we prove that the generalized sine-Gordon's $z\_{,xt}=F(z)$ and $z\_{,xt}^{\prime}=G(z^{\prime})$ have an invertible Backlund transformation between solutions $z$ and $z^{\prime}$ if and only if $F$ and $G$ are solutions of $\ddot{F}=\alpha ^{2}F,\ddot{G}=\beta ^{2}G$ where, in general, $\beta =\alpha h^{-1},\alpha $ is a complex number and $h^{2}(\neq 0)$ is real. In case $h$ = 1 and $F$ and $G$ are the same function $z\_{,xt}=F(z)$ has an auto-Backlund transformation if and only if $\ddot{F}=\alpha ^{2}F$. We exhibit the B.ts and a.B.ts in these cases as well as the other B.ts for the generalized sine-Gordon. We conclude that the multiple sine-Gordon's do not have a.B.ts and infer that, despite the soliton character of the numerical solutions, the multiple sine-Gordon's are not soluble by present simplest formulations of the two by two inverse scattering method.

R. K. Dodd

Papers

Solitons and Nonlinear Wave Equations

A New Hierarchy of Korteweg-De Vries Equations

Polynomial conserved densities for the sine-Gordon equations

The prolongation structure of a higher order Korteweg-de Vries equation

Backlund Transformations for the Sine-Gordon Equations