The doctoral dissertation in the field of Mathematics will be examined at the Faculty of Science, Forestry and Technology, Joensuu campus.
What is the topic of your doctoral research? Why is it important to study the topic?
My doctoral research topic is the existence and growth of meromorphic solutions of certain differential-difference equations. I investigated meromorphic solutions of the complex Schrödinger equation with delay and the complex Schrödinger equation with q-shift.
The meromorphic solutions of the complex Schrödinger equation with delay and the complex Schrödinger equation with q-shift have not been studied and these equations are related to Malmquist theorem.
What are the key findings or observations of your doctoral research?
When some assumptions are satisfied, my research results generalize the celebrated Malmquist theorem. If the growth of the meromorphic solution to a generalized complex Schrödinger equation with delay is subnormal, then I show that the equation reduces to one of four forms.
In addition, for two forms out of four, I obtain an accurate asymptotic relation for the distribution of zeros and poles of the meromorphic solutions. Based on this distribution, I demonstrate that the two forms of the differential-difference equation can be reduced to a particular Riccati differential equation.
I also study meromorphic solutions of the complex Schrödinger equation with q-shift. If the meromorphic solutions of the equation are of zero order and all coefficients are small functions relative to the meromorphic solutions, then I obtain some necessary conditions for the equation to admit a meromorphic solution. If all coefficients are rational functions, then I derive more precise results.
Moreover, I also consider the case where all coefficients are constant and without imposing any restriction on the growth order of meromorphic solutions. In this case, I prove the existence of meromorphic (or entire) solutions of the equationand further study their number and explicit forms.
What are the key research methods and materials used in your doctoral research?
I present some new definitions when I use the iterative method of the zeros and poles of a meromorphic solution of functional equations, which was developed by Halburd and Korhonen (2016). By using the method, I obtain some results which are contained in the thesis.
Also, motivated by the work of Gundersen et al. (2002), I combine formal power series, Cauchy-Hadamard theorem, and mathematical induction to obtain some sufficient conditions for the complex Schrödinger equation with q-shift to admit a meromorphic solution.
The doctoral dissertation of Wenlong Liu, MSc, entitled The existence and growth of meromorphic solutions of certain differential-difference equations will be examined at the Faculty of Science, Forestry and Technology, Joensuu campus. The opponent will be Professor Kazuya Tohge, Kanazawa University, and the custos will be Professor Risto Korhonen, University of Eastern Finland. Language of the public defence is English.
For further information, please contact:
Wenlong Liu, [email protected]
