We use the integrable Kaup-Boussinesq shallow water system, modified by a small viscous term, to model the formation of an undular bore with a steady profile. The description is made in terms of the corresponding integrable Whitham system, also appropriately modified by friction. This is derived in Riemann variables using a modified finite-gap integration technique for the AKNS scheme. The Whit...

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and Applied Analysis 3 (2) If 0 < g < g0, we get a solitary wave solution u6 ( x, y, t ) c (√ 2 α − 1 β cosh θ ( x y − ct )) α − 1 ( 1 β − cosh 2θ ( x y − ct )) α ( −1 α β − α − 1 cosh 2θ ( x y − ct )) , 2.4 and two blow-up solutions u7± ( x, y, t ) c ( α ( 2 β ) − 2 − 2 α − 1 cosh θ ( x y − ct ) ± β 3/2 coth θ/2 ( x y − ct )) 2α ( −1 α β − α − 1 cosh θ ( x y − ct )) , 2.5 where β 6 − 6α α2 and...

Abstract The generalized Broer-Kaup-Kupershmidt (generalized BKK) isospectral problem, including the x-derivative of potential, is considered based on Lie algebra A1. The variational trace identity is extended to construct Hamiltonian structure of generalized BKK system. The Lie algebra A1 is extended to the non-semi-simple Lie algebra of 4 × 4 matrix form, from which a hierarchy of soliton equ...