# Korteweg–de Vries Soliton Solutions

Soliton solutions of the KdV equation normally require retraining a physics-informed neural network whenever the dispersion parameter changes, and long-time extrapolation tends to drift. The scaled TW-PINN first normalizes the equation, trains once in normalized coordinates, and inverse-scales back to the original variables; the one-soliton solver uses a traveling-wave coordinate, and the two-soliton case trains in two steps, learning individual speeds first and the interaction second. In numerical experiments the two-soliton model preserved profiles ten times beyond the training domain, and a three-soliton experiment reproduced collision and separation.

Soliton solutions of the KdV equation normally require retraining a physics-informed neural network whenever the dispersion parameter changes, and long-time extrapolation tends to drift. The scaled TW-PINN first normalizes the equation, trains once in normalized coordinates, and inverse-scales back to the original variables; the one-soliton solver uses a traveling-wave coordinate, and the two-soliton case trains in two steps, learning individual speeds first and the interaction second. In numerical experiments the two-soliton model preserved profiles ten times beyond the training domain, and a three-soliton experiment reproduced collision and separation.