摘 要：本文對(duì)文獻(xiàn)[1]中提出的一類(lèi)二階駐定方程[G/G=j=0mpjG/Gj]的求法進(jìn)行探討，提出了全新的求解方法——特征伴隨方程法，通過(guò)該求法得到這類(lèi)方程當(dāng)[m]取不同非負(fù)整數(shù)（本文僅討論[m=2]）時(shí)的通解，并相應(yīng)給出該方程作為求解非線性偏微分方程的輔助方程時(shí)所需要的[G/G]的解析表達(dá)式；同時(shí)給出一個(gè)作為常微分方程中駐定方程的應(yīng)用實(shí)例，以及該方程作為非線性偏微分方程的輔助方程時(shí)利用擴(kuò)展[G/G]展開(kāi)法的求解例子，通過(guò)該實(shí)例給出方程的精確行波解。

關(guān)鍵詞：二階駐定方程；特征方程；指數(shù)變換；特征伴隨方程（法）；降階法；異型通解

中圖分類(lèi)號(hào)：O175 DOI：10.16375/j.cnki.cn45-1395/t.2024.01.018

參考文獻(xiàn)

[1] 韓松，何曉瑩，周紅衛(wèi)，等．一個(gè)解非線性方程的必要條件及動(dòng)態(tài)齊次平衡法[J]. 廣西科技大學(xué)學(xué)報(bào)，2019，30（4）：95-104．

[2] WANG M L. Solitary wave solutions for variant Boussinesq equations[J]. Physics Letters A，1995，199（3-4）：169-172.

[3] WANG M L，LI X Z，ZHANG J L，et al. The（G′/G）-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics[J]. Phy- sics Letters A，2008，372（4）：417-423.

[4] HAN S，ZHAO Z H， WANG Q. The （G（k+1）/G（k））-expand method and travelling wave solutions of KdV equation[J]. Mathematic Applicata，2014，27（4）：906-912.

[5] WANG M L，LI X Z. Applications of F-expansion to periodic wave solutions for a new Hamiltonian amplitude equation[J]. Chaos，Solitons and Fractals，2005，24（5）：1257-1268.

[6] FAN E G. Extended tanh-function method and its applications to "nonlinear equations[J]. Physics Letters A，2000，277（4-5）：212-218.

A new method for solving specific second order stationary

equations：characteristic adjoint equation method

HAN Song， ZHANG Mingjun， HE Xiaoying

（College of Science， Guangxi University of Science and Technology， Liuzhou 545006， China）

Abstract：The characteristic adjoint equation method is presented for solving a type of second-order

stationary equation， which is denoted as [G/G=j=0mpjG/Gj]，[m=0，1，2，…]，and was initially proposed in paper [1]. With the new method， the general solution of the equation can be obtained when [m] is equal to different non-negative integers （we only discuss [m=2] in this text）. The analytical expression [G/G] of the equation can also be obtained when the second-order stationary equation is used as an auxiliary equation for solving nonlinear partial differential equations. Two application examples are shown. In the first case， the second-order stationary equation is used as a stationary equation in the ordinary differential equation; in the second example， it is regarded as an auxiliary equation for solving the nonlinear partial differential equation by extending [G/G] using the expansion method， which obtains the corresponding exact traveling wave solution.

Keywords：second-order stationary equation; characteristic equation; exponential transformation; characteristic adjoint equation（method）; reduction method; heterotypic general solution

（責(zé)任編輯：羅小芬