(b) Write out the equations for the time dependence of ρ11, ρ22, ρ12 and ρ21 assuming that a light field interaction V = -μ12Ee iωt + c.c. couples only levels |1> and |2>, and that the excited levels exhibit spontaneous decay. (8 marks) (c) Under steady-state conditions, find the ratio of populations in states |2> and |3>. (3 marks) (d) Find the slowly varying amplitude ̃ ρ 12 of the polarization ρ12 = ̃ ρ 12e iωt . (6 marks) (e) In the limiting case that no decay is possible from intermediate level |3>, what is the ground state population ρ11(∞)? (2 marks) 2. (15 marks total) In a 2-level atom system subjected to a strong field, dressed states are created in the form |D1(n)> = sin θ |1,n> + cos θ |2,n-1> |D2(n)> = cos θ |1,n> sin θ |2,n-1>

The Quantum Theory of Light

Extended tanh-function method and its applications to nonlinear equations

The potential for sea ice-albedo feedback to give rise to nonlinear climate change in the Arctic Ocean – defined as a nonlinear relationship between polar and global temperature change or, equivalently, a time-varying polar amplification – is explored in IPCC AR4 climate models. Five models supplying SRES A1B ensembles for the 21 st century are examined and very linear relationships are found between polar and global temperatures (indicating linear Arctic Ocean climate change), and between polar temperature and albedo (the potential source of nonlinearity). Two of the climate models have Arctic Ocean simulations that become annually sea ice-free under the stronger CO 2 increase to quadrupling forcing. Both of these runs show increases in polar amplification at polar temperatures above-5 o C and one exhibits heat budget changes that are consistent with the small ice cap instability of simple energy balance models. Both models show linear warming up to a polar temperature of-5 o C, well above the disappearance of their September ice covers at about-9 o C. Below-5 o C, surface albedo decreases smoothly as reductions move, progressively, to earlier parts of the sunlit period. Atmospheric heat transport exerts a strong cooling effect during the transition to annually ice-free conditions. Specialized experiments with atmosphere and coupled models show that the main damping mechanism for sea ice region surface temperature is reduced upward heat flux through the adjacent ice-free oceans resulting in reduced atmospheric heat transport into the region.

Geophysical fluid dynamics.

This article offers a comprehensive survey of results obtained for solitons and complex nonlinear wave patterns supported by nonlinear lattices (NLs), which represent a spatially periodic modulation of the local strength and sign of the nonlinearity, and their combinations with linear lattices. A majority of the results obtained, thus far, in this field and reviewed in this article are theoretical. Nevertheless, relevant experimental settings are also surveyed, with emphasis on perspectives for implementation of the theoretical predictions in the experiment. Physical systems discussed in the review belong to the realms of nonlinear optics (including artificial optical media, such as photonic crystals, and plasmonics) and Bose-Einstein condensation. The solitons are considered in one, two, and three dimensions. Basic properties of the solitons presented in the review are their existence, stability, and mobility. Although the field is still far from completion, general conclusions can be drawn. In particular, a novel fundamental property of one-dimensional solitons, which does not occur in the absence of NLs, is a finite threshold value of the soliton norm, necessary for their existence. In multidimensional settings, the stability of solitons supported by the spatial modulation of the nonlinearity is a truly challenging problem, for theoretical and experimental studies alike. In both the one-dimensional and two-dimensional cases, the mechanism that creates solitons in NLs in principle is different from its counterpart in linear lattices, as the solitons are created directly, rather than bifurcating from Bloch modes of linear lattices.

Solitons in nonlinear lattices

The main purpos e of this chapter is to present a direct and systematic way of finding exact solutions and Backlund transformations of a certain class of nonlinear evolution equations. The nonlinear evolution equations are transformed, by changing the dependent variable(s), into bilinear differential equations of the following special form 
 
$$ F\left( {\frac{\partial }{{\partial t}} - \frac{\partial }{{\partial {t^1}}},\frac{\partial }{{\partial x}} - \frac{\partial }{{\partial {x^1}}}} \right)f(t,x)f({t^1},{x^1}){|_{t = {t^1},x = {x^1}}} = 0 $$ 
 
, which we solve exactly using a kind of perturbational approach.

Direct Methods in Soliton Theory (非線形現象の取扱いとその物理的課題に関する研究会報告)

Starting from a known Lax pair, one can get some infinitely many coupled Lax pairs, infinitely many nonlocal symmetries and infinitely many new integrable models in some different ways. In this paper, taking the well known Kadomtsev–Petviashvili (KP) equation as a special example, we show that infinitely many nonhomogeneous linear Lax pairs can be obtained by using infinitely many symmetries, differentiating the spectral functions with respect to the inner parameters. Using a known Lax pair and the Darboux transformations (DT), infinitely many nonhomogeneous nonlinear Lax pairs can also be obtained. By means of the infinitely many Lax pairs, DT and the conformal invariance of the Schwartz form of the KP equation, infinitely many new nonlocal symmetries can be obtained naturally. Infinitely many integrable models in (1+1)-dimensions, (2+1)-dimensions, (3+1)-dimensions and even in higher dimensions can be obtained by virtue of symmetry constraints of the KP equation related to the infinitely many Lax pairs.

Infinitely many Lax pairs and symmetry constraints of the KP equation

In nonlinear science, it is very difficult to find exact interaction solutions among solitons and other kinds of complicated waves such as cnoidal waves and Painlevwaves. Actually, even if for the most well-known prototypical models such as the Kortewet-de Vries (KdV) equation and the Kadomtsev-Petviashvili (KP) equation, this kind of problem has not yet been solved. In this paper, the explicit analytic interaction solutions betweensolitarywavesandcnoidalwavesareobtainedthroughthelocalizationprocedureofnonlocalsymmetries which are related to Darboux transformation for the well-known KdV equation. The same approach also yields some other types of interaction solutions among different types of solutions such as solitary waves, rational solutions, Bessel function solutions, and/or general Painlev´ e II solutions.

http://faculty.ecnu.edu.cn/picture/article/202/f0/c8/84fed854482b8df4528d33705ee8/df073856-4c9b-4d32-a8bf-2cbe5fd3b6ff.pdf.x

Explicit solutions from eigenfunction symmetry of the Korteweg-de Vries equation.

A note on the new similarity reductions of the Boussinesq equation

A consistent Riccati expansion (CRE) is proposed for solving nonlinear systems with the help of a Riccati equation. A system having a CRE is then defined to be CRE solvable. The CRE solvability is demonstrated quite universal for various integrable systems including the Korteweg–de Vries, Kadomtsev–Petviashvili, Ablowitz–Kaup–Newell–Segur (and then nonlinear Schrodinger), sine-Gordon, Sawada–Kotera, Kaup–Kupershmidt, modified asymmetric Nizhnik–Novikov–Veselov, Broer–Kaup, dispersive water wave, and Burgers systems. In addition, it is revealed that many CRE solvable systems share a similar determining equation describing the interactions between a soliton and a cnoidal wave. They have a common nonlocal symmetry expression and they possess a formally universal once Backlund transformation.

Sen-Yue Lou

Papers

Infinitely many Lax pairs and symmetry constraints of the KP equation

Explicit solutions from eigenfunction symmetry of the Korteweg-de Vries equation.

A note on the new similarity reductions of the Boussinesq equation

Consistent Riccati Expansion for Integrable Systems

Solutions of Nonlocal Equations Reduced from the AKNS Hierarchy