Continuous limit of the difference second Painlevé equation and its asymptotic solutions
From the discrete second Painlev\(\acute{\text{e}}\) equation dP\(_{\amalg}\), the author obtains the equation NEWLINE\[NEWLINE v(x+i\varepsilon)+v(x-i\varepsilon)=\frac{(2-\varepsilon^2x)v(x)-\varepsilon^2\alpha}{1+\varepsilon^2v(x)^2}. NEWLINE\]NEWLINE Regarding dP\(_{\amalg}\) as a difference equation, the author presents a certain asymptotic solution that reduces to a triply-truncated solution of P\(_{\amalg}\) in this continuous limit. In a special case, the solution corresponds to a rational one of dP\(_{\amalg}\). Furthermore, the author describes one-parameter families of solutions having sequential limits to truncated solutions of P\(_{\amalg}\). So, the main results in this paper are new and important. These results can be applied to future studies.
scientific article; zbMATH DE number 6090091
scientific article