Sine-Gordon-equation

The Hamiltonian, Eq. (25), can be minimized exactly. The ground state satisfies the time-independent sine Gordon equation ... 

One more important property of the self-dual Yang-Mills equations is that they are equivalent to the compatibility conditions of some overdetermined system of linear partial differential equations [11,12]. In other words, the selfdual Yang-Mills equations admit the Lax representation and, in this sense, are integrable. For this very reason it is possible to reduce Eq. (2) to the widely studied solitonic equations, such as the Euler-Amold, Burgers, and Devy-Stuardson equations [13,14] and Liouville and sine-Gordon equations [15] by use of the symmetry reduction method. 

The Aharonov-Bohm effect requires topological consideration [1], (i.e., a structured vacuum), and there exist conservation laws of topological origin, the simplest one is given by the sine-Gordon equation, which also appears in the discussion of 0(3) electrodynamics by Evans and Crowell [5]. 

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... 

The nonlinear wave equation thus obtained is the famous sine-Gordon equation, which is well known from soliton theory (see, for example, Dodd et al. [1982] and Rajaraman [1982]). The long wave approximation used to replace the discrete rotor angle < by continuous variable 4>(x, t)... 

That is, the simple classical picture we described applies to the quantum mechanical case provided the masses of solitons and breathers are changed. This remarkably simple result is due to the special characteristics of the sine-Gordon equation. The quantization reduces to factorization of the action to the classical action and a constant factor that is independent of the soliton velocity and the breather frequency. 

Replacing differences by differentials eq. (153) is reduced to the sine-Gordon-equation (Frank and Van der Merwe, 1949a, b),... 

Solitary wave can also be defined as the traveling wave whose transition from one constant asymptotic state as —oo to another as - -oo is localized in [6]. An example would be a solution of sine-Gordon equation... 

In the stationary regime, for the balance of the elastic and electric torques we have a sine-Gordon equation [16] ... 

Even within the single>elastic>constant approximation the Euler equations (2.13a) and (2.13b) are still very complicated. For r>ro the Euler equation (2.13b) is relatively simple but has a form of the Sine-Gordon equation for which no general solution exists except by numerical means. For a review on the Sine-Gordon equation, see, for example A. Barone, F. Esposito C. J. Magef and A. C. Scott, Nuovo Cimento 1, 227 (1971). 

INTERACTION OF A PAIR OF IMPURITIES AND KINKS IN THE SINE-GORDON EQUATION... 

The kink d5mamics of the sine-Gordon equation is studied in the model of the loealized spatial modulation of the periodic potential. A case of two identical areas (or impurities) of the spatial modulation of the periodic potential is considered. It is shown that observing the collective effects of impurity influence is possible and depends on the distance between the impurities. A definite critical value of impurity distances causing two quite different ways of the d5mamic kink behavior is demonstrated. The structure and properties of three-kink solutions of the sine-Gordon equation in the impurity area are studied. 

Spatial modulation of the periodic potential (or impurity) is of great interest as well [3, 6], The problem of the sine-Gordon equation kinks and impurity interaction for the one-dimensional case has been long discussed in literature [6-9], For example, the model of the classical particle for the kink and impurity interaction is applied in case the impurity is devoid of a mode as a localized vibrational state on the impurity [3], Importance of impurity modes in kink and impurity interaction is demonstrated in the work of Ref. [9], Much attention is drawn to multisoliton solutions of the sine-Gordon equation [10, 11],... 

Let us consider the modified sine-Gordon equation of the following type [6, 12]... 

There are multisoliton solutions of the sine-Gordon equation, for example, in Refs. [10, 11] an interesting three-kink solution of a wobble t5q)e is described ... 

We examine the case of the kink pinning on one of the impurities. Note that the presence of two impurity areas enables to find multisoliton solutions of the sine-Gordon equation. To be definite enough, we take W = 1, AST = 1.2, Xj = -7 and the distance d between the impurities may vary widely. In all cases hereinafter the initial kink velocity was selected so that the kink was pinned in the second impurity area. There are some difficulties in the numerical study of the task considered. As the interaction of the solitons excited may lead to oscillation mode appearance characterized by energy transfer from the kink to the breather and eonversely (similar to the beat regime for harmonic oscillators), the oscillation frequency may change in the eourse of time. So further, we resort to isotropic oscillations and stationary frequencies that are set overtime. 

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. 

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