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Related Concept Videos

Tangent Planes to Surfaces01:19

Tangent Planes to Surfaces

In multivariable calculus, the concept of a tangent plane plays a central role in approximating curved surfaces. When dealing with a surface defined by a function of two variables, such as z = f(x, y), the tangent plane at a given point provides the best linear approximation to the surface near that point. This local linearization allows complex, nonlinear geometries to be treated using simpler, planar models.The construction of the tangent plane involves taking vertical slices of the surface...
Tangent Planes to Level Surfaces01:31

Tangent Planes to Level Surfaces

A level surface consists of all points in space where a function of three variables takes the same fixed value. If a point lies on this surface, understanding the surface’s geometry there requires more than just knowing the point’s coordinates; it requires describing how the surface is oriented, or how it tilts, near that point.To probe this local geometry, imagine tracing a path that stays entirely on the level surface and passes through the point of interest. This path can be described as a...
Equipotential Surfaces and Field Lines01:29

Equipotential Surfaces and Field Lines

Electric potential can be pictorially represented as a three-dimensional surface. On such a surface, the electric potential is constant everywhere. The equipotential surface is always perpendicular to the electric field lines, and while it is three-dimensional, it can be treated as an equipotential line in a two-dimensional case. These equipotential lines are also always perpendicular to electric field lines. The term equipotential is often used as a noun, referring to an equipotential line or...
Tangent Planes to a Parametric Surface01:22

Tangent Planes to a Parametric Surface

A tangent plane provides a linear approximation to a curved surface at a specific point, capturing the local behavior of the surface. It can be understood as the plane that just touches the surface at that point and is defined by the tangent directions of curves lying on the surface. These tangent directions arise naturally when the surface is described parametrically, allowing systematic construction of the plane.For a surface expressed in parametric form, the position of any point is...
Electrostatic Boundary Conditions01:16

Electrostatic Boundary Conditions

Consider an external electric field propagating through a homogeneous medium. When the electric field crosses the surface boundary of the medium, it undergoes a discontinuity. The electric field can be resolved into normal and tangential components. The amount by which the field changes at any boundary is given by the difference between the field components above and below the surface boundary.
The surface integral of an electric field is given by Gauss's law in integral form and is related to...
Boundary Conditions: Lossless Lines01:21

Boundary Conditions: Lossless Lines

Consider a single-phase, two-wire, lossless transmission line terminated by an impedance at the receiving end and a source with Thevenin voltage and impedance at the sending end. The line, with length, has a surge impedance and wave velocity determined by the line's inductance and capacitance.
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Fabrication, Operation and Flow Visualization in Surface-acoustic-wave-driven Acoustic-counterflow Microfluidics
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Published on: August 27, 2013

Vector mixed-gap surface solitons.

Yaroslav V Kartashov, Fangwei Ye, Lluis Torner

    Optics Express
    |June 12, 2009
    PubMed
    Summary
    This summary is machine-generated.

    We discovered stable vector surface solitons in nonlinear optical lattices. These mixed-gap solitons, emerging from different spectral gaps, achieve stability through mutual trapping and cross-coupling effects.

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    The Preparation of Electrohydrodynamic Bridges from Polar Dielectric Liquids
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    Area of Science:

    • Nonlinear optics
    • Condensed matter physics
    • Photonics

    Background:

    • Optical lattices create periodic potentials for light propagation.
    • Solitons are self-reinforcing wave packets that maintain their shape.
    • Surface waves propagate along an interface between two media.

    Purpose of the Study:

    • To investigate the properties of mixed-gap vector surface solitons.
    • To understand the formation and stability mechanisms of these novel soliton states.
    • To explore the unique characteristics arising from the combination of surface and gap soliton features.

    Main Methods:

    • Theoretical analysis of nonlinear wave propagation.
    • Numerical simulations of soliton dynamics.
    • Investigation of the interface between a uniform medium and an optical lattice in a Kerr nonlinear medium.

    Main Results:

    • Formation of stable vector surface solitons by components emerging from different lattice gaps.
    • Stabilization of unstable soliton components through cross-coupling with stable components.
    • Demonstration of a unique combination of vectorial surface wave and gap soliton properties.

    Conclusions:

    • Mixed-gap vector surface solitons represent a new class of nonlinear optical waves.
    • Mutual trapping and cross-coupling are key to their stability.
    • These solitons offer potential for novel applications in photonics and optical information processing.