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Project 2013-2014

Novikov-Veselov Equation

PhD research focused on developing novel computational methods for the Novikov-Veselov equation, advancing our understanding of nonlinear wave phenomena and multi-dimensional integrable systems.

Numerical Methods PDE Theory Scientific Computing MATLAB C++

Project Overview

This PhD research represents a significant advancement in our understanding of the Novikov-Veselov (NV) equation, a (2+1)-dimensional integrable system that generalizes the famous Korteweg-de Vries (KdV) equation. While (1+1)-dimensional soliton equations like KdV have been extensively studied, their multidimensional counterparts remain less explored due to their inherent complexity and the challenges in developing appropriate analytical and numerical tools.

The work combines analytical methods with novel numerical approaches to explore soliton solutions, stability analysis, and the inverse scattering transform in multiple dimensions. This research addresses fundamental questions about the nature of integrability in higher dimensions and provides new tools for studying complex nonlinear systems.

Historical Context

The Novikov-Veselov equation is part of a broader family of multidimensional integrable systems that extend well-known one-dimensional models. Key historical developments include:

Kadomtsev-Petviashvili (KP) Equation (1970)

The first widely studied 2D extension of the KdV equation, represented as:

$$ (u_t + 6uu_x + u_{xxx})_x + 3\alpha^2u_{yy} = 0 $$

Davey-Stewartson System (1974)

A 2D extension of the nonlinear Schrödinger equation:

$$ \begin{align} iq_t + q_{xx} + q_{yy} + (|q|^2 + 2\phi)q &= 0\\ \phi_{xx} - \phi_{yy} &= (|q|^2)_{xx} \end{align} $$

The Novikov-Veselov equation, developed in the late 1980s, represents a more sophisticated extension that preserves complete integrability in multiple dimensions, a property that makes it particularly valuable for theoretical and applied studies.

The Novikov-Veselov Equation System

Core Equations

The NV equation system consists of three coupled equations:

Evolution equation:

$$ u_t = \partial_x^3u + \partial_y^3u + 3\partial_x(uv) + 3\partial_y(uw) $$

Auxiliary equations:

$$ \partial_x v = \partial_y w,\quad \partial_y v = -\partial_x w $$

Mathematical Structure

The system can be written in terms of complex derivatives:

$$ u_t = \partial_z^3 u + \partial_{\bar{z}}^3 u + 3\partial_z(u\omega) + 3\partial_{\bar{z}}(u\bar{\omega}) $$
$$ \partial_{\bar{z}}\omega = \partial_z u $$

where

$$ \begin{align} z &= x + iy\\ \omega &= v + iw \end{align} $$

Lax Representation

The NV equation admits a Lax representation:

$$ \begin{align} L &= -\partial_z\partial_{\bar{z}} + u\\ A &= \partial_z^3 + \partial_{\bar{z}}^3 + 3\omega\partial_z + 3\bar{\omega}\partial_{\bar{z}} \end{align} $$

with the Lax equation:

$$ \frac{dL}{dt} = [A,L] $$

Conservation Laws and Symmetries

The system possesses an infinite hierarchy of conservation laws, including:

Mass:

$$ E_1 = \iint u \, dx \, dy $$

Energy:

$$ E_2 = \iint u^2 \, dx \, dy $$

Momentum:

$$ E_3 = \iint (|\partial_z u|^2 + |\partial_{\bar{z}} u|^2 - 2u^3) \, dx \, dy $$

Symmetry Group Analysis

  • Translational invariance in x, y, and t
  • Rotational invariance in the x-y plane
  • Galilean invariance
  • Scaling symmetry: (x,y,t,u) → (λx,λy,λ³t,λ⁻²u)

Research Contributions

1. Novel Traveling Wave Solutions

I discovered and analyzed new soliton solutions, including line soliton solutions of the form:

$$ u(x,y,t) = 2\kappa^2 \text{sech}^2(\kappa(x\cdot\cos\theta + y\cdot\sin\theta - ct)) $$

where κ determines amplitude, θ direction, and c velocity.

2. Numerical Spectral Method

I developed the first comprehensive numerical implementation for the NV system: 

% Core implementation of the Fourier spectral method for NV equation
function [u, v, w] = evolve_nv(u0, tmax, dt, nx, ny, Lx, Ly)
    % Set up grid and wavenumbers
    dx = Lx/nx;
    dy = Ly/ny;
    x = (0:nx-1)*dx;
    y = (0:ny-1)*dy;
    [X, Y] = meshgrid(x, y);

    % Define wavenumbers for spectral method
    kx = 2*pi/Lx * [0:nx/2-1, -nx/2:-1];
    ky = 2*pi/Ly * [0:ny/2-1, -ny/2:-1];
    [KX, KY] = meshgrid(kx, ky);

    % Initial conditions
    u = u0;
    v = zeros(size(u));
    w = zeros(size(u));

    % Time evolution using RK4
    t = 0;
    while t < tmax
        % RK4 step
        [k1u, k1v, k1w] = nv_rhs(u, v, w, KX, KY);
        [k2u, k2v, k2w] = nv_rhs(u + dt/2*k1u, v + dt/2*k1v, w + dt/2*k1w, KX, KY);
        [k3u, k3v, k3w] = nv_rhs(u + dt/2*k2u, v + dt/2*k2v, w + dt/2*k2w, KX, KY);
        [k4u, k4v, k4w] = nv_rhs(u + dt*k3u, v + dt*k3v, w + dt*k3w, KX, KY);

        u = u + dt/6 * (k1u + 2*k2u + 2*k3u + k4u);
        v = v + dt/6 * (k1v + 2*k2v + 2*k3v + k4v);
        w = w + dt/6 * (k1w + 2*k2w + 2*k3w + k4w);

        t = t + dt;

        % Compute and display conservation laws
        E1 = sum(sum(u))*dx*dy;
        E2 = sum(sum(u.^2))*dx*dy;
        fprintf('t = %.4f, Mass = %.6e, Energy = %.6e\n', t, E1, E2);
    end
end

% Right-hand side of the NV system in Fourier space
function [dudt, dvdt, dwdt] = nv_rhs(u, v, w, KX, KY)
    % Compute derivatives in Fourier space
    uhat = fft2(u);
    vhat = fft2(v);
    what = fft2(w);

    % Nonlinear terms
    uvhat = fft2(u .* v);
    uwhat = fft2(u .* w);

    % Compute right-hand side
    dudt_hat = 1i^3 * KX.^3 .* uhat + 1i^3 * KY.^3 .* uhat + ...
               3 * 1i * KX .* uvhat + 3 * 1i * KY .* uwhat;
    dvdt_hat = 1i * KY .* what;
    dwdt_hat = -1i * KX .* vhat;

    % Transform back to physical space
    dudt = real(ifft2(dudt_hat));
    dvdt = real(ifft2(dvdt_hat));
    dwdt = real(ifft2(dwdt_hat));
end

3. Soliton Stability Analysis

I applied the direct K-method to analyze stability:

Perturbation analysis using:

$$ u(x,y,t) = u_0(x,y,t) + \varepsilon v(x,y,t) $$

Leading to the linearized stability equation:

$$ v_t = \mathcal{L}v + \mathcal{O}(\varepsilon) $$

4. Closed-Form Solutions

I discovered several new classes of solutions:

The Inverse Scattering Transform

The inverse scattering transform (IST) represents a nonlinear analog of the Fourier transform, providing a systematic method for solving certain nonlinear PDEs. For the NV equation, the transform involves several sophisticated mathematical structures and concepts:

The Direct Scattering Problem

The associated 2D Schrödinger equation at fixed time:

$$ (-\Delta + q)\psi = 0, \quad \Delta = \partial_x^2 + \partial_y^2 $$

Complex geometric optics (CGO) solutions are sought in the form:

$$ \psi(z,k) = e^{ikz}(1 + \mu(z,k)) \\ \mu(z,k) \rightarrow 0 \text{ as } |z| \rightarrow \infty $$

where z = x + iy is the complex coordinate and k is the spectral parameter. The CGO solutions satisfy:

$$ \partial_{\bar{z}}\mu = -e^{-ikz}q(z)e^{ikz}(1 + \mu) $$

Time Evolution

The time evolution of the scattering data is remarkably simple:

$$ t(k,t) = t(k,0)e^{4ik^3t} $$

This linear evolution is a key feature of integrable systems, analogous to how Fourier modes evolve linearly for linear PDEs.

Significance and Applications

The IST formalism for the NV equation has significant implications:

Impact and Publications

This research has contributed to the mathematical understanding of multidimensional integrable systems and has applications in various fields, including:

Key Publications

Liu, R., Croke, R., & Perry, P. (2019)

Computational Methods for the Novikov-Veselov Equation

Journal of Physics: Conference Series, 1187, 042074

View Publication

The numerical methods developed in this research have been incorporated into open-source libraries for studying nonlinear wave equations, providing valuable tools for the scientific community.