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Introduction

The cubic nonlinear Schrödinger equation (NLS) is one of the most important equations in mathematical physics. This comprehensive analysis includes numerical simulations, comparative data, and complete references.

The Cubic Schrödinger Equation

The one-dimensional cubic nonlinear Schrödinger equation is:

$$i\frac{\partial \psi}{\partial t} + \frac{\partial^2 \psi}{\partial x^2} + 2|\psi|^2 \psi = 0 \quad \text{…(1.1)}$$

Conserved Quantities

The cubic Schrödinger equation admits several conserved quantities:

Mass (L² norm):

$$M = \int_{-\infty}^{\infty} |\psi(x,t)|^2 dx = \text{constant}$$

Energy (Hamiltonian):

$$H = \int_{-\infty}^{\infty} \left[\left|\frac{\partial \psi}{\partial x}\right|^2 – |\psi|^4\right] dx = \text{constant}$$

Numerical Simulation Results

Plot 1: Soliton Evolution

The soliton solution $|\psi(x)|^2 = 2\eta^2 \operatorname{sech}^2(\sqrt{2}\eta x)$ for amplitude $\eta = 1.0$ is shown below. This solution exhibits the remarkable property of solitons: they maintain their shape and amplitude indefinitely as they propagate through the nonlinear medium.

Cubic Schrödinger Soliton Solution

Plot 2: Space-Time Evolution of Cubic Schrödinger Soliton

The density plot below shows the evolution of $|\psi(x,t)|^2$ over time (0 to 5 time units) and space (-8 to 8 spatial units). The soliton structure is preserved as it propagates, with the peak maintaining constant amplitude and moving at constant velocity. The color intensity represents the probability density at each point in space-time.

Space-Time Evolution of Cubic Schrödinger Soliton

Comparative Analysis Table

The table below compares the key properties of focusing and defocusing nonlinear Schrödinger equations, highlighting how the sign of the nonlinearity coefficient determines the physical behavior:

Comparative Analysis: Focusing vs Defocusing NLS

Numerical Implementation Example

Here is a Python example implementing the soliton solution computation:

import numpy as np
import matplotlib.pyplot as plt
class SolitonSolver:

    “””Compute cubic Schrödinger soliton solutions”””
    def __init__(self, eta=1.0):

        self.eta = eta

        self.x = np.linspace(-10, 10, 200)
    def compute_soliton(self):

        # Soliton amplitude squared

        psi_squared = 2 * self.eta**2 / np.cosh(np.sqrt(2) * self.eta * self.x)**2

        return psi_squared
    def plot_solution(self):

        psi = self.compute_soliton()

        plt.figure(figsize=(10, 6))

        plt.plot(self.x, psi, ‘b-‘, linewidth=2.5)

        plt.fill_between(self.x, psi, alpha=0.3)

        plt.xlabel(‘Position (x)’, fontsize=12)

        plt.ylabel(‘|ψ(x)|²’, fontsize=12)

        plt.title(‘Cubic Schrödinger Soliton’, fontsize=14)

        plt.grid(True, alpha=0.3)

        plt.show()
# Verify conservation of mass

solver = SolitonSolver(eta=1.0)

psi_squared = solver.compute_soliton()

mass = np.trapz(psi_squared, solver.x)

print(f”Conserved mass: {mass:.4f}”)

solver.plot_solution()

Physical Applications

1. Nonlinear Optics in Fiber

In optical fibers, the cubic NLS describes light pulse propagation:

$$i\frac{\partial A}{\partial z} = -\frac{\beta_2}{2}\frac{\partial^2 A}{\partial t^2} + \gamma \|A\|^2 A$$

where $A$ is the electric field envelope, $\beta_2$ is dispersion, $\gamma$ is the nonlinearity coefficient.

2. Bose-Einstein Condensates

The Gross-Pitaevskii equation for ultracold atoms:

$$i\hbar\frac{\partial \Phi}{\partial t} = -\frac{\hbar^2}{2m}\nabla^2\Phi + V(\mathbf{r})\Phi + g|\Phi|^2\Phi$$

is a direct analog of the cubic NLS in 3D, with $g$ as the interaction strength.

3. Deep Water Waves

Rogue wave formation is explained by focusing NLS dynamics. The Davey-Stewartson system reduces to NLS in certain limits.

Numerical Methods

Split-Step Fourier Method

The most accurate method for NLS:

$$\psi^{n+1} = e^{-i\Delta t L_2} e^{-i\Delta t L_1} e^{-i\Delta t L_2} \psi^n + O(\Delta t^3)$$

where $L_1 = 2|\psi|^2$ (nonlinear), $L_2 = \frac{\partial^2}{\partial x^2}$ (dispersive).

Advantages:
– Spectral accuracy
– Explicit time stepping
– Energy conservative (with proper implementation)

References

[1] Ablowitz, M. J., & Clarkson, P. A. (2011). Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, 1st edition.

[2] Sulem, C., & Sulem, P.-L. (1999). The nonlinear Schrödinger equation: Self-focusing and wave collapse. Applied Mathematical Sciences, 139, 1–350.

[3] Hasegawa, A., & Tappert, F. (1973). Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. Applied Physics Letters, 23(3), 142–144.

[4] Kevrekidis, P. G. (2015). The Discrete Nonlinear Schrödinger Equation. Springer, Berlin, Heidelberg.

[5] Erdoğan, M. B., & Tzirakis, N. (2016). Dispersive partial differential equations: Wellposedness and applications. London Mathematical Society Student Texts, 86.

Conclusion

The cubic Schrödinger equation:
– Governs wave phenomena in optics, quantum gases, and fluids
– Is integrable (exactly solvable via inverse scattering)
– Admits soliton solutions with remarkable stability properties
– Exhibits focusing/defocusing behavior depending on nonlinearity sign
– Has rich mathematical structure and diverse applications

The numerical methods (split-step Fourier, Runge-Kutta) and conserved quantities make it an ideal testbed for understanding nonlinear wave dynamics.

The Cubic Schrödinger Equation: Complete Analysis with Plots and Data