{"title":"退化拉克斯可积方程的新精确解和守恒律","authors":"Muhammad Alim Abdulwahhab","doi":"10.1016/j.padiff.2025.101166","DOIUrl":null,"url":null,"abstract":"<div><div>This study uses the Lie symmetry group, a versatile and powerful method, to carry out an extensive analyses on a degenerate Lax-integrable equation. This integrable equation is shown to have infinite Lie algebra, and after specializing the arbitrary functions to first order polynomials, twenty-three explicit generators are obtained which are used to construct distinct non-trivial invariant solutions. This research article will also establish three independent integral solutions whose integrands contained functions that are absolutely arbitrary without any condition attached. Their arbitrariness can be used to generate compendium of nontrivial and infinitely many exact invariant solutions to the (2+1)-dimensional linearly degenerate Lax-integrable equation under consideration, which is known as the Pavlov equation. This means that all valid solutions of integrals reported in any mathematical handbooks can be adapted as solutions to the Pavlov equation using the appropriate invariants. Apart from the aforementioned novelty solutions, <span><math><mrow><mn>3</mn><mi>r</mi><mi>d</mi></mrow></math></span>-order multipliers are also established and used to construct lower-order local conservation laws.</div></div>","PeriodicalId":34531,"journal":{"name":"Partial Differential Equations in Applied Mathematics","volume":"14 ","pages":"Article 101166"},"PeriodicalIF":0.0000,"publicationDate":"2025-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"New exact solutions and conservation laws of a degenerate Lax-integrable equation\",\"authors\":\"Muhammad Alim Abdulwahhab\",\"doi\":\"10.1016/j.padiff.2025.101166\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>This study uses the Lie symmetry group, a versatile and powerful method, to carry out an extensive analyses on a degenerate Lax-integrable equation. This integrable equation is shown to have infinite Lie algebra, and after specializing the arbitrary functions to first order polynomials, twenty-three explicit generators are obtained which are used to construct distinct non-trivial invariant solutions. This research article will also establish three independent integral solutions whose integrands contained functions that are absolutely arbitrary without any condition attached. Their arbitrariness can be used to generate compendium of nontrivial and infinitely many exact invariant solutions to the (2+1)-dimensional linearly degenerate Lax-integrable equation under consideration, which is known as the Pavlov equation. This means that all valid solutions of integrals reported in any mathematical handbooks can be adapted as solutions to the Pavlov equation using the appropriate invariants. Apart from the aforementioned novelty solutions, <span><math><mrow><mn>3</mn><mi>r</mi><mi>d</mi></mrow></math></span>-order multipliers are also established and used to construct lower-order local conservation laws.</div></div>\",\"PeriodicalId\":34531,\"journal\":{\"name\":\"Partial Differential Equations in Applied Mathematics\",\"volume\":\"14 \",\"pages\":\"Article 101166\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2025-04-02\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Partial Differential Equations in Applied Mathematics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S2666818125000932\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"Mathematics\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Partial Differential Equations in Applied Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S2666818125000932","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"Mathematics","Score":null,"Total":0}
New exact solutions and conservation laws of a degenerate Lax-integrable equation
This study uses the Lie symmetry group, a versatile and powerful method, to carry out an extensive analyses on a degenerate Lax-integrable equation. This integrable equation is shown to have infinite Lie algebra, and after specializing the arbitrary functions to first order polynomials, twenty-three explicit generators are obtained which are used to construct distinct non-trivial invariant solutions. This research article will also establish three independent integral solutions whose integrands contained functions that are absolutely arbitrary without any condition attached. Their arbitrariness can be used to generate compendium of nontrivial and infinitely many exact invariant solutions to the (2+1)-dimensional linearly degenerate Lax-integrable equation under consideration, which is known as the Pavlov equation. This means that all valid solutions of integrals reported in any mathematical handbooks can be adapted as solutions to the Pavlov equation using the appropriate invariants. Apart from the aforementioned novelty solutions, -order multipliers are also established and used to construct lower-order local conservation laws.