The dispersion-relation curvature Ω"(k0) – representing group velocity dispersion – is always negative for water waves under the action of gravity, for any water depth. For waves on the water surface of deep water, the coefficients of importance for the nonlinear Schrödinger equation are: so where g is the acceleration due to gravity at the Earth's surface. In the original (x0, t0) coordinates the nonlinear Schrödinger equation for water waves reads: with (i.e. the complex conjugate of ) and So for deep water waves. Gauge equivalent counterpart NLSE (1) is gauge equivalent to the following isotropic Landau-Lifshitz equation (LLE) or Heisenberg ferromagnet equation Note that this equation admits several integrable and non-integrable generalizations in 2 + 1 dimensions like the Ishimori equation and so on. Zero-curvature formulation The NLSE is equivalent to the curvature of a particular -connection on being equal to zero. Explicitly, with coordinates on , the connection components are given by where the are the Pauli matrices. Then the zero-curvature equation is equivalent to the NLSE . The zero-curvature equation is so named as it corresponds to the curvature being equal to zero if it is defined . The pair of matrices and are also known as a Lax pair for the NLSE, in the sense that the zero-curvature equation recovers the PDE rather than them satisfying Lax's equation. Relation to vortices showed that the work of on vortex filaments is closely related to the nonlinear Schrödinger equation. Subsequently, used this correspondence to show that breather solutions can also arise for a vortex filament. See also AKNS system Eckhaus equation Quartic interaction for a related model in quantum field theory Soliton (optics) Logarithmic Schrödinger equation Notes References Notes Other External links Tutorial lecture on Nonlinear Schrodinger Equation (video). Nonlinear Schrodinger Equation with a Cubic Nonlinearity at EqWorld: The World of Mathematical Equations. Nonlinear Schrodinger Equation with a Power-Law Nonlinearity at EqWorld: The World of Mathematical Equations. Nonlinear Schrodinger Equation of General Form at EqWorld: The World of Mathematical Equations. Mathematical aspects of the nonlinear Schrödinger equation at Dispersive Wiki Partial differential equations Exactly solvable models Schrödinger equation Integrable systems