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A 2+1 dimensional Volterra type system with nonzero boundary conditions via Dbar dressing method

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Abstract

A 2+1 dimensional Volterra type system with nonzero boundary conditions is considered. The Dbar dressing method based on a single-valued variable is discussed. The explicit solutions, including line-solitons and rational solutions of the Volterra type system are considered on certain new variables.

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  1. School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, 450001, Henan, China

    Tengfei Liu

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Appendix

Appendix

Here, we deduce the 2+1 dimensional Volterra type system (2). Acting E and \(E^{-1}\) to (14) follow that

$$\begin{aligned} \begin{aligned} \partial _y\ln E\chi ^{(0)}_n&=\frac{2E^2\chi ^{(1)}_n}{E^2\chi ^{(0)}_n}-\frac{2E\chi ^{(1)}_n}{E\chi ^{(0)}_n},\\ \partial _y\ln E^{-1}\chi ^{(0)}_n&=\frac{2\chi ^{(1)}_n}{\chi ^{(0)}_n}-\frac{2E^{-1}\chi ^{(1)}_n}{E^{-1}\chi ^{(0)}_n}. \end{aligned} \end{aligned}$$
(A.1)

Substituting (14) and (A.1) into (12), we have

$$\begin{aligned} b_n=a_n\partial _y[\ln (\chi ^{(0)}_n\cdot E\chi ^{(0)}_n)], \end{aligned}$$
(A.2)

and it is clear to find that

$$\begin{aligned} a_nb_{n-1}-b_na_{n-1}=(a_na_{n-1})_y, \end{aligned}$$

which is the second equation of (2).

Acting E to (15) follows that

$$\begin{aligned}&\frac{2\partial _yE\chi ^{(1)}_n}{E\chi ^{(0)}_n}+\frac{\chi ^{(0)}_n}{E^2\chi ^{(0)}_n}\nonumber \\&+\frac{4E\chi ^{(2)}_n}{E\chi ^{(0)}_n}-\frac{4E^2\chi ^{(2)}_n}{E^2\chi ^{(0)}_n}-1=0, \end{aligned}$$
(A.3)

substituting (15) and (A.3) into (16), we have

$$\begin{aligned}&\partial _t\ln \chi ^{(0)}_n+2-\frac{2E^{-1}\chi ^{(0)}_n}{E\chi ^{(0)}_n} -\frac{2\partial _y\chi ^{(1)}_n}{\chi ^{(0)}_n}-\frac{2\partial _yE\chi ^{(1)}_n}{E\chi ^{(0)}_n}\nonumber \\&+\frac{4E\chi ^{(1)}_n}{E\chi ^{(0)}_n}{[\frac{E^2\chi ^{(1)}_n}{E^2\chi ^{(0)}_n}-\frac{\chi ^{(1)}_n}{\chi ^{(0)}_n}]}=0, \end{aligned}$$
(A.4)

noting that

$$\begin{aligned} \begin{aligned} \partial _y(\frac{\chi ^{(1)}_n}{\chi ^{(0)}_n})&=\frac{\partial _y\chi ^{(1)}_n}{\chi ^{(0)}_n}-\frac{\chi ^{(1)}_n}{\chi ^{(0)}_n}\cdot \partial _y\ln \chi ^{(0)}_n,\\ \partial _y(\frac{E\chi ^{(1)}_n}{E\chi ^{(0)}_n})&=\frac{\partial _yE\chi ^{(1)}_n}{E\chi ^{(0)}_n}-\frac{E\chi ^{(1)}_n}{E\chi ^{(0)}_n}\cdot \partial _y\ln E\chi ^{(0)}_n. \end{aligned} \end{aligned}$$
(A.5)

Substituting (A.1), (A.5) into (A.4) and acting \(1-E\) to (A.4) afterwards, we have

$$\begin{aligned} \begin{aligned}&\partial _t\ln (\frac{\chi ^{(0)}_n}{E\chi ^{(0)}_n})+\frac{2\chi ^{(0)}_n}{E^2\chi ^{(0)}_n}-\frac{2E^{-1}\chi ^{(0)}_n}{E\chi ^{(0)}_n}\\&+2\partial _y(\frac{E^2\chi ^{(1)}_n}{E^2\chi ^{(0)}_n}-\frac{\chi ^{(1)}_n}{\chi ^{(0)}_n})\\&+(\frac{2E\chi ^{(1)}_n}{E\chi ^{(0)}_n}-\frac{2\chi ^{(1)}_n}{\chi ^{(0)}_n})\partial _y\ln \chi ^{(0)}_n\\&+(\frac{2E\chi ^{(1)}_n}{E\chi ^{(0)}_n}-\frac{2E^2\chi ^{(1)}_n}{E^2\chi ^{(0)}_n})\partial _y\ln E\chi ^{(0)}_n=0, \end{aligned} \end{aligned}$$
(A.6)

according to (14) and (A.1), we have

$$\begin{aligned} \begin{aligned}&(\frac{2E\chi ^{(1)}_n}{E\chi ^{(0)}_n}-\frac{2\chi ^{(1)}_n}{\chi ^{(0)}_n})\partial _y\ln \chi ^{(0)}_n\\&+(\frac{2E\chi ^{(1)}_n}{E\chi ^{(0)}_n}-\frac{2E^2\chi ^{(1)}_n}{E^2\chi ^{(0)}_n})\partial _y\ln E\chi ^{(0)}_n\\ =&(\frac{2E^2\chi ^{(1)}_n}{E^2\chi ^{(0)}_n}-\frac{2\chi ^{(1)}_n}{\chi ^{(0)}_n})(\partial _y\ln \chi ^{(0)}_n-\partial _y\ln E\chi ^{(0)}_n). \end{aligned} \end{aligned}$$
(A.7)

Substituting (A.7) and (11) into (A.6), we get

$$\begin{aligned}&\partial _t\ln a_n+2a_na_{n+1}-2a_na_{n-1}+2\partial _y(\frac{E^2\chi ^{(1)}_n}{E^2\chi ^{(0)}_n}\nonumber \\&-\frac{\chi ^{(1)}_n}{\chi ^{(0)}_n})+(\frac{2E\chi ^{(1)}_n}{E\chi ^{(0)}_n}-\frac{2\chi ^{(1)}_n}{\chi ^{(0)}_n})\partial _y\ln a_n=0, \end{aligned}$$
(A.8)

then (A.8) multiplies \(a_n\) and substituting (12) into the result, we can obtain

$$\begin{aligned} a_{n,t}+b_{n,y}=2a^2_n(a_{n-1}-a_{n+1}), \end{aligned}$$

which is the first equation of (2).

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Liu, T. A 2+1 dimensional Volterra type system with nonzero boundary conditions via Dbar dressing method. Nonlinear Dyn 111, 671–682 (2023). https://doi.org/10.1007/s11071-022-07855-z

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