ONE-DIMENSIONAL

A wave that starts smooth and continuous can, in a short time, deform and collapse into an abrupt front, forming a shock, a simple phenomenon that remarkably mimics the behavior of waves in real fluids.

A wave that could dissipate or break apart instead reorganizes into stable structures called solitons, which travel long distances without losing their shape, a phenomenon discovered in the 19th century and still considered fascinating today.

When multiple solitons meet, they interact in a nontrivial way: they collide, overlap, and then continue their journey almost without losing their shape, undergoing only a phase shift. This "dance" of waves is one of the most remarkable signatures of the KdV equation and one of the reasons why solitons became so famous.

A complex initial wave can, through the dynamics of the KdV equation, naturally decompose into a "soliton train": several solitary waves emerge with different amplitudes and velocities, traveling in an organized manner like a convoy of stable waves.

A solitary wave that travels stably through the domain may, upon reaching the boundary, lose its integrity: the soliton breaks apart and returns as several small waves propagating in the opposite direction, revealing the dispersive nature and dynamic richness of the model.

In the modified version of the KdV (mKdV) equation, breathers emerge, localized structures that oscillate in both time and space. They appear in physical contexts such as optical fibers and plasma dynamics, where they describe pulses that interact in a periodic and stable manner.