New integrable (2+1)- and (3+1)-dimensional shallow water wave equations: multiple soliton solutions and lump solutions

Abstract

PurposeThis study aims to develop two integrable shallow water wave equations, of higher-dimensions, and with constant and time-dependent coefficients, respectively. The author derives multiple soliton solutions and a class of lump solutions which are rationally localized in all directions in space.Design/methodology/approachThe author uses the simplified Hirota’s method and lump technique for determining multiple soliton solutions and lump solutions as well. The author shows that the developed (2+1)- and (3+1)-dimensional models are completely integrable in in the Painlené sense.FindingsThe paper reports new Painlevé-integrable extended equations which belong to the shallow water wave medium.Research limitations/implicationsThe author addresses the integrability features of this model via using the Painlevé analysis. The author reports multiple soliton solutions for this equation by using the simplified Hirota’s method.Practical implicationsThe obtained lump solutions include free parameters; some parameters are related to the translation invariance and the other parameters satisfy a non-zero determinant condition.Social implicationsThe work presents useful algorithms for constructing new integrable equations and for the determination of lump solutions.Originality/valueThe paper presents an original work with newly developed integrable equations and shows useful findings of solitary waves and lump solutions.