Modifications of the nonlinear Schrödinger (MNLS) model [Formula: see text] where [Formula: see text] and [Formula: see text], are considered. We show that the MNLS models possess infinite towers of quasi-conservation laws for soliton-type configurations with a special complex conjugation, shifted parity and delayed time reversion ([Formula: see text]) symmetry. Infinite towers of anomalous charges appear even in the standard NLS model for [Formula: see text] invariant [Formula: see text]-bright solitons. The true conserved charges emerge through some kind of anomaly cancellation mechanism. Our analytical results are supported by numerical simulations of two-bright-soliton scatterings with potential [Formula: see text]. Our numerical simulations show the elastic scattering of bright solitons for a wide range of values of the set [Formula: see text] and a variety of amplitudes and relative velocities. The MNLS-type systems are quite ubiquitous, and so, our results may find potential applications in several areas of nonlinear physics, such as Bose–Einstein condensation, superconductivity, soliton turbulence and the triality among gauge theories, integrable models and gravity theories.

We consider a set of equations of the form Pj (x, y) = (10x + mj )(10y + nj ), x ≥ 0, y ≥ 0, j = 1,2,3, such that {m 1 = 7, n 1 = 3}, {m 2 = n 2 = 9} and {m 3 = n 3 = 1}, respectively. It is shown that if ( a ( p j ) , b ( p j ) ) ∈ ℕ × ℕ is a solution of the j’th equation one has the inequality p j 100 ≤ A ( p j ) B ( p j ) ≤ 121 10 4 p j , where A ( p j ) ≡ a ( p j ) + 1 , B ( p j ) ≡ b ( p j ) + 1 and pj is a natural number ending in 1, such that { A ( p 1 ) ≥ 4 , B ( p 1 ) ≥ 8 } , { A ( p 2 ) ≥ 2 , B ( p 2 ) ≥ 2 } , and { A ( p 3 ) ≥ 10 , B ( p 3 ) ≥ 10 } hold, respectively. Moreover, assuming the previous result we show that 1 ≤ ( A ( p j + 10 ) B ( p j + 10 ) A ( p j ) B ( p j ) ) 1 / 100 ≤ e 0 , 000201 × ( 1 + 10 p j ) ( 0 , 101 ) 2 , with { A ( p 1 ) ≥ 31 , B ( p 1 ) ≥ 71 } , { A ( p 2 ) ≥ 11 , B ( p 2 ) ≥ 11 } , and { A ( p 3 ) ≥ 91 , B ( p 3 ) ≥ 91 } , respectively. Finally, we present upper and lower bounds for the relevant positive integer solution of the equation defined by pj = (10A + mj )(10B + nj ), for each case j = 1, 2, 3, respectively.

In this work we prove some theorems that allow us to find integer solutions of quadratic equations in two variables that represent a natural number.

In this paper, a dual Riccati-type pseudo-potential formulation is introduced for a modified AKNS system (MAKNS) and infinite towers of novel anomalous conservation laws are uncovered. In addition, infinite towers of exact nonlocal conservation laws are uncovered in a linear formulation of the system. It is shown that certain modifications of the nonlinear Schrödinger model (MNLS) can be obtained through a reduction process starting from the MAKNS model. So, the novel infinite sets of quasi-conservation laws and related anomalous charges are constructed by an unified and rigorous approach based on the Riccati-type pseudo-potential method, for the standard NLS and modified MNLS cases, respectively. The nonlocal properties, the complete list of towers of infinite number of anomalous charges and the (nonlocal) exact conservation laws of the quasi-integrable systems, such as the deformed Bullough–Dodd, Toda, KdV and SUSY sine-Gordon systems, can be studied in the framework presented in this paper. Our results may find many applications since the AKNS-type system arises in several branches of nonlinear physics such as Bose–Einstein condensation, superconductivity and soliton turbulence.

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