In study of solitons, one can try a WKB-style approximation to nonlinear wave equation (cf. eikonal equation, Maslov index etc.). Stokes has observed that when trying to connect the local solutions, one has discontinuities along certain lines, now called Stokes lines. This is called the Stokes phenomenon. Similar issues appear in study of isomonodromic deformation?s of nonlinear ODE-s in complex plane, what is also relevant in soliton theory, and integrable systems, and special functions like Painlevé transcendents. This has especially much been studied by Kyoto school (Jimbo, Miwa, Sato, Kashiwara etc.), including using D-modules and microlocal analysis. Kyoto school found a connection of isomonodromic theory to the so-called holonomic quantum fields.
The solutions of meromorphic differential equations can be expressed in terms of meromorphic connections. Then the slopes related to the solutions can be viewed as features of particular objects in a category of $D$-modules. More generally, slope filtrations are structures which appear in many other additive categories, e.g. in Hodge theory, theory of Dieudonné modules and so on. Many of those are related to the stability of the objects, which is important in the construction of moduli spaces.
In algebraic geometry, Grothendieck has shown how to correctly define and construct some fundamental moduli spaces, like Hilbert schemes and Quot schemes for coherent sheaves. The work has been continued by David Mumford who geometrized classical invariant theory into geometric invariant theory. To keep moduli under control, one needs to impose stability conditions on objects and also look at classes with some fixed data: those involve slopes or equivalently phase factors. This is thus similar to the phases of eikonal in the case of Stokes phenomenon. Cf. also Harder-Narasimhan filtration, Castelnuovo-Mumford regularity? (cf. wikipedia) etc.
In some physical situations solutions can be quite constrained by some special symmetry or representation theoretic conditions. One example are BPS states, which can be often counted or measured in some way. Another example are the moduli spaces of Higgs bundles, studied by Carlos Simpson and others, which have special cases with interpretations both in geometry and in the gauge theory (instantons). It appears that sometimes they can be linked to the geometric picture. Riemann-Hilbert correspondence, spectral transform and similar correspondences again play a major role.
Surely, one often works at the derived level. An adaptation of the notion of stability into the setup of triangulated categories has been introduced by Bridgeland. Bridgeland stability for the derived categories of (boundary conditions of) D-branes (B-model) are relevant for string theory.
We should also mention the related wall crossing in representation theory. Symmetries related to Weyl groups, Weyl chambers and chamber walls are involved (what is sometimes also in BPS setup above, see the paper by Cheng and Verlinde below). A priori wall crossing functors in representation theory (introduced in 1970s by Russian school, Gelfand, Bernstein etc.) are about certain functors which in take as input an infinite-dimensional representation, tensor it with finite-dimensional and look for certain pieces in the decomposition, where the business of chamber walls is crucial. Cf. A. Beilinson, V. Ginzburg, Wall-crossing functors and -modules, Representation Theory 3 (electronic), 1–31 (1999) pdf.
(future) Kontsevich in Aarhus, August 2010, master class on wall crossing; we will keep a nlab page on it
(past) Focus Week on New Invariants and Wall Crossing, May 18-22, 2009, Kashiwa Campus of the University of Tokyo
(past) Wall-crossing in Mathematics and Physics, May 24-28, 2010, Department of Mathematics, University of Illinois at Urbana-Champaign
Description of seminar on stability conditions and Stokes factors in Bonn, pdf
Maxim Kontsevich, Yan Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, arXiv:0811.2435
Arend Bayer, Yuri I. Manin, Stability conditions, wall-crossing and weighted Gromov-Witten invariants, math.AG/0607580, Mosc. Math. J. 9 (1), 2009.
Mina Aganagic, Hirosi Ooguri, Cumrun Vafa, Masahito Yamazaki, Wall crossing and M-theory, arxiv/0908.1194
M. Aganagic, Wall crossing, quivers and crystals, arxiv/1006.2113
S. Cecotti, C. Vafa, BPS wall crossing and topological strings, arXiv/0910.2615
Davide Gaiotto, Gregory W. Moore, Andrew Neitzke, Wall-crossing, Hitchin systems, and the WKB approximation, arxiv/0907.3987
E. Diaconescu, G. W. Moore, Crossing the wall: branes vs. bundles, arXiv/0706.3193
E. Andriyash, F. Denef?, D. L. Jafferis, G. W. Moore, Wall-crossing from supersymmetric galaxies, arxiv/1008.0030
M. Kontsevich, Y. Soibelman, Motivic Donaldson-Thomas invariants: summary of results, 0910.4315
Tom Bridgeland, Valerio Toledano-Laredo, Stability conditions and Stokes factors, arxiv/0801.3974
M. C. N. Cheng, E. P. Verlinde, Wall crossing, discrete attractor flow and Borcherds algebra, SIGMA 4 (2008), 068, 33 pages, pdf
Masahito Yamazaki, Crystal melting and wall crossing phenomena, Ph.D. thesis, arxiv/1002.1709
Michele Cirafici, Annamaria Sinkovics, Richard J. Szabo, Instanton counting and wall-crossing for orbifold quivers, arxiv/1108.3922
H.-Y. Chen, N. Dorey, K. Petunin, Moduli space and wall-crossing formulae in higher-rank gauge theories, JHEP 11 (2011) 020, doi; Wall crossing and instantons in compactified gauge theory, JHEP 06 (2010) 024 arXiv:1004.0703
Related entries BPS state, D-module, cluster algebra, quiver, representation theory, Donaldson-Thomas invariant.