Duc, I do not know if you understood my last statement. You will never find it in publications, and it took me several years to arrive to those conclusions.

Half of economists are quite happy to live in the world without space and time. Perhaps 30% use time, but mostly in discrete form and often for stochastic processes. Maybe 20% use space (regional science and urban economics). But less than 1% use space and time simultaneously. There are 3 major courses for the beginners: microeconomics (I gave you a book), macroeconomics and econometrics. Mathematicians with specialization in statistics or fixed point theory are highly welcome to corresponding branches (of course, after they listen to basic courses), but I do not know any specialist in wave theory (even linear) who has done a career in economics based on that knowledge. There are many hidden doors that are closed and not visible before one approaches them. But if you want to get your own experience, just study basic economics (it will take a couple of years) and then derive your own PDE consistent with basic assumptions. I guess that you will find no supervisor, and if they see you as a smart student, they will open many other doors to work on their problems, but not on PDE. Wave theory requires both local interaction (which in economics is often non-local and too fast) and no dissipation (while in economics there is hardly a value preserved over time; even global wealth can drop by 10% after a bad day on stock exchange).

If you want to stay as applied mathematician in the world of PDE, you better look for other application. This may be welcome in biology, urban traffic (by the way, waves of cars are even in the book of Whitham), demography (I also tried to write something like wave theory here; see https://www.researchgate.net/publication/275019458_Economic_Consequences_of_Demographic_Waves ), perhaps environment. It is easy to derive a system of non-linear equations for interaction between economics and nature; see https://www.researchgate.net/publication/278110040_Low-Dimensional_Nonlinear_Dynamical_Systems_as_a_Tool_for_Socio-Economic_Modelling . It gives some non-linear dynamics, but not via PDE and still not much welcome by pure economists.

Conference Paper Economic Consequences of Demographic Waves

Conference Paper Low-Dimensional Nonlinear Dynamical Systems as a Tool for So...

Dear Duc, I also had a background of PDE with physical applications before coming to economics. As far as I know, financial economists use only parabolic equation of dissipation type to describe movement of prices. I think that there is a problem in understanding both math and economics to derive new PDE with economic applications. Most likely, they will differ from known equations in mathematical physics (KdV, Schoridinger, Burgers). I have a couple of such experiences before understanding that this will never become an economic mainstream. But maybe I am not aware of all relevant literature. Anyway, I present two links - to my paper and conference presentation. Maybe they will help you to find proper sub-fields in economics where assumptions are natural for such derivations. Note that the conference AGSOE was for physicists who are interested in socio-economic problems.

https://www.researchgate.net/publication/279537697_Spatial_Price_Dynamics

https://www.researchgate.net/publication/277017625_Potential_and_Spatial_Evolution_of_Location_Patterns

Conference Paper Potential and Spatial Evolution of Location Patterns

Conference Paper Spatial Price Dynamics

Dear Duc

You can read

the book of Whitem

Linear and nonlinear waves

Some of my papers

Some of my papers

1- A. R. Seadawy, Fractional solitary wave solutions of the

nonlinear higher-order extended KdV equation in a stratified

shear flow: Part I, Comp. and Math. Appl. \textbf{70} (2015)

345–352.

2- Khater, A. H., Callebaut D. K., Malfliet, W. and Seadawy A. R., Nonlinear Dispersive

Rayleigh-Taylor Instabilities in Magnetohydro-dynamic Flows,

Physica Scripta, 64 (2001) 533-547.

3- Khater, A. H., Callebaut D.

K. and Seadawy A. R., "Nonlinear Dispersive Kelvin-Helmholtz

Instabilities in Magnetohydrodynamic Flows" Physica Scripta, 67

(2003) 340-349.

4- A.R. Seadawy, Exact solutions of a two-dimensional

nonlinear Schrodinger equation, Appl. Math. Lett. 25 (2012) 687.

5- A.R. Seadawy, Stability analysis for Zakharov-Kuznetsov equation of weakly

nonlinear ion-acoustic waves in a plasma, Computers and

Mathematics with Applications 67 (2014) 172-180.

6- A.R. Seadawy, Stability analysis for two-dimensional ion-acoustic waves in

quantum plasmas, PHYSICS OF PLASMAS 21 (2014) 052107.

7- Helal, M. A. and Seadawy A. R., Variational method for the

derivative nonlinear Schrodinger equation with computational

applications, Physica Scripta, 80, (2009) 350-360.

8- Helal, M. A. and Seadawy A. R., Exact

soliton solutions of an D-dimensional nonlinear Schrodinger

equation with damping and diffusive terms, Z. Angew. Math. Phys.

(ZAMP) 62 (2011), 839-847.

9- Khater, A. H., Callebaut D. K. and Seadawy A. R., General

soliton solutions of an n-dimensional Complex Ginzburg-Landau

equation, Physica Scripta, Vol. 62 (2000) 353-357.

10- Khater, A. H., Helal M. A. and Seadawy A. R.,

General soliton solutions of n-dimensional nonlinear Schrodinger

equation" IL Nuovo Cimento 115B, (2000) 1303-1312.

11- Khater, A. H., Callebaut D. K., Helal, M. A. and Seadawy A.

R., Variational Method for the Nonlinear Dynamics of an Elliptic

Magnetic Stagnation Line, The European Physical Journal D, 39,

(2006) 237-245.

12- Khater, A. H., Callebaut D. K.,

Helal, M. A. and Seadawy A. R., General Soliton Solutions for

Nonlinear Dispersive Waves in Convective Type Instabilities,

Physica Scripta, 74, (2006) 384.

13- Seadawy A. R., Three-dimensional nonlinear modified Zakharov–Kuznetsov

equation of ion-acoustic waves in a magnetized plasma, Computers

and Mathematics with Applications 71 (2016) 201-212.

14- M.A. Helal and A.R. Seadawy, Benjamin-Feir-instability in

nonlinear dispersive waves, Computers and Mathematics with

Applications 64 (2012) 3557-3568.

15- Seadawy, A.R., and El-Rashidy, K., Traveling wave solutions for some coupled nonlinear

evolution equations by using the direct algebraic method, Math.

and Comp. model. \textbf{57} (2013) 1371.

16- A. R. Seadawy, New exact solutions for the KdV equation with higher order nonlinearity

by using the variational method, Comp. and Math. Appl. 62 (2011)

3741-3755.

17- Seadawy A. R, Stability analysis solutions for nonlinear

three-dimensional modified Korteweg-de Vries-Zakharov-Kuznetsov

equation in a magnetized electron-positron plasma" Physica A:

Statistical Mechanics and its Applications Physica A 455 (2016)

44-51.

18- A.R. Seadawy, Nonlinear wave solutions of the three-dimensional

Zakharov–Kuznetsov–Burgers equation in dusty plasma, Physica A

439 (2015) 124–131.

19- A.R. Seadawy, Stability analysis of traveling wave

solutions for generalized coupled nonlinear KdV equations, Appl.

Math. Inf. Sci. 10 (1) (2016) 209–214.

Best regards

Prof Aly seadawy

Dear Ali, Whitham  (see it on-line https://archive.org/details/LinearAndNonlinearWaves ) is a good book for physicists, but I guess that no economist has ever read it. Maybe apart from those with education in physics, who later started to work in finance and design programs to win in financial markets (but those algorithms are never published).

You list your papers in physics. Overall there are plenty, I have also: https://www.researchgate.net/publication/275581998_Evolution_of_long_nonlinear_waves_on_shelves

But they are not relevant to this question. Have you ever tried to derive some PDE based on economic assumptions?

Duc, the problem exists also at the fundamental level. Neoclassical economics assumes agents to be fully rational, and this kills all dynamics. Economic growth theory (Barro & Sala-i-Martin) includes dynamics but as on optimal path for maximization of some Hamiltonian; the result is the system of ordinary DE, often non-linear, but not PDE. I also have several papers of this type. It is classics for economics, but not PDE, as you wish.

There might be a room to derive Schrodinger type equation for an evolution of a cloud of tourists in the potential based on attractive points of a city or shopping centers, but so far I know nobody who has worked on that. Economists often use stochasticity, but they do not think in terms of population density. I tried to make some elaborations in this chapter (it should be clear for an applied mathematician): https://www.researchgate.net/publication/283908757_Role_of_Density_and_Field_in_Spatial_Economics

Conference Paper Evolution of long nonlinear waves on shelves

Chapter Role of Density and Field in Spatial Economics

Thank you Yuri, you have given tremendous information, and actually re-triggered my interest in combining PDEs and Economics. From what I have seen, there are works trying to (1) relax the rational expectation which is more realistic, and (2) to integrate PDEs intro Economics models. 

Could you please elaborate more about the statements "Neoclassical economics assumes agents to be fully rational, and this kills all dynamics"?

Duc, if you open Microeconomic textbooks for students (for example, Varian, here on-line: http://mileslight.com/armenia/Varian-MicroeconomicAnalysis.pdf ), you will find that time is mentioned only in chapter 19, and for discrete number of periods. Equilibrium theory is about what static prices emerge if all know all information and maximize utility. Dynamic models in continuous time come only in growth theory.

Spatial models in economics are also not common, but if they emerge, mostly in static form (see urban economics). That is why it is almost impossible to combine spatial and temporal variables to get PDE, like in physics. But there are some way out, like local interaction combined with non-storability of good, that I used in my paper above. (By the way, did you find Burgers equation there?)

Another escape is in stochastic models, because random shocks destroy rationality (in the sense that I know what all know and what will happen in future in equilibrium), because future cannot be fully forecasted.

Duc, I do not know if you understood my last statement. You will never find it in publications, and it took me several years to arrive to those conclusions.

Half of economists are quite happy to live in the world without space and time. Perhaps 30% use time, but mostly in discrete form and often for stochastic processes. Maybe 20% use space (regional science and urban economics). But less than 1% use space and time simultaneously. There are 3 major courses for the beginners: microeconomics (I gave you a book), macroeconomics and econometrics. Mathematicians with specialization in statistics or fixed point theory are highly welcome to corresponding branches (of course, after they listen to basic courses), but I do not know any specialist in wave theory (even linear) who has done a career in economics based on that knowledge. There are many hidden doors that are closed and not visible before one approaches them. But if you want to get your own experience, just study basic economics (it will take a couple of years) and then derive your own PDE consistent with basic assumptions. I guess that you will find no supervisor, and if they see you as a smart student, they will open many other doors to work on their problems, but not on PDE. Wave theory requires both local interaction (which in economics is often non-local and too fast) and no dissipation (while in economics there is hardly a value preserved over time; even global wealth can drop by 10% after a bad day on stock exchange).

If you want to stay as applied mathematician in the world of PDE, you better look for other application. This may be welcome in biology, urban traffic (by the way, waves of cars are even in the book of Whitham), demography (I also tried to write something like wave theory here; see https://www.researchgate.net/publication/275019458_Economic_Consequences_of_Demographic_Waves ), perhaps environment. It is easy to derive a system of non-linear equations for interaction between economics and nature; see https://www.researchgate.net/publication/278110040_Low-Dimensional_Nonlinear_Dynamical_Systems_as_a_Tool_for_Socio-Economic_Modelling . It gives some non-linear dynamics, but not via PDE and still not much welcome by pure economists.

Conference Paper Economic Consequences of Demographic Waves

Conference Paper Low-Dimensional Nonlinear Dynamical Systems as a Tool for So...

Very interesting opinions Mr. Yuri Yegorov.

Thank you, Mr. Gryllakis, for your nice comment. Dear discussants, if you are interested, I have opened a question linked to our discussion here: https://www.researchgate.net/post/Why_for_a_specialist_in_wave_theory_it_is_almost_impossible_to_do_a_second_career_in_economics