I am attempting to solve the complex nonlinear equation with space dependent variable coefficients. The usual algebraic methods say jacobi elliptic, tanh, sine-cosine etc couldn't work for this equation and also many variable coefficient nonlinear equations I came across so far are time dependent which can be easily solvable and methods are available. I do no have knowledge on Lie symmetries. (See the file attached). Thanks in well advance.
yes generalized solution. In case if possible for solution through expansion or perturbation, that too I acept. The equation I shown is first order perturbed fields of magnetization for a ferromagnetic medium. I am expecting soliton solutions because this eq is just extended form of integrable derivative NLS equation when the variable coefficient f and h are zero.
I Think that your problem could be solved using a linearization method or an approximate scheme ( i.e using finite difference techniques). Nevertheless, i have searched for other advice I may give you. I found this URL http://www.ias.ac.in/mathsci/vol122/aug2012/pmsc-d-11-00013.pdf. There appears the main problem subject to an equation which is almost equal to that you had shown.
I hope will be useful my answer
Firstly, you should clearly formulate boundary or transmission conditions for the problem you consider. In the most cases this is unnecessary and even useless to look for a general solution for a nonlinear PDE.
Secondly, it is clear that the problem is difficult to solve, I would suggest to look for solutions for simpler problems, if the original problem can be simplified. For example, you can try to use an ansatz $\psi = e^{i\gamma\tau}\phi$, where $\gamma$ is a real unknown constant, $\phi$ is a new unknown function w.r.t. $z$ (or you can use some other ansatz in order to reach the same effect). Then you obtain an ordinary differential equation. However, I would prefer to simplify the problem more. Is it possible to consider the simplified problem with constant coefficients? If yes, then this is a good way to understand the problem better. As the matter of fact, even in the most simplified form you get a nonlinear autonomous ordinary differential equation. If you are unable to investigate such a simplified problem, then there is nothing to say about the original problem. You can also try to "freeze" nonconstant coefficients if it is possible from physical point of view. However, the first paragraph is of fundamental importance, without rigorous statement there is nothing to do.
If you don't wont to simplify, then you can try to use the Green functions' approach. You reduce your problem to an equivalent nonlinear integral equation and the using functional analysis to investigate it. In some cases it is possible using the contracting mapping principle to prove existence and uniqueness of solutions. However, without all necessary boundary or transmission conditions this method can not be applied.
I also stress, if you use a perturbation theory or another linearization procedure you can find only that solutions which are close to the solutions of corresponding linear problem. However, in nonlinear problems there can exist solutions that do not reduce to the solutions of corresponding linearized problem. And these very solutions are very interesting and important. Existence of perturbed solutions are expectable in may problems without deep investigations.
If you are looking for numerical solutions, you should give a try to the finite element method (or finite-difference as @Xiomara Josefina Contreras has mentioned). For FE method you will need to know boundary conditions. If numerical solutions are acceptable to you and you know boundary conditions, I highly recommend this software: http://www.comsol.com/comsol-multiphysics as you can specify your own equations that you want to solve.
@Saravanan Manickam
You can use, in particular, a so called Hirota method for finding a wide enough class of exact analytical solutions to this Nonlinear Schrodinger type equation.
One can only try to find special solutions making use a well known Painleve approach - it often also works well in such nonlinear cases.
- Badges
- Science topic
- Similar topics
- Mathematics
- Applied Mathematics
- Differential Equations