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

Abstract

AbstractWe 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 $$\xi = (k x +l y-\omega t)$$ ξ = ( k x + l y - ω t ) . We found the soliton solutions in the form $$A\,\text {sech}^{2}(\xi )$$ A sech 2 ( ξ ) , periodic solutions in the form $$A\,\text {cn}^{2}(\xi ,m)$$ A cn 2 ( ξ , m ) and superposition solutions in the form $$\frac{A}{2}[\text {dn}^{2}(\xi ,m)\pm \sqrt{m}\,\text {cn}(\xi ,m)\text {dn} (\xi ,m)]$$ A 2 [ dn 2 ( ξ , m ) ± m cn ( ξ , m ) dn ( ξ , m ) ] 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.