Soliton, periodic and superposition solutions to nonlocal (2+1)-dimensional, extended KdV equation derived from the ideal fluid model

Abstract

We present new (2+1)-dimensional extended KdV (KdV2) equation derived within an ideal fluid model. Next, we show several families of analytic solutions to this equation. The solutions are expressed by functions of argument ξ=(kx+ly-ωt)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi = (k x +l y-\\omega t)$$\\end{document}. We found the soliton solutions in the form Asech2(ξ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A\\,\ ext {sech}^{2}(\\xi )$$\\end{document}, periodic solutions in the form Acn2(ξ,m)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A\\,\ ext {cn}^{2}(\\xi ,m)$$\\end{document} and superposition solutions in the form A2[dn2(ξ,m)±mcn(ξ,m)dn(ξ,m)]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\frac{A}{2}[\ ext {dn}^{2}(\\xi ,m)\\pm \\sqrt{m}\\,\ ext {cn}(\\xi ,m)\ ext {dn} (\\xi ,m)]$$\\end{document} analogous to the solutions of (1+1)-dimensional, extended KdV equation and to the solutions to ordinary Korteweg-de Vries equation. On the other hand, the existence of these families of analytical solutions for the highly nonlinear non-local (2+1)-dimensional, extended KdV equation is astounding. The existence of essentially one-dimensional solutions to the (2+1)-dimensional extended KdV equation explains the enormous success of the one-dimensional nonlinear wave equations for the shallow water problem.