All Questions
Tagged with nonlinear-dynamics partial-differential-equations
48
questions
2
votes
0
answers
47
views
Which nonlinear PDEs can be converted into linear PDEs?
In Section 4.4 of Partial Differential Equations by Evans, the author describes several techniques for converting certain nonlinear equations into linear equations. First, the author introduces the ...
- 356
1
vote
1
answer
62
views
Unsolvable characteristic system ODE as a part of PDE solution?
I'm trying to solve the following PDE:
$$F(x_1,x_2,u,p_1,p_2)=\text{ln}(x_2)p_1+x_2up_2-u=0 \ \ \ \ \ \ p_i=\partial_iu(x_1,x_2)$$
Where the initial conditions are:
$$\begin{cases}x_1(t)=t+1 \\x_2(t)=...
- 379
0
votes
0
answers
13
views
L2-preserving discretization of inviscid Burgers’ equation
I’m looking for a stable discretization of the inviscid Burgers’ equation that exactly preserves the L2-norm of the solution. Does such a discretization exist?
I’d appreciate any insight/references!
- 53
0
votes
0
answers
21
views
Complex valued Hamilton Jacobi equation
Let $g_{ij}(t,x)$ be a metric tensor with dependence on t,x. Consider
$$\partial_t u(t,x) = i\sqrt{\sum_{i,j} g_{ij}\partial_iu\partial_ju},u(0,x)=u_0(x).$$
Where $u(t,x):\mathbb{R}\times\mathbb{R}^n\...
- 1
0
votes
0
answers
45
views
Are there any examples of diffusion PDEs with nonlinear complications, that would possess analytical solutions?
I need an example (at least one, but more are welcome) of nonlinear PDEs in one space dimension (finite interval), containing transient diffusional terms plus some nonlinear complications, with ...
- 91
1
vote
0
answers
21
views
Realizing a modified transport equation
Stated somewhat informally, the continuity equation or transport equation $\partial_t\rho_t = -\nabla\cdot(\rho_t v_t)$ describes the evolution of a density where each particle flows along a vector ...
- 610
0
votes
0
answers
71
views
Mathematical theory of plasma
I am working on a heavily mathematically project about plasma. In particular, I want to find references that treat the problem from microscopic models that include relativistic and magnetic effects (...
- 113
0
votes
1
answer
82
views
Strang splitting for the NLS
The continuous cubic-focusing NLS is $$i \partial_t u(t,x) = -u_{xx} - |u|^2 u$$
A split-step scheme can be performed. Consider the linear split equation: $\partial_t u = -iu_{xx}$
It can be solved ...
- 451
0
votes
0
answers
24
views
How to know if a set of equations are Lorentz Invariant Spinors
I'm currently busy with a course in QFT and am completely baffled by Spinors. In particular there are two parts, that while I mostly understand the theory, struggle to show mathematically (especially ...
- 39
0
votes
0
answers
59
views
Finding fixed points of a specific constrained nonlinear PDE
I'm looking for attracting fixed points of the following differential equation in a vector $F$ of length $n$, and $M$ is a known $n \times n$ square matrix:
\begin{align}
\frac{dF}{dt} = I(F), \...
- 1
1
vote
1
answer
148
views
Reaction Diffusion Equation general solution
I have been struggling to find the general solution of the following BVP of reaction-diffusion equation:
$$\frac{\partial N}{\partial t}=\frac{\partial^2 N}{\partial x^2}+N(1-N)-\sigma N$$
$$N(0,x)=...
- 11
2
votes
0
answers
61
views
How to solve partial differential equation of the form $\partial_x f\ (dx/dt)+\partial_t f =0$
In some of my clases, they teach me some function which describe infiltration in the soil. The equation is known as Kostiakov's equation. We denote the Kostiakov's equation as $I(t)$, which maps $I:\...
- 495
1
vote
0
answers
45
views
What is the name of this PDE $\partial_t u = F(u) - \mathcal{L}_{\mathcal{V}}u$?
I've come across the following PDE
$$
\partial_t u = F(u) - \mathcal{L}_{\mathcal{V}} u
$$
where the solution $u : \mathbb{R}^3 \times \mathbb{R} \to \mathbb{R}^3$ is a vector field in space $\mathbb{...
- 193
0
votes
0
answers
51
views
book reference: beginner book on nonlinear pdes with phase transitions in reaction-diffusion systems
I am looking for suggestions on a good introductory book to nonlinear PDEs and the ideas of phase transitions or pattern formation.
My background in PDEs is mostly self-taught. So I worked through ...
- 2,531
2
votes
1
answer
222
views
Fourier transform of a nonlinear Schrödinger soliton
The nonlinear Schrödinger equation
$$
i\frac{\partial\psi}{\partial t} = -\frac{1}{2}\frac{\partial^2\psi}{\partial x^2}- |\psi|^2\psi
$$
has the single-soliton solution
$$
\psi(x,t)=A \frac{e^{iv(x-...
- 179