Related Experiment Video
Updated: May 31, 2026
11:03
An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids
Published on: December 4, 2017
Anomalous scaling and solitary waves in systems with nonlinear diffusion
Alex Hansen1, Bo-Sture Skagerstam, Glenn Tørå
1Department of Physics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway. Alex.Hansen@ntnu.no
Summary
This study investigates a nonlinear convective-diffusive equation, revealing anomalous subdiffusive scaling and the emergence of stable solitary waves. These findings suggest general features in nonlinear convective-diffusive dynamics.
Area of Science:
- Mathematical Physics
- Nonlinear Dynamics
- Fluid Dynamics
Background:
- Examines a nonlinear convective-diffusive equation modeling wetting film thickness dynamics.
- Highlights the role of nonlinear diffusion in forming fronts and shock fronts.
Purpose of the Study:
- Investigate anomalous subdiffusive scaling in the absence of memory effects.
- Analyze the emergence and stability of solitary waves due to nonlinear diffusion and convection.
Main Methods:
- Analysis of a nonlinear convective-diffusive equation.
- Exploration of solutions exhibiting anomalous subdiffusive scaling.
- Identification of solitary wave formation and topological stability.
Main Results:
- Demonstrates anomalous subdiffusive scaling in pure nonlinear diffusion.
- Shows the appearance of solitary waves through a balance of nonlinear diffusion and convection.
- Confirms the merging of solitary waves into a single, topologically stable wave over time.
Conclusions:
- The study identifies anomalous diffusion and stable solitary waves as general features in nonlinear convective-diffusive dynamics.
- Numerical simulations support the broad applicability of these findings beyond the specific equation studied.
Related Concept Videos
01:26Linear Approximation in Frequency Domain
Linear systems are characterized by two main properties: superposition and homogeneity. Superposition allows the response to multiple inputs to be the sum of the responses to each individual input. Homogeneity ensures that scaling an input by a scalar results in the response being scaled by the same scalar.
In contrast, nonlinear systems do not inherently possess these properties. However, for small deviations around an operating point, a nonlinear system can often be approximated as linear.
In contrast, nonlinear systems do not inherently possess these properties. However, for small deviations around an operating point, a nonlinear system can often be approximated as linear.
01:20Types of Damping
If the amount of damping in a system is gradually increased, the period and frequency start to become affected because damping opposes, and hence slows, the back and forth motion (the net force is smaller in both directions). If there is a very large amount of damping, the system does not even oscillate; instead, it slowly moves toward equilibrium. In brief, an overdamped system moves slowly towards equilibrium, whereas an underdamped system moves quickly to equilibrium but will oscillate about...
01:26Scaling
In designing and analyzing filters, resonant circuits, or circuit analysis at large, working with standard element values like 1 ohm, 1 henry, or 1 farad can be convenient before scaling these values to more realistic figures. This approach is widely utilized by not employing realistic element values in numerous examples and problems; it simplifies mastering circuit analysis through convenient component values. The complexity of calculations is thereby reduced, with the understanding that...
01:21Linear Approximation in Time Domain
Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length, the...
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length, the...
01:07Damped Oscillations
In the real world, oscillations seldom follow true simple harmonic motion. A system that continues its motion indefinitely without losing its amplitude is termed undamped. However, friction of some sort usually dampens the motion, so it fades away or needs more force to continue. For example, a guitar string stops oscillating a few seconds after being plucked. Similarly, one must continually push a swing to keep a child swinging on a playground.
Although friction and other non-conservative...
Although friction and other non-conservative...
01:19Limits with Oscillating Discontinuities
An oscillating discontinuity is a type of discontinuity in which a function’s values fluctuate infinitely often as the input approaches a particular point. Unlike jump discontinuities, where the function suddenly shifts between two values, or infinite discontinuities, where the function diverges without bound, an oscillating discontinuity arises from rapid back-and-forth variation. Because the function never stabilizes toward a single value, no finite limit exists at that point.One of the most...