New exact periodical solutions of mKP-1 equation via $\overline{\partial}$-dressing

V. G. Dubrovsky; A. V. Topovsky

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arXiv · 2003.07227

New exact periodical solutions of mKP-1 equation via $\overline{\partial}$-dressing

V. G. Dubrovsky·A. V. Topovsky

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Abstract

We proposed general scheme for construction of exact real periodical solutions of mKP-1 equation via Zakharov-Manakov $\overline{\partial}$-dressing method, derived convenient determinant formula for calculation of such solutions and demonstrated how reality and boundary conditions for the field $u(x,y,t)$ can be satisfied. We calculated the new classes of exact periodical solutions of mKP-1 equation: 1. the class of nonsingular one-periodic solutions or nonlinear plane monochromatic waves; 2. the class of two-periodic solutions without imposition of any boundary condition; 3. the class of two-periodic solutions with integrable boundary condition $u(x,y,t)\mid_{y=0}=0$. We interpreted the third class of two-periodic solutions with integrable boundary condition obtained by the use of special nonlinear superpositions of two simple one-periodical waves as eigenmodes of oscillations of the field $u(x,y,t)$ in semi-plane $y\geq 0$, the analogs of standing waves on the string with fixed endpoints.

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V. G. Dubrovsky, A. V. Topovsky. 2020-03-13. New exact periodical solutions of mKP-1 equation via $\overline{\partial}$-dressing. https://arxiv.org/abs/2003.07227

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