The nonlinear terms, which depend on the nature of the nonlinearity, cause plane waves to focus as the amplitude varies across the wavefront. Equations describing the evolution of the amplitude, frequency and wavenumber are generated using mean Lagrangian techniques. The Lagrangian and associated variational principle will be the most important tool in the analysis of the non-linear phenomena.
The special cases of linear and circular polarization are discussed and various special forms of the non-linear terms are studied in detail. The characteristics of the general two-dimensional and three-dimensional radially symmetric cases are discussed, and the nearly linear problem is used for illustration. The shape of the power series expansion of U(R) is crucial in determining the effects that occur.
1] indicate some of the physical effects responsible for the nonlinear terms and list the quantities. The substitution of J in Maxwell's equations 1 yields the terms that normally arise from the constitutive properties of the medium.
CHAPTER II
Substituting (II. 7) into (H. 5), we examine the form of the functions to illustrate the periodic characteristics of the exact solutions. The nearly linear equations are found by putting potential (II. 7) ir::.to the equations (II.1}. Putting the first two remaining terms of each series in (II.9} and coefficients of sentence 9 and equating sense 39 results in the equations .. II.10) is the almost linear distribution relation that must be the.
Expansion of this complete elliptic integral for small '(in the san1e. II.10) again as expected. These integrated equations are placed in the third and fourth equations of (II.15), eliminating A. We now examine the special case of potential (II.?) which leads to a cubic central force:.
In the case of circular polarization, we use the special form of V(P) in (II. 24) to obtain the dispersion relation. Compared with (II. 1 0), the nearly linear dispersion relation for linear polarization, we see that the dispersion relation for circular polarization differs only by a nwneric factor in the amplitude-dependent term.
CHAPTER III
Equations like (IV.l2) can be transformed into linear equations by the hodograph transformation in which a and {3 are treated as functions of p and e. Since the set (IV. 12) is homogeneous in derivatives, the Jacobian cancels out by leaving the equations. The second of {IV. 25) has sinusoidal solutions so that elementary solutions for the separated factors of X take the form.
From the first equation, however, we see that the energy density inside the beam tubes decreases as the inverse distance from the beam axis. Similarly, when the beam is directed away from the beam axis, the amplitude decreases disproportionately relative to the two-dimensional case. Solutions to the problem are sought in such a form that the derivatives ¢ are small corrections tc of the mean wave number K for the wave propagating in the x direction.
The set (IV.29) and (IV.30} are the thin beam equations that have been analyzed by Russian researchers [10]. The essential difference between this method and the one described in Section IV.2 using beam coordinates is seen from ( IV.28) nearly constant, but the direction was allowed to vary appreciably. Here, in the case of the thin beam, the derivatives of cf> are all small; the beams are nearly parallel, and the phase velocity does not vary appreciably over all space.
This group is familiar in the context of shallow water theory, but differs in the sign of T. In contrast, the split solution for thicker beams (IV.l8) is with finite Vlidth being forced to zero by a function fold in ray coordinates. space. When and p are placed in equations (IV.33), equating similar powers of r produces the following relations:. finally f0 are defined in terms of £1 • We assign the following initial conditions to tra.:.
This integral is now explicitly evaluated for m = 0 and m = 1 and. one must consider the sign of C. for any sign of C) and the application of the boundary condition and rearrangement results. which can be used to recover p and cf> in closed form. In the first case for thin beams, the amplitude was finite at the focal point, but became .. complex beyond the focal point. rays, since the conservation of energy inside the ray tubes is only approximate.). For R > 0, the rays are divergent at x = 0. • As a result, the sign of x changes in the already found solutions.
CHAPTER V
The periodic forms (III. 2) are modified by an additional small term that does not appear in the periodic plane wave solutions. We are then left with a Lagrangian in terms of a, K and w that is used in the normal way to generate the dispersion relation and wave action. It is important to keep in mind that the solutions found in Chapter II are the fast oscillations, while here we are dealing with variations in the envelope of these oscillations.
The integral (V. 8) diverges when a goes to the value of a double or higher root, so all the roots are extreme values, and multiple roots can be reached only in the limit as y -• ± ao and these solutions are not oscillatory. In the present context, the envelope of a solitary wave is a very trivial example of a localized beam. There is an oscillatory solution in the defocusing medium in case {e), which also represents parallel rays.
In heuristic terms, a be3.m in the foc-1.1sing medium tends to focus due to non-linearity, but to go out of focus due to dispersion. We now substitute the shape of a circularly polarized plane wave propagating in the x-direction to this Lagrangian. As in the two-dimensional case, the shapes from Chapter III are modified by additional small terms.
In the same way as for linear polarization, we observe the form of the Lagrangian to simplify the process of extracting extraneous amplitudes. First, we consider the solutions of linearly polarized wave equations (V. 4) and (V. 5), which are independent of y and z . but they vary as a function of time in the x direction. We use the same definition of T] in the equations of circularly polarized waves, (V. 14) and (V. 15), and continue in the same way.
If the solution is localized in the radial variable, we have a pulse that propagates with. We make the substitution Tl = x - Vt, where V is the same magnitude as in V. ii, in the equations for circular polarization. 13] have shown through phase plane arguments that this equation has analytic, symmetric eigensolutions that are assyrr1ptotic to zero at infinity and have one, two, three, etc., zero crossings as in the case of solutions to (V. 17) .
This method quickly produces integrals of the Euler equations so that certain variables in the Lagrangian can be eliminated. In the terms we have called "boundary terms 11", there is always a factor ff from the integrand of (A. 6).
